math 110 sec 14-4 lecture: the normal distribution the normal distribution describes many real-life...
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MATH 110 Sec 14-4 Lecture: The Normal Distribution
The normal distribution describes many real-life data sets.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The normal distribution describes many real-life data sets.The histogram shown gives an idea of
the shape of a normal distribution.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The normal distribution describes many real-life data sets.Although the normal distribution is a continuous distribution whose graph is a smooth curve, an appropriate histogram can
give a very good approximation to the actual normal graph.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
Properties of the Normal Distribution
MATH 110 Sec 14-4 Lecture: The Normal Distribution
Properties of the Normal Distribution is the mean and is the standard deviation of the distribution.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
Properties of the Normal Distribution
1. A normal curve is bell-shaped. is the mean and is the standard deviation of the distribution.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
Properties of the Normal Distribution
1. A normal curve is bell-shaped.2. The highest point on the curve is at the mean.
is the mean and is the standard deviation of the distribution.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
Properties of the Normal Distribution
1. A normal curve is bell-shaped.2. The highest point on the curve is at the mean.3. The mean, median and mode are equal.
is the mean and is the standard deviation of the distribution.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
Properties of the Normal Distribution
1. A normal curve is bell-shaped.2. The highest point on the curve is at the mean.3. The mean, median and mode are equal.4. The curve is symmetric with respect to its mean.
is the mean and is the standard deviation of the distribution.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
Properties of the Normal Distribution
1. A normal curve is bell-shaped.2. The highest point on the curve is at the mean.3. The mean, median and mode are equal.4. The curve is symmetric with respect to its mean.5. The total area under the curve is 1.
is the mean and is the standard deviation of the distribution.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
Properties of the Normal Distribution
1. A normal curve is bell-shaped.2. The highest point on the curve is at the mean.3. The mean, median and mode are equal.4. The curve is symmetric with respect to its mean.5. The total area under the curve is 1.6. Roughly 68% of the data values are within 1 standard deviation of the mean, 95% are within 2 standard deviations of the mean and 99.7% are within 3 standard deviations of the mean.
is the mean and is the standard deviation of the distribution.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
Properties of the Normal Distribution
1. A normal curve is bell-shaped.2. The highest point on the curve is at the mean.3. The mean, median and mode are equal.4. The curve is symmetric with respect to its mean.5. The total area under the curve is 1.6. Roughly 68% of the data values are within 1 standard deviation of the mean, 95% are within 2 standard deviations of the mean and 99.7% are within 3 standard deviations of the mean.
is the mean and is the standard deviation of the distribution.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
Properties of the Normal Distribution
1. A normal curve is bell-shaped.2. The highest point on the curve is at the mean.3. The mean, median and mode are equal.4. The curve is symmetric with respect to its mean.5. The total area under the curve is 1.6. Roughly 68% of the data values are within 1 standard deviation of the mean, 95% are within 2 standard deviations of the mean and 99.7% are within 3 standard deviations of the mean.
is the mean and is the standard deviation of the distribution.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
Properties of the Normal Distribution
1. A normal curve is bell-shaped.2. The highest point on the curve is at the mean.3. The mean, median and mode are equal.4. The curve is symmetric with respect to its mean.5. The total area under the curve is 1.6. Roughly 68% of the data values are within 1 standard deviation of the mean, 95% are within 2 standard deviations of the mean and 99.7% are within 3 standard deviations of the mean.
is the mean and is the standard deviation of the distribution.
The 68-95-99.7 Rule for Normal Distributions
MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 1000 students on an intelligence test is a
normal distribution with mean 450 and standard deviation of 25. What percent of scores would be expected to fall between 425 & 475?
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The 68-95-99.7 Rule68% of the values are within 1 std. deviation of the mean,
95% are within 2 standard deviations of the mean and 99.7% are with 3 standard deviations of the mean.
The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25.
What percent of scores would be expected to fall between 425 & 475?
The 68-95-99.7 Rule68% of the values are within 1 std. deviation of the mean,
95% are within 2 standard deviations of the mean and 99.7% are with 3 standard deviations of the mean.
425 475
The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25.
What percent of scores would be expected to fall between 425 & 475?
The 68-95-99.7 Rule68% of the values are within 1 std. deviation of the mean,
95% are within 2 standard deviations of the mean and 99.7% are with 3 standard deviations of the mean.
425 475
The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25.
What percent of scores would be expected to fall between 425 & 475?
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The 68-95-99.7 Rule68% of the values are within 1 std. deviation of the mean,
95% are within 2 standard deviations of the mean and 99.7% are with 3 standard deviations of the mean.
425 475
The shaded area gives the probability of a score falling in the 425 β 475 range.
The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25.
What percent of scores would be expected to fall between 425 & 475?
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The 68-95-99.7 Rule 68% of the values are within 1 std. deviation of the mean,
95% are within 2 standard deviations of the mean and 99.7% are with 3 standard deviations of the mean.
Here (mean) and (std deviation)
450425 475
The shaded area gives the probability of a score falling in the 425 β 475 range.
The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25.
What percent of scores would be expected to fall between 425 & 475?
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The 68-95-99.7 Rule 68% of the values are within 1 std. deviation of the mean,
95% are within 2 standard deviations of the mean and 99.7% are with 3 standard deviations of the mean.
Here (mean) and (std deviation)
450425 475
The shaded area gives the probability of a score falling in the 425 β 475 range.
The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25.
What percent of scores would be expected to fall between 425 & 475?
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The 68-95-99.7 Rule 68% of the values are within 1 std. deviation of the mean,
95% are within 2 standard deviations of the mean and 99.7% are with 3 standard deviations of the mean.
Here (mean) and (std deviation)
450425 475
The shaded area gives the probability of a score falling in the 425 β 475 range.
450 β 425 = 2525
The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25.
What percent of scores would be expected to fall between 425 & 475?
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The 68-95-99.7 Rule 68% of the values are within 1 std. deviation of the mean,
95% are within 2 standard deviations of the mean and 99.7% are with 3 standard deviations of the mean.
Here (mean) and (std deviation)
450425 475
The shaded area gives the probability of a score falling in the 425 β 475 range.
475 β 450 = 2525 25
The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25.
What percent of scores would be expected to fall between 425 & 475?
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The 68-95-99.7 Rule 68% of the values are within 1 std. deviation of the mean,
95% are within 2 standard deviations of the mean and 99.7% are with 3 standard deviations of the mean.
Here (mean) and (std deviation)
450425 475
The shaded area gives the probability of a score falling in the 425 β 475 range.
25 25
The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25.
What percent of scores would be expected to fall between 425 & 475?
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The 68-95-99.7 Rule 68% of the values are within 1 std. deviation of the mean,
95% are within 2 standard deviations of the mean and 99.7% are with 3 standard deviations of the mean.
Here (mean) and (std deviation)
450425 475
The shaded area gives the probability of a score falling in the 425 β 475 range.
25 25But 25 is the standard deviation.
The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25.
What percent of scores would be expected to fall between 425 & 475?
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The 68-95-99.7 Rule 68% of the values are within 1 std. deviation of the mean,
95% are within 2 standard deviations of the mean and 99.7% are with 3 standard deviations of the mean.
Here (mean) and (std deviation)
425 475
25 25But 25 is the standard deviation.The interval shown consists of all scores
within 1 standard deviation of the mean.
450The shaded area gives the probability of
a score falling in the 425 β 475 range.
The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25.
What percent of scores would be expected to fall between 425 & 475?
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The 68-95-99.7 Rule 68% of the values are within 1 std. deviation of the mean,
95% are within 2 standard deviations of the mean and 99.7% are with 3 standard deviations of the mean.
Here (mean) and (std deviation)
425 475
25 25
450The shaded area gives the probability of
a score falling in the 425 β 475 range.
The 68-95-99.7 Rule tells us that 68% of the scores are within 1 standard deviation of the mean.
The interval shown consists of all scores within 1 standard deviation of the mean.
The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25.
What percent of scores would be expected to fall between 425 & 475?
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The 68-95-99.7 Rule 68% of the values are within 1 std. deviation of the mean,
95% are within 2 standard deviations of the mean and 99.7% are with 3 standard deviations of the mean.
Here (mean) and (std deviation)
425 475
25 25
450The shaded area gives the probability of
a score falling in the 425 β 475 range.
The 68-95-99.7 Rule tells us that 68% of the scores are within 1 standard deviation of the mean.
68%The interval shown consists of all scores within 1 standard deviation of the mean.
The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25.
What percent of scores would be expected to fall between 425 & 475?
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The 68-95-99.7 Rule 68% of the values are within 1 std. deviation of the mean,
95% are within 2 standard deviations of the mean and 99.7% are with 3 standard deviations of the mean.
Here (mean) and (std deviation)
425 475
25 25
450The shaded area gives the probability of
a score falling in the 425 β 475 range.
The 68-95-99.7 Rule tells us that 68% of the scores are within 1 standard deviation of the mean.
68%The interval shown consists of all scores within 1 standard deviation of the mean.
The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25.
What percent of scores would be expected to fall between 425 & 475?
About 68% of scores
should fall between
425 & 475.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The 68-95-99.7 Rule 68% of the values are within 1 std. deviation of the mean,
95% are within 2 standard deviations of the mean and 99.7% are with 3 standard deviations of the mean.
Here (mean) and (std deviation)
425 475
25 25
450The shaded area gives the probability of
a score falling in the 425 β 475 range.
The 68-95-99.7 Rule tells us that 68% of the scores are within 1 standard deviation of the mean.
68%The interval shown consists of all scores within 1 standard deviation of the mean.
The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25.
What percent of scores would be expected to fall between 425 & 475?
About 68% of scores
should fall between
425 & 475.
How many of the 1000 scores would we expect to be between 425 & 475?
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The 68-95-99.7 Rule 68% of the values are within 1 std. deviation of the mean,
95% are within 2 standard deviations of the mean and 99.7% are with 3 standard deviations of the mean.
Here (mean) and (std deviation)
425 475
25 25
450The shaded area gives the probability of
a score falling in the 425 β 475 range.
The 68-95-99.7 Rule tells us that 68% of the scores are within 1 standard deviation of the mean.
68%The interval shown consists of all scores within 1 standard deviation of the mean.
The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25.
What percent of scores would be expected to fall between 425 & 475?
If there are 1000 scores, we would expect about 68% of them to be between 425 and 475.
About 68% of scores
should fall between
425 & 475.
How many of the 1000 scores would we expect to be between 425 & 475?
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The 68-95-99.7 Rule 68% of the values are within 1 std. deviation of the mean,
95% are within 2 standard deviations of the mean and 99.7% are with 3 standard deviations of the mean.
Here (mean) and (std deviation)
425 475
25 25
450The shaded area gives the probability of
a score falling in the 425 β 475 range.
The 68-95-99.7 Rule tells us that 68% of the scores are within 1 standard deviation of the mean.
68%The interval shown consists of all scores within 1 standard deviation of the mean.
If there are 1000 scores, we would expect about 68% of them to be between 425 and 475.1000 (68% )=1000 (0.68 )=680
The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25.
What percent of scores would be expected to fall between 425 & 475?
About 68% of scores
should fall between
425 & 475.
How many of the 1000 scores would we expect to be between 425 & 475?
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The 68-95-99.7 Rule 68% of the values are within 1 std. deviation of the mean,
95% are within 2 standard deviations of the mean and 99.7% are with 3 standard deviations of the mean.
Here (mean) and (std deviation)
425 475
25 25
450So, we expect about 680 scores to be in
the 425 β 475 range.
The 68-95-99.7 Rule tells us that 68% of the scores are within 1 standard deviation of the mean.
68%The interval shown consists of all scores within 1 standard deviation of the mean.
If there are 1000 scores, we would expect about 68% of them to be between 425 and 475.1000 (68% )=1000 (0.68 )=680
The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25.
What percent of scores would be expected to fall between 425 & 475?
About 68% of scores
should fall between
425 & 475.
How many of the 1000 scores would we expect to be between 425 & 475?
MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 1000 students on an intelligence test is a
normal distribution with mean 450 and standard deviation of 25. What percent of scores would be expected to fall above 500?
MATH 110 Sec 14-4 Lecture: The Normal Distribution
450
This is exactly the same distribution as before.
The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25.
What percent of scores would be expected to fall above 500?
The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25.
What percent of scores would be expected to fall above 500?
MATH 110 Sec 14-4 Lecture: The Normal Distribution
450
This is exactly the same distribution as before.Only the question has changed.
500
The shaded area is the probability of a score being above 500.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
450
This is exactly the same distribution as before.
First we need to find how many standard deviations 500 is from the mean.
Only the question has changed.
500
The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25.
What percent of scores would be expected to fall above 500?
The shaded area is the probability of a score being above 500.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
450
This is exactly the same distribution as before.
First we need to find how many standard deviations 500 is from the mean.
Only the question has changed.
500 β 450 = 5050
500
The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25.
What percent of scores would be expected to fall above 500?
The shaded area is the probability of a score being above 500.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
450
This is exactly the same distribution as before.
First we need to find how many standard deviations 500 is from the mean.
Only the question has changed.
500 β 450 = 5050
But 25 is the standard deviation.
500
The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25.
What percent of scores would be expected to fall above 500?
The shaded area is the probability of a score being above 500.
The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25.
What percent of scores would be expected to fall above 500?
MATH 110 Sec 14-4 Lecture: The Normal Distribution
450
This is exactly the same distribution as before.
First we need to find how many standard deviations 500 is from the mean.
Only the question has changed.
500 β 450 = 5050
But 25 is the standard deviation.So 500 is 2 standard deviations () from the mean.
500
The shaded area is the probability of a score being above 500.
The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25.
What percent of scores would be expected to fall above 500?
MATH 110 Sec 14-4 Lecture: The Normal Distribution
450
This is exactly the same distribution as before.
First we need to find how many standard deviations 500 is from the mean.
Only the question has changed.
50500
The 68-95-99.7 Rule 68% of the values are within 1 std. deviation of the mean,
95% are within 2 standard deviations of the mean and 99.7% are with 3 standard deviations of the mean.
So 500 is 2 standard deviations () from the mean.
The shaded area is the probability of a score being above 500.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
450
This is exactly the same distribution as before.
First we need to find how many standard deviations 500 is from the mean.
Only the question has changed.
50500
The 68-95-99.7 Rule 68% of the values are within 1 std. deviation of the mean,
95% are within 2 standard deviations of the mean and 99.7% are with 3 standard deviations of the mean.
The 68-95-99.7 Rule tells us that 95% of the scores are within 2 standard deviations of the mean.
So 500 is 2 standard deviations () from the mean.
The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25.
What percent of scores would be expected to fall above 500?
The shaded area is the probability of a score being above 500.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
450
This is exactly the same distribution as before.
First we need to find how many standard deviations 500 is from the mean.
Only the question has changed.
50500
The 68-95-99.7 Rule 68% of the values are within 1 std. deviation of the mean,
95% are within 2 standard deviations of the mean and 99.7% are with 3 standard deviations of the mean.
The 68-95-99.7 Rule tells us that 95% of the scores are within 2 standard deviations of the mean.
95%50
400
So 500 is 2 standard deviations () from the mean.
The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25.
What percent of scores would be expected to fall above 500?
The shaded area is the probability of a score being above 500.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
450
This is exactly the same distribution as before.
First we need to find how many standard deviations 500 is from the mean.
Only the question has changed.
50500
The 68-95-99.7 Rule 68% of the values are within 1 std. deviation of the mean,
95% are within 2 standard deviations of the mean and 99.7% are with 3 standard deviations of the mean.
The 68-95-99.7 Rule tells us that 95% of the scores are within 2 standard deviations of the mean.
95%50
400
That leaves 5% to be split between the
two tails.
So 500 is 2 standard deviations () from the mean.
The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25.
What percent of scores would be expected to fall above 500?
The shaded area is the probability of a score being above 500.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
450
This is exactly the same distribution as before.
First we need to find how many standard deviations 500 is from the mean.
Only the question has changed.
50500
The 68-95-99.7 Rule 68% of the values are within 1 std. deviation of the mean,
95% are within 2 standard deviations of the mean and 99.7% are with 3 standard deviations of the mean.
The 68-95-99.7 Rule tells us that 95% of the scores are within 2 standard deviations of the mean.
95%50
400
That leaves 5% to be split between the
two tails.
Half of 5% is 2.5%.So the orange shaded area is 2.5%.
So 500 is 2 standard deviations () from the mean.
The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25.
What percent of scores would be expected to fall above 500?
The shaded area is the probability of a score being above 500.
The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25.
What percent of scores would be expected to fall above 500?
MATH 110 Sec 14-4 Lecture: The Normal Distribution
450
This is exactly the same distribution as before.
First we need to find how many standard deviations 500 is from the mean.
Only the question has changed.
50500
The 68-95-99.7 Rule 68% of the values are within 1 std. deviation of the mean,
95% are within 2 standard deviations of the mean and 99.7% are with 3 standard deviations of the mean.
The 68-95-99.7 Rule tells us that 95% of the scores are within 2 standard deviations of the mean.
95%50
400
That leaves 5% to be split between the
two tails.
2.5%2.5%
So 500 is 2 standard deviations () from the mean.
Half of 5% is 2.5%.So the orange shaded area is 2.5%.
The shaded area is the probability of a score being above 500.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
450
This is exactly the same distribution as before.
First we need to find how many standard deviations 500 is from the mean.
Only the question has changed.
