2.2: Normal Distributions

Homework for 2.1 Day 1: 41, 43, 45, 47, 49, 51

1) To use the 68-95-99.7 rule to estimate the percent of observations from a Normal Distribution that fall in an interval

2) Use the standard Normal distribution to calculate the proportion of values in a specified interval

3) Use the standard Normal distribution to determine a z-score from a percentile

4) Use Table A to find the percentile of a value from any Normal distribution

5) Make an appropriate graph to determine if a distribution is bell-shaped

6) Use the 68-95-99.7 rule to assess normality of a data set

7) Interpret a normal probability plot

Goals: Over the next three days…

A. A specific type of density curve is a Normal Curve

B. The distributions they describe are called Normal distributions

C. Characteristics:a) Have the same overall shape: symmetric,

single peaked, bell-shapedb) Mean- and standard deviation is c) Mean is in the center and so is median (think

back to previous section and symmetrical denisity curves)

d) controls the spread

1.) To use the 68-95-99.7

On the board

Keep in mind: these are special properties of normal distributions NOT ALL DENSITY CURVES!!!!

1.) Cont…Finding

Example: The mean of MLB batting averages is 0.261 with a standard deviation of 0.034. Suppose that the distribution is normal (this is key to know)with

a) Sketch a normal density curve for this distribution of batting averages. (Take notice of points that are 1,2,or 3 standard deviations away from the mean)

b) What percent of the batting averages are above 0.329?

c) What percent of the batting averages are between 0.193 and 0.295?

Let’s draw a 68-95-99.7

Check your understanding:Page 114

Z-Scores!!Check out the box on page 115Now, let’s look up our Table A in the back of the

book…These are the values of the z scores, the area under

the curve to the left of zExample: Suppose we wanted to find the proportion

of observations in a Normal distribution that were more than 1.53 standard deviations above the mean.

Ok, so we want to know what proportion of observations in the standard Normal distribution are greater than, z=1.53

2.) Use the standard Normal Distribution

First, find the area to the left of z=1.53 in table A

What is that value?Z=1.53 0.9370Now, this is the value to the left of 1.53, or

you can think of it as 1.53If I want to know greater than 1.53 I should…1-0.9370=0.0630Why?AREA UNDER THE CURVE ALWAYS = 1

2.) Cont…

Example: Find the proportion of observations from the standard Normal distribution that are between -0.58 and 1.79Look up the z-scores for those values:Z=-0.58Z=1.79 0.2810Values between: I should subtract them!Answer: 0.6823

2.) Cont…

Look at the beige box on page 120Example: In the 2008 Wimbledon tennis

tournament, Rafael Nadal averaged 115 MPH on his first serves. Assume that the distribution of his first serve speeds is Normal with a mean of 115 mph, and a standard deviation of 6 mph. About what portion of his speeds would you expect to exceed 120 mph?

Conclude: About 20% of Nadal’s first serves will travel more than 120 mph.

4.) Use Table A to find the percentile of a value from any Normal distribution

What percent of Rafael Nadal’s first serves are between 100 and 110 mph?

Conclude: About 20% of Nadal’s first serves will travel between 100 and 110 mph

Another look at that example

The heights of three-year old females are approximately Normally distributed with a mean of 94.5 cm and a standard deviation of 4 cm. What is the third quartile of this distribution?

Hint: Look at the z-chart in reverse!!Conclude: The third quartile of three-year old females’ heights is 97.18 cm

Now, you’re given N( , ) and find x

5.) Determine if a distribution is bell-shaped…Is it normal?

1. Plot the data- you can use a dotplot, stemplot, or

histogram. See if the shape is a bell2. Check to see if the data follows the 68-

95-99.7 RuleA. Find the mean and standard deviationB. Calculate 1, 2, and 3 standard

deviations to the right and leftC. Find the percent of the data that lies

between those standard deviations

How to work it out

12.9, 13.7, 14.1, 14.2, 14.5, 14.5, 14.6, 14.7, 15.1, 15.2, 15.3, 15.3, 15.3, 15.3, 15.5, 15.6, 15.8, 16.0, 16.0, 16.2, 16.2, 16.3, 16.4, 16.5, 16.6, 16.6, 16.6, 16.8, 17.0, 17.0, 17.2, 17.4, 17.4, 17.9, 18.4

The measurements listed below describe the usable capacity of a sample of 36 side-by-side refrigerators. Are the data close to Normal?

12.9

14.1

14.5

14.7

15.2

15.5

15.8

16.2

16.4

16.6 17

17.4

18.4

0

1

2

3

4

No space in the fridge?

Usable Capacity

Fre

qu

en

cy

Combined with the graph, and the fact that these numbers are extremely close to our rule, we have good evidence to believe this is a Normal Distribution

Conclusion: