Z-ScoresZ-Score: It is a measure of a the position specific value in a data set relative to mean in terms of the standard deviation units. It is sometimes called the Standard Score.

Value of 92 is 1.45 standard deviation units above the mean

Value of 72 is 0.35 standard deviation units below the mean

1Section 2.5, Page 45

Z Score Problems

2Problems, Page 52

Shelia and Joan competed in two events at a track meet. In the long jump, Shelia’s jump was 0.5 meters above the mean of the group and Joan’s jump was 0.3 meters beyond the mean. The standard deviation was 0.25 meters.

In the 100 meter run Shelia’s time was 5 seconds faster than the mean and Joan’s time was 8 seconds faster than the mean. The standard deviation was 3 seconds.

Who had the best combined performance? Why?

2.50

3

Normal Model

Values on the x-axis extend from -∞ to + ∞.The total area under the curve equals 1.0.The area above any interval on the x-axis equals the percent of data values in the interval.

Section 6.1

4

Normal Models

Section 6.1

Standard Normal Curve μ = 0, σ = 1

€

f (x) =e−(x−μ )2

2σ 2

σ 2π

Probabilities and Normal Curves

5Section 6.3, Page 124

To demonstrate the process of finding probabilities related to populations that can be described by a normal curve, let’s consider IQ scores. IQ scores are normally distributed with a mean of 100 and a standard deviation of 16. If a person is picked at random, what is the probability tat her IQ is between 100 and 115: that is, what is P(100<x<115)?

Finding Areas with theTI-83

PRGM – NORMDIST1 – ENTER100 ENTER: Enters Lower Bound115 ENTER: Enters Upper Bound100 ENTER: Enters Mean16 ENTER: Enters Standard DeviationDisplays results:

The area above the interval is .3257. Therefore the probability of selecting a person at random with an IQ between 100 and 115 is .3257 or 32.57%.

6Section 6.3, Page 124

Finding Areas with the TI-83

Find P(x > 90)

PRGM – NORMDIST – ENTER1 – ENTER90 ENTER2ND EE99 – ENTER: On the TI 83, 2nd E99 = ∞100 – ENTER16 – ENTERDisplay:

P(x > 90) = .7340

7Section 6.3, Page 125

90 100σ=16

Problems

8Problems, Page 133

Problems

9Problems, Page 133

Standard Normal CurveZ-Curve

z-axis

μ=0 σ=1

The standard normal curve or distribution is simply a normal distribution with a mean of 0 and a standard deviation of 1. The values on the horizontal axis are z-values.

Before computers, all problems had to be converted to a standard normal distribution and then solved with tables in the book that relate the z-values to the areas above the intervals.

Example: Find the area above the z-axis bounded by -1.5<z<2.1NORMDIST 1LOWER BOUND = -1. 5UPPER BOUND = 2.1Mean = 0STANDARD DEVIATION = 1Answer: AREA = .9153

10Section 6.2, Page 122

Problems

11Problems, Page 132

Determine Data Values with TI-83

Suppose that in a large class, your instructor tells you that you need to be in the top 10% of the class to get an A. For a particular exam, the mean is 72 and the standard deviation is 13. (Assume a normal distribution) What grade is required for an A?

Area = .10

Score = ?

Top 10%

Area from the left = 1-.10=.90

μ = 72, σ = 13

PRGM – NORMDIST – 2.90 – ENTER72 – ENTER13 – ENTERDisplay:

A test score of 88.66 or above is required to be in the top 10%

12Section 6.3, Page 125

Problems

13Problems, Page 133

6.51 IQ scores are normally distributed with a mean of 100 and a standard deviation of 16. Find the following:a.The 66th percentile.b.The 80th percentile.c.The minimum score required to be in the top 10%.d.The minimum score to be in the top 25%.

6.52 Find the two z-scores that bound the middle 30% of the standard normal distribution.