50500
The 68-95-99.7 Rule 68% of the values are within 1 std. deviation of the mean,
95% are within 2 standard deviations of the mean and 99.7% are with 3 standard deviations of the mean.
The 68-95-99.7 Rule tells us that 95% of the scores are within 2 standard deviations of the mean.
95%50
400
That leaves 5% to be split between the
two tails.
2.5%2.5%
So 500 is 2 standard deviations () from the mean.
The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25.
What percent of scores would be expected to fall above 500?
About 2.5% of scores should be above 500.
Half of 5% is 2.5%.So the orange shaded area is 2.5%.
The shaded area is the probability of a score being above 500.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
450
This is exactly the same distribution as before.
First we need to find how many standard deviations 500 is from the mean.
Only the question has changed.
50500
The 68-95-99.7 Rule 68% of the values are within 1 std. deviation of the mean,
95% are within 2 standard deviations of the mean and 99.7% are with 3 standard deviations of the mean.
The 68-95-99.7 Rule tells us that 95% of the scores are within 2 standard deviations of the mean.
95%50
400
That leaves 5% to be split between the
two tails.
2.5%2.5%
So 500 is 2 standard deviations () from the mean.
The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25.
What percent of scores would be expected to fall above 500?
About 2.5% of scores should be above 500.
How many of the 1000 scores would we expect to be above 500?
Half of 5% is 2.5%.So the orange shaded area is 2.5%.
The shaded area is the probability of a score being above 500.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
450
This is exactly the same distribution as before.
First we need to find how many standard deviations 500 is from the mean.
Only the question has changed.
50500
The 68-95-99.7 Rule 68% of the values are within 1 std. deviation of the mean,
95% are within 2 standard deviations of the mean and 99.7% are with 3 standard deviations of the mean.
The 68-95-99.7 Rule tells us that 95% of the scores are within 2 standard deviations of the mean.
95%50
400
That leaves 5% to be split between the
two tails.
2.5%2.5%
The shaded area is the probability of a score being above 500.
So 500 is 2 standard deviations () from the mean.
The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25.
What percent of scores would be expected to fall above 500?
About 2.5% of scores should be above 500.
We expect about 2.5% of them to be above 500.
Half of 5% is 2.5%.So the orange shaded area is 2.5%.
How many of the 1000 scores would we expect to be above 500?
MATH 110 Sec 14-4 Lecture: The Normal Distribution
450
This is exactly the same distribution as before.
First we need to find how many standard deviations 500 is from the mean.
Only the question has changed.
50500
The 68-95-99.7 Rule 68% of the values are within 1 std. deviation of the mean,
95% are within 2 standard deviations of the mean and 99.7% are with 3 standard deviations of the mean.
The 68-95-99.7 Rule tells us that 95% of the scores are within 2 standard deviations of the mean.
95%50
400
That leaves 5% to be split between the
two tails.
2.5%2.5%
The shaded area is the probability of a score being above 500.
So 500 is 2 standard deviations () from the mean.
The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25.
What percent of scores would be expected to fall above 500?
About 2.5% of scores should be above 500.
We expect about 2.5% of them to be above 500.
Half of 5% is 2.5%.So the orange shaded area is 2.5%.
How many of the 1000 scores would we expect to be above 500?
1000 (2.5% )=1000 (0.025 )=25
MATH 110 Sec 14-4 Lecture: The Normal Distribution
450
This is exactly the same distribution as before.
First we need to find how many standard deviations 500 is from the mean.
Only the question has changed.
50500
The 68-95-99.7 Rule 68% of the values are within 1 std. deviation of the mean,
95% are within 2 standard deviations of the mean and 99.7% are with 3 standard deviations of the mean.
The 68-95-99.7 Rule tells us that 95% of the scores are within 2 standard deviations of the mean.
95%50
400
That leaves 5% to be split between the
two tails.
2.5%2.5%
The shaded area is the probability of a score being above 500.
So 500 is 2 standard deviations () from the mean.
The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25.
What percent of scores would be expected to fall above 500?
About 2.5% of scores should be above 500.
We expect about 2.5% of them to be above 500.
Half of 5% is 2.5%.So the orange shaded area is 2.5%.
How many of the 1000 scores would we expect to be above 500?
So, we expect about 25 scores to be above 500.1000 (2.5% )=1000 (0.025 )=25
MATH 110 Sec 14-4 Lecture: The Normal Distribution This procedure can be somewhat simplified by noticing that the number
of standard deviations of the interval endpoints from the mean determines the percentages under the normal curve.
MATH 110 Sec 14-4 Lecture: The Normal Distribution This procedure can be somewhat simplified by noticing that the number
of standard deviations of the interval endpoints from the mean determines the percentages under the normal curve.
π+ππβπ π
of scores
When endpoints are 1 standard deviation from the mean
MATH 110 Sec 14-4 Lecture: The Normal Distribution This procedure can be somewhat simplified by noticing that the number
of standard deviations of the interval endpoints from the mean determines the percentages under the normal curve.
π+ππβπ π
of scores34% 34%
When endpoints are 1 standard deviation from the mean
MATH 110 Sec 14-4 Lecture: The Normal Distribution This procedure can be somewhat simplified by noticing that the number
of standard deviations of the interval endpoints from the mean determines the percentages under the normal curve.
π+ππβπ π
of scores34% 34%16% 16%
When endpoints are 1 standard deviation from the mean
MATH 110 Sec 14-4 Lecture: The Normal Distribution This procedure can be somewhat simplified by noticing that the number
of standard deviations of the interval endpoints from the mean determines the percentages under the normal curve.
When endpoints are 2 standard deviations from the mean
π+2ππβ2π π
of scores
MATH 110 Sec 14-4 Lecture: The Normal Distribution This procedure can be somewhat simplified by noticing that the number
of standard deviations of the interval endpoints from the mean determines the percentages under the normal curve.
When endpoints are 2 standard deviations from the mean
π+2ππβ2π π
of scores47.5% 47.5%
MATH 110 Sec 14-4 Lecture: The Normal Distribution This procedure can be somewhat simplified by noticing that the number
of standard deviations of the interval endpoints from the mean determines the percentages under the normal curve.
When endpoints are 2 standard deviations from the mean
π+2ππβ2π π
of scores47.5%2.5% 2.5%47.5%
MATH 110 Sec 14-4 Lecture: The Normal Distribution This procedure can be somewhat simplified by noticing that the number
of standard deviations of the interval endpoints from the mean determines the percentages under the normal curve.
π+3ππβ3π π
of scores
When endpoints are 3 standard deviations from the mean
MATH 110 Sec 14-4 Lecture: The Normal Distribution This procedure can be somewhat simplified by noticing that the number
of standard deviations of the interval endpoints from the mean determines the percentages under the normal curve.
π+3ππβ3π π
of scores49.85% 49.85%
When endpoints are 3 standard deviations from the mean
MATH 110 Sec 14-4 Lecture: The Normal Distribution This procedure can be somewhat simplified by noticing that the number
of standard deviations of the interval endpoints from the mean determines the percentages under the normal curve.
π+3ππβ3π π
of scores49.85%0 .15% 49.85% 0 .15%
When endpoints are 3 standard deviations from the mean
of scores49.85%0 .15% 49.85% 0 .15%
When endpoints are 3 standard deviations from the mean
When endpoints are 2 standard deviations from the mean
π+2ππβ2π π
of scores47.5%2.5% 2.5%47.5%
π+3ππβ3π π
π+ππβπ π
of scores34% 34%16% 16%
When endpoints are 1 standard deviation from the mean
SUMMARY
How each percentage
of the 68-95-99.7 rule breaks down
underneaththe Normal Curve
MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 1000 students on an intelligence test is a
normal distribution with mean 450 and standard deviation of 25. What percent of scores would be expected to fall below 375?
MATH 110 Sec 14-4 Lecture: The Normal Distribution
450
This is exactly the same distribution as before.
The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25.
What percent of scores would be expected to fall below 375?
MATH 110 Sec 14-4 Lecture: The Normal Distribution
450
This is exactly the same distribution as before.Only the question has changed.
The shaded area gives the probability of a score falling below 375.
375
The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25.
What percent of scores would be expected to fall below 375?
MATH 110 Sec 14-4 Lecture: The Normal Distribution
450
This is exactly the same distribution as before.Only the question has changed.
First we need to find how many standard deviations 375 is from the mean.
375
The shaded area gives the probability of a score falling below 375.
The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25.
What percent of scores would be expected to fall below 375?
MATH 110 Sec 14-4 Lecture: The Normal Distribution
450
This is exactly the same distribution as before.Only the question has changed.
375
First we need to find how many standard deviations 375 is from the mean.
450 β 375 = 7575
The shaded area gives the probability of a score falling below 375.
The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25.
What percent of scores would be expected to fall below 375?
MATH 110 Sec 14-4 Lecture: The Normal Distribution
450
This is exactly the same distribution as before.Only the question has changed.
375
First we need to find how many standard deviations 375 is from the mean.
450 β 375 = 75
The shaded area gives the probability of a score falling below 375.
75
But 25 is the standard deviation.
The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25.
What percent of scores would be expected to fall below 375?
MATH 110 Sec 14-4 Lecture: The Normal Distribution
450
This is exactly the same distribution as before.Only the question has changed.
375
First we need to find how many standard deviations 375 is from the mean.
450 β 375 = 75
The shaded area gives the probability of a score falling below 375.
75
But 25 is the standard deviation.So 375 is 3 standard deviations (75) from the mean.
The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25.
What percent of scores would be expected to fall below 375?
MATH 110 Sec 14-4 Lecture: The Normal Distribution
450
This is exactly the same distribution as before.Only the question has changed.
375
First we need to find how many standard deviations 375 is from the mean.
The shaded area gives the probability of a score falling below 375.
75
So 375 is 3 standard deviations (75) from the mean.
The 68-95-99.7 Rule 68% of the values are within 1 std. deviation of the mean,
95% are within 2 standard deviations of the mean and 99.7% are with 3 standard deviations of the mean.
The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25.
What percent of scores would be expected to fall below 375?
MATH 110 Sec 14-4 Lecture: The Normal Distribution
450
This is exactly the same distribution as before.Only the question has changed.
375
First we need to find how many standard deviations 375 is from the mean.
The shaded area gives the probability of a score falling below 375.
75
So 375 is 3 standard deviations (75) from the mean.
The 68-95-99.7 Rule 68% of the values are within 1 std. deviation of the mean,
95% are within 2 standard deviations of the mean and 99.7% are with 3 standard deviations of the mean.
The 68-95-99.7 Rule tells us that 99.7% of the scores are within 3 standard deviations of the mean.
The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25.
What percent of scores would be expected to fall below 375?
MATH 110 Sec 14-4 Lecture: The Normal Distribution
450
This is exactly the same distribution as before.Only the question has changed.
375
First we need to find how many standard deviations 375 is from the mean.
The shaded area gives the probability of a score falling below 375.
75
So 375 is 3 standard deviations (75) from the mean.
The 68-95-99.7 Rule 68% of the values are within 1 std. deviation of the mean,
95% are within 2 standard deviations of the mean and 99.7% are with 3 standard deviations of the mean.
The 68-95-99.7 Rule tells us that 99.7% of the scores are within 3 standard deviations of the mean.
This time, letβs take
advantage of the summary
sheet we developed.
The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25.
What percent of scores would be expected to fall below 375?
MATH 110 Sec 14-4 Lecture: The Normal Distribution
450
This is exactly the same distribution as before.Only the question has changed.
375
First we need to find how many standard deviations 375 is from the mean.
The shaded area gives the probability of a score falling below 375.
75
So 375 is 3 standard deviations (75) from the mean.
