Probability and Statistics – Mrs. Leahy

Unit 6: Normal Curves and Sampling Distributions Unit 6: Section 1 --- Graphs of Normal Probability Distributions What is a Normal Distribution and a Normal Curve?

Example 1: Two Normal Curves, A & B Do these two distributions have the same mean? If so, what is it? One curve has a standard deviation σ=1 and the other has a standard deviation σ=3. Which is which?

Area under a Normal Curve

Example 2: What percentage of the area under the normal curve lies: a) to the left of µ ? b) between µ - σ and µ + σ ? c) between µ - 3σ and µ + 3σ ? Example 3: The scores of a recent SAT test are normally distributed with a mean of µ=1100 and a standard deviation σ=200.

a) What is the probability that a score selected at random will be between 1100 and 1500? b) Is the probability that the score is between 700 and 1100 the same as the probability that the score is between 1100 and 1500? c) What is the probability that the score is between 1100 and 1300? d) What is the probabilty that the score is between 1300 and 1500? Example 4:

0.15% 0.15%

MEMORIZE THIS!!!

Unit 6: Section 2 --- Standard Units and Areas Under the Standard Normal Distribution

What is a Z-score and a Raw Score? Suppose Tina and jack are in two different sections of the same course. Each section is quite large and the scores on the midterm exams of each section follow a normal distribution. In Tina’s section, the average (mean) was 64 and her score was 74. In Jack’s section, the mean was 72 and his score was 82. Tina and Jack were pleased that their scores were each 10 points above the mean of their sections. Looking at the normal curves at the right, who did better with respect to the other students in their section? One way to express this numerically is with a “z value” or “z score” Example 1:

Example 2:

Sometimes it is also useful to be able to convert a z score into a raw score.

Example 3: Example 4:

Standard Normal Distribution: Areas under the Standard normal Curve

Example 5: Let z be a random variable with a standard normal distribution. Find the indicated probability and shade the corresponding area under the normal curve.

a) 1 00( . )P z b) 2 14( . )P z

c) 1 98( . )P z d) 0 42( . )P z

e) 1 33( . )P z f) 0 45( . )P z

g) 0 25 1 35( . . )P z h) 1 68 2 59( . . )P z

.

Unit 6: Section 3 --- Areas under any Normal Curve A

Normal Curve vs. Standard Normal Curve Normal curves: Standard Normal curves: Show mean and standard deviation values Show z-scores (#standard deviations from the mean) Generally not a table of areas for EVERY Has a table of areas for EVERY z value (to 2 dec) value of x to reference and find probabilities. to reference to find probabilities. In other words:

8 2 10 5( . . )P x = 1 35 1 80( . . )P z =

To find the probability that x will fall into an interval from a to b on a normal curve, we must _______________ the original measurements for x, a, and b into z values. Example 1: Suppose x has a normal distribution with µ=2 and σ=3.

Find 7( )P x

Example 2: Suppose x has a normal distribution with µ=2 and σ=3.

Find 4( )P x

Example 3: Assume that x has a normal distribution with the specified mean and standard deviation. Find the indicated probabilities. a)

Inverse Normal Distribution

How to find x, given a desired probability. 1. Use your table (from yesterday) to locate the probability. If you can’t find it exactly, use the closest answer to estimate z 2. Translate your z answer into x using the equation: x z

Case 1: Left Tail Find z by looking up probability “A” in table.

Example 4. Example 5:

Case 2: Right Tail Find z by looking up the probability “1 – A” in table.

Example 6: Example 7:

Case 3: Symmetrical Center Tail Find z by looking up the probability “(1 – A)/2” in table.

Example 8:

Unit 6: Section 4 --- The Central Limit Theorem A

Probability of a MEAN rather than a SINGLE SCORE Example 1: Children between the ages of 2 and 5 watch an average of 25 hours of TV per week with a standard deviation of 3 hours. If a sample of 20 children are selected, find the probability that the mean number of hours watched per week will be greater than 26.3? Variables in this problem: µ = mean = ______ n = sample size = ______ σ = standard deviation = _______ �̅� = mean of sample = _______ Central Limit Theorem: Standard Error: The standard deviation of a

sampling distribution.

standard error: x n

Conclusion we can draw:

xz

n

Our example: 26 3( . )P x

Step 1: Convert to z-score Step 2: Find Probability from Standard Normal Distribution Table

Example 3:

Example 4: