INTRODUCTORY STATISTICSChapter 6 THE NORMAL DISTRIBUTION

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FIGURE 6.1

If you ask enough people about their shoe size, you will find that your graphed data is shaped like a bell curve and can be described as normally distributed.

(credit: Ömer Ünlü)

A NORMAL DISTRIBUTION

NORMAL DISTRIBUTION AND THE 68-95-99.7 RULE

The mean, µ, is in the center. In general,

• 68% of data falls within one standard deviation

• 95% of data falls within two standard deviations

• 99.7% of data falls within three standard deviations

Z-SCORES

Z-scores are used to evaluate how far something is from the mean with normal distributions often.

Recall: a value’s z-score =

Example:

If heights are normally distributed with a mean of 67 inches and a standard deviation of 4, calculate the z-scores for the following heights:

a) 62 in.

b) 70 in.

c) 75 in.

Which height would be considered the most unusual?

EXAMPLE:

Scores on a standardized test are normally distributed with a mean of 54 and a standard deviation of 7. If X = a student’s score, then we would write . Calculate the z-scores for the following results:

a) 35

b) 77

c) 90

d) You received an 62 on the particular test. Your parents are upset. Justify mathematically to them why your score isn’t so bad.

PRACTICE:

Olivia’s school has an average GPA of 3.0 with a 0.7 standard deviation. She has a GPA of 3.2. Sam’s school has an average GPA of 2.7 with a 0.3 standard deviation. Sam has a GPA of 2.9. Who has the best GPA and why?

SECTION 6.3: USING THE NORMAL DISTRIBUTION

Distribution is written as

TOTAL AREA ADDS UP TO 1

Probability to the right is always 1 - P(X<x)

Ex. P(X>65) = 1- P(X<65)

USING A CALCULATOR TO COMPUTE PROBABILITY

Go to 2nd Dist and choose normalcdf(

Enter normalcdf(lower value, uppervalue, mean, standard deviation)

Ex. For an exam, scores are normally distributed with a mean of 71 and standard deviation of 5. Calculate:

a) The probability of scoring less than 60.

Normalcdf(0,60,71,5)=0.139

b) The probability of scoring less than 80.

Normalcdf(0,80,71,5)=0.964

c) The probability of scoring more than 80.

1-0.964=0.036 or Normalcdf(80,1EE99,71,5)=0.036

d) The probability of scoring more than 90

Normalcdf(90,1EE99,71,5)=7.237x10-5 (which is practically 0)

USING A CALCULATOR TO FIND PERCENTILES

The invNorm( function allows you to find a percentile value.

Enter invNorm(area to the left, mean, standard deviation)

Ex. In this example, find the 90th percentile if the mean is 63 and the standard deviation is 9. (

invNorm(0.9,63,9)=74.534

EXAMPLE

Heights for a school are normally distributed with a mean of 65 inches and a standard deviation of 7inches.

a) Write the distribution

b) Find the probability that a student is over 70 inches. Sketch the graph.

c) Find the probability that a student is under 63 inches. Sketch the graph.

d) Find the 80th percentile of heights.

EXAMPLE: THE MEAN IS 2 AND THE STANDARD DEVIATION IS 0.3

Find P(1.8<X<2.75)

STADIUM CAPACITIES