The 68-95-99.7 Rule 68% of the values are within 1 std. deviation of the mean,
95% are within 2 standard deviations of the mean and 99.7% are with 3 standard deviations of the mean.
The 68-95-99.7 Rule tells us that 99.7% of the scores are within 3 standard deviations of the mean.
This time, letβs take
advantage of the summary
sheet we developed.
The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25.
What percent of scores would be expected to fall below 375?
of scores49.85%0 .15% 49.85% 0 .15%
When endpoints are 3 standard deviations from the mean
When endpoints are 2 standard deviations from the mean
π+2ππβ2π π
of scores47.5%2.5% 2.5%47.5%
π+3ππβ3π π
π+ππβπ π
of scores34% 34%16% 16%
When endpoints are 1 standard deviation from the mean
SUMMARY
How each percentage
of the 68-95-99.7 rule breaks down
underneaththe Normal Curve
of scores49.85%0 .15% 49.85% 0 .15%
When endpoints are 3 standard deviations from the mean
When endpoints are 2 standard deviations from the mean
π+2ππβ2π π
of scores47.5%2.5% 2.5%47.5%
π+3ππβ3π π
π+ππβπ π
of scores34% 34%16% 16%
When endpoints are 1 standard deviation from the mean
SUMMARY
How each percentage
of the 68-95-99.7 rule breaks down
underneaththe Normal Curve
We figured out that 375 is 3 standard deviations from the mean, so the
bottom graph is the one that we need.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
450
This is exactly the same distribution as before.Only the question has changed.
375
First we need to find how many standard deviations 375 is from the mean.
The shaded area gives the probability of a score falling below 375.
75
So 375 is 3 standard deviations (75) from the mean.
The 68-95-99.7 Rule 68% of the values are within 1 std. deviation of the mean,
95% are within 2 standard deviations of the mean and 99.7% are with 3 standard deviations of the mean.
The 68-95-99.7 Rule tells us that 99.7% of the scores are within 3 standard deviations of the mean.
This time, letβs take
advantage of the summary
sheet we developed.
π+3ππβ3π π
of scores49.85%0 .15% 49.85% 0 .15%
When endpoints are 3 standard deviations from the mean
The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25.
What percent of scores would be expected to fall below 375?
MATH 110 Sec 14-4 Lecture: The Normal Distribution
450
This is exactly the same distribution as before.Only the question has changed.
375
First we need to find how many standard deviations 375 is from the mean.
The shaded area gives the probability of a score falling below 375.
75
So 375 is 3 standard deviations (75) from the mean.
The 68-95-99.7 Rule 68% of the values are within 1 std. deviation of the mean,
95% are within 2 standard deviations of the mean and 99.7% are with 3 standard deviations of the mean.
The 68-95-99.7 Rule tells us that 99.7% of the scores are within 3 standard deviations of the mean.
This time, letβs take
advantage of the summary
sheet we developed.
π+3ππβ3π π
of scores49.85%0 .15% 49.85% 0 .15%
When endpoints are 3 standard deviations from the mean
The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25.
What percent of scores would be expected to fall below 375?
MATH 110 Sec 14-4 Lecture: The Normal Distribution
450
This is exactly the same distribution as before.Only the question has changed.
375
First we need to find how many standard deviations 375 is from the mean.
The shaded area gives the probability of a score falling below 375.
75
So 375 is 3 standard deviations (75) from the mean.
The 68-95-99.7 Rule 68% of the values are within 1 std. deviation of the mean,
95% are within 2 standard deviations of the mean and 99.7% are with 3 standard deviations of the mean.
The 68-95-99.7 Rule tells us that 99.7% of the scores are within 3 standard deviations of the mean.
This time, letβs take
advantage of the summary
sheet we developed.
π+3ππβ3π π
of scores49.85%0 .15% 49.85% 0 .15%
When endpoints are 3 standard deviations from the mean
The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25.
What percent of scores would be expected to fall below 375?
MATH 110 Sec 14-4 Lecture: The Normal Distribution
450
This is exactly the same distribution as before.Only the question has changed.
375
First we need to find how many standard deviations 375 is from the mean.
The shaded area gives the probability of a score falling below 375.
75
So 375 is 3 standard deviations (75) from the mean.
The 68-95-99.7 Rule 68% of the values are within 1 std. deviation of the mean,
95% are within 2 standard deviations of the mean and 99.7% are with 3 standard deviations of the mean.
The 68-95-99.7 Rule tells us that 99.7% of the scores are within 3 standard deviations of the mean.
This time, letβs take
advantage of the summary
sheet we developed.
π+3ππβ3π π
of scores49.85%0 .15% 49.85% 0 .15%
When endpoints are 3 standard deviations from the mean
So the orange shaded area is 0.15%.
The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25.
What percent of scores would be expected to fall below 375?
MATH 110 Sec 14-4 Lecture: The Normal Distribution
450
This is exactly the same distribution as before.Only the question has changed.
375
First we need to find how many standard deviations 375 is from the mean.
The shaded area gives the probability of a score falling below 375.
75
So 375 is 3 standard deviations (75) from the mean.
The 68-95-99.7 Rule 68% of the values are within 1 std. deviation of the mean,
95% are within 2 standard deviations of the mean and 99.7% are with 3 standard deviations of the mean.
The 68-95-99.7 Rule tells us that 99.7% of the scores are within 3 standard deviations of the mean.
This time, letβs take
advantage of the summary
sheet we developed.
π+3ππβ3π π
of scores49.85%0 .15% 49.85% 0 .15%
When endpoints are 3 standard deviations from the mean
So the orange shaded area is 0.15%.
The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25.
What percent of scores would be expected to fall below 375?
So, we expect about 0.15%of the scores to be below 375.
MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 2000 students on an intelligence test is a
normal distribution with mean 150 and standard deviation of 10. How many scores do we expect to fall between 150 and 180?
MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 2000 students on an intelligence test is a
normal distribution with mean 150 and standard deviation of 10. How many scores do we expect to fall between 150 and 180?
Given what we have
learned, letβs try to solve
this as efficiently as
possible.
MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 2000 students on an intelligence test is a
normal distribution with mean 150 and standard deviation of 10. How many scores do we expect to fall between 150 and 180?
Given what we have
learned, letβs try to solve
this as efficiently as
possible.
MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 2000 students on an intelligence test is a
normal distribution with mean 150 and standard deviation of 10. How many scores do we expect to fall between 150 and 180?
Given what we have
learned, letβs try to solve
this as efficiently as
possible.First, we must decide which part of the
β68-95-99.7 Ruleβ applies.
MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 2000 students on an intelligence test is a
normal distribution with mean 150 and standard deviation of 10. How many scores do we expect to fall between 150 and 180?
Given what we have
learned, letβs try to solve
this as efficiently as
possible.
To decide the percent to use, we must find the number of standard
deviations there are between180 and the mean (150).
First, we must decide which part of the
β68-95-99.7 Ruleβ applies.
MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 2000 students on an intelligence test is a
normal distribution with mean 150 and standard deviation of 10. How many scores do we expect to fall between 150 and 180?
Given what we have
learned, letβs try to solve
this as efficiently as
possible.180β150=30180β150=30180β150=30
To decide the percent to use, we must find the number of standard
deviations there are between180 and the mean (150).
180β150=30 First, we must decide which part of the
β68-95-99.7 Ruleβ applies.
MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 2000 students on an intelligence test is a
normal distribution with mean 150 and standard deviation of 10. How many scores do we expect to fall between 150 and 180?
Given what we have
learned, letβs try to solve
this as efficiently as
possible.
To decide the percent to use, we must find the number of standard
deviations there are between180 and the mean (150).
180β150=30
30 (10+10+10) is3 standard deviations
First, we must decide which part of the
β68-95-99.7 Ruleβ applies.
MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 2000 students on an intelligence test is a
normal distribution with mean 150 and standard deviation of 10. How many scores do we expect to fall between 150 and 180?
Given what we have
learned, letβs try to solve
this as efficiently as
possible.
To decide the percent to use, we must find the number of standard
deviations there are between180 and the mean (150).
180β150=30So, 180 is 3 standard deviations
from the mean.
30 (10+10+10) is3 standard deviations
First, we must decide which part of the
β68-95-99.7 Ruleβ applies.
MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 2000 students on an intelligence test is a
normal distribution with mean 150 and standard deviation of 10. How many scores do we expect to fall between 150 and 180?
Given what we have
learned, letβs try to solve
this as efficiently as
possible.
To decide the percent to use, we must find the number of standard
deviations there are between180 and the mean (150).
180β150=30So, 180 is 3 standard deviations
from the mean.
First, we must decide which part of the
β68-95-99.7 Ruleβ applies.
MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 2000 students on an intelligence test is a
normal distribution with mean 150 and standard deviation of 10. How many scores do we expect to fall between 150 and 180?
Given what we have
learned, letβs try to solve
this as efficiently as
possible.
To decide the percent to use, we must find the number of standard
deviations there are between180 and the mean (150).
180β150=30So, 180 is 3 standard deviations
from the mean.
So, the3 standard
deviation part of the
β68-95-99.7 Ruleβ applies.
π+3ππβ3π π
of scores49.85%0 .15% 49.85% 0 .15%
When endpoints are 3 standard deviations from the mean
MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 2000 students on an intelligence test is a
normal distribution with mean 150 and standard deviation of 10. How many scores do we expect to fall between 150 and 180?
Given what we have
learned, letβs try to solve
this as efficiently as
possible.
To decide the percent to use, we must find the number of standard
deviations there are between180 and the mean (150).
180β150=30So, 180 is 3 standard deviations
from the mean.
So, the3 standard
deviation part of the
β68-95-99.7 Ruleβ applies.
π+3ππβ3π π
of scores49.85%0 .15% 49.85% 0 .15%
When endpoints are 3 standard deviations from the mean
150 180
MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 2000 students on an intelligence test is a
normal distribution with mean 150 and standard deviation of 10. How many scores do we expect to fall between 150 and 180?
Given what we have
learned, letβs try to solve
this as efficiently as
possible.
To decide the percent to use, we must find the number of standard
deviations there are between180 and the mean (150).
180β150=30So, 180 is 3 standard deviations
from the mean.
So, the3 standard
deviation part of the
β68-95-99.7 Ruleβ applies.
π+3ππβ3π π
of scores49.85%0 .15% 49.85% 0 .15%
When endpoints are 3 standard deviations from the mean
150 180
49.85% is the percent between 150 and 180.
MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 2000 students on an intelligence test is a
normal distribution with mean 150 and standard deviation of 10. How many scores do we expect to fall between 150 and 180?
Given what we have
learned, letβs try to solve
this as efficiently as
possible.
To decide the percent to use, we must find the number of standard
deviations there are between180 and the mean (150).
180β150=30So, 180 is 3 standard deviations
from the mean.
So, the3 standard
deviation part of the
β68-95-99.7 Ruleβ applies.
π+3ππβ3π π
of scores49.85%0 .15% 49.85% 0 .15%
When endpoints are 3 standard deviations from the mean
150 180
49.85% is the percent between 150 and 180.2000 (49.85% )=2000 (0.4985 )=997
MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 2000 students on an intelligence test is a
normal distribution with mean 150 and standard deviation of 10. How many scores do we expect to fall between 150 and 180?
Given what we have
learned, letβs try to solve
this as efficiently as
possible.
To decide the percent to use, we must find the number of standard
deviations there are between180 and the mean (150).
180β150=30So, 180 is 3 standard deviations
from the mean.
So, the3 standard
deviation part of the
β68-95-99.7 Ruleβ applies.
π+3ππβ3π π
of scores49.85%0 .15% 49.85% 0 .15%
When endpoints are 3 standard deviations from the mean
150 180
49.85% is the percent between 150 and 180.2000 (49.85% )=2000 (0.4985 )=997 So, we expect about 997 scores to
be between 150 and 180.
MATH 110 Sec 14-4 Lecture: The Normal Distribution A normal distribution has a mean of 36 and a standard deviation of 5.
What percentage of values do we expect to be below 26?
MATH 110 Sec 14-4 Lecture: The Normal Distribution A normal distribution has a mean of 36 and a standard deviation of 5.
What percentage of values do we expect to be below 26?Letβs use this exercise to try to get the answer while showing even less work.
MATH 110 Sec 14-4 Lecture: The Normal Distribution A normal distribution has a mean of 36 and a standard deviation of 5.
What percentage of values do we expect to be below 26?Letβs use this exercise to try to get the answer while showing even less work.
Step 1. Find the difference between 26 and 36 (mean).
MATH 110 Sec 14-4 Lecture: The Normal Distribution A normal distribution has a mean of 36 and a standard deviation of 5.
What percentage of values do we expect to be below 26?Letβs use this exercise to try to get the answer while showing even less work.
Step 1. Find the difference between 26 and 36 (mean).ππβππ=ππ
MATH 110 Sec 14-4 Lecture: The Normal Distribution A normal distribution has a mean of 36 and a standard deviation of 5.
What percentage of values do we expect to be below 26?Letβs use this exercise to try to get the answer while showing even less work.
Step 1. Find the difference between 26 and 36 (mean).
Step 2. Express that difference in terms of standard deviations.
ππβππ=ππ
MATH 110 Sec 14-4 Lecture: The Normal Distribution A normal distribution has a mean of 36 and a standard deviation of 5.
What percentage of values do we expect to be below 26?Letβs use this exercise to try to get the answer while showing even less work.
Step 1. Find the difference between 26 and 36 (mean).
Step 2. Express that difference in terms of standard deviations.Standard deviation is 5 so 10 is 2 standard deviations from the mean.
ππβππ=ππ
MATH 110 Sec 14-4 Lecture: The Normal Distribution A normal distribution has a mean of 36 and a standard deviation of 5.
What percentage of values do we expect to be below 26?Letβs use this exercise to try to get the answer while showing even less work.
Step 1. Find the difference between 26 and 36 (mean).
Step 2. Express that difference in terms of standard deviations.Standard deviation is 5 so 10 is 2 standard deviations from the mean.
ππβππ=ππ
Step 3. Use the percent diagram for the β2 standard deviationβ case.
MATH 110 Sec 14-4 Lecture: The Normal Distribution A normal distribution has a mean of 36 and a standard deviation of 5.
What percentage of values do we expect to be below 26?Letβs use this exercise to try to get the answer while showing even less work.
Step 1. Find the difference between 26 and 36 (mean).
Step 2. Express that difference in terms of standard deviations.Standard deviation is 5 so 10 is 2 standard deviations from the mean.
ππβππ=ππ
Step 3. Use the percent diagram for the β2 standard deviationβ case.When endpoints are 2 standard
deviations from the mean
π+2ππβ2π π
of scores47.5%2.5% 2.5%47.5%
MATH 110 Sec 14-4 Lecture: The Normal Distribution A normal distribution has a mean of 36 and a standard deviation of 5.
What percentage of values do we expect to be below 26?Letβs use this exercise to try to get the answer while showing even less work.
Step 1. Find the difference between 26 and 36 (mean).
Step 2. Express that difference in terms of standard deviations.Standard deviation is 5 so 10 is 2 standard deviations from the mean.
ππβππ=ππ
Step 3. Use the percent diagram for the β2 standard deviationβ case.When endpoints are 2 standard
deviations from the mean
π+2ππβ2π π
of scores47.5%2.5% 2.5%47.5%
Step 4. Find the percent that goes with βbelow 26β.
MATH 110 Sec 14-4 Lecture: The Normal Distribution A normal distribution has a mean of 36 and a standard deviation of 5.
What percentage of values do we expect to be below 26?Letβs use this exercise to try to get the answer while showing even less work.
Step 1. Find the difference between 26 and 36 (mean).
Step 2. Express that difference in terms of standard deviations.Standard deviation is 5 so 10 is 2 standard deviations from the mean.
ππβππ=ππ
Step 3. Use the percent diagram for the β2 standard deviationβ case.When endpoints are 2 standard
deviations from the mean
π+2ππβ2π π
of scores47.5%2.5% 2.5%47.5%
Step 4. Find the percent that goes with βbelow 26β. 26
MATH 110 Sec 14-4 Lecture: The Normal Distribution A normal distribution has a mean of 36 and a standard deviation of 5.
What percentage of values do we expect to be below 26?Letβs use this exercise to try to get the answer while showing even less work.
Step 1. Find the difference between 26 and 36 (mean).
Step 2. Express that difference in terms of standard deviations.Standard deviation is 5 so 10 is 2 standard deviations from the mean.
ππβππ=ππ
Step 3. Use the percent diagram for the β2 standard deviationβ case.When endpoints are 2 standard
deviations from the mean
π+2ππβ2π π
of scores47.5%2.5% 2.5%47.5%
Step 4. Find the percent that goes with βbelow 26β. 26
MATH 110 Sec 14-4 Lecture: The Normal Distribution A normal distribution has a mean of 36 and a standard deviation of 5.
What percentage of values do we expect to be below 26?Letβs use this exercise to try to get the answer while showing even less work.
Step 1. Find the difference between 26 and 36 (mean).
Step 2. Express that difference in terms of standard deviations.Standard deviation is 5 so 10 is 2 standard deviations from the mean.
ππβππ=ππ
Step 3. Use the percent diagram for the β2 standard deviationβ case.When endpoints are 2 standard
deviations from the mean
π+2ππβ2π π
of scores47.5%2.5% 2.5%47.5%
Step 4. Find the percent that goes with βbelow 26β. 26
About 2.5% of values will be below 26.Step 5. Answer: