1. WARM – UP Find the z-scores for the following observations:
1. x = 19; With a population mean = 21 and Standard Deviation = 3.5
2. x = 111; With a population mean = 100 and Standard Deviation = 8.8
3. National gas prices are at $2.72 a gallon with Std. Dev. at $ Would it be unusual to find a gas station charging $3.00 a gallon?
z =
z = 1.25
z = YES!

2. National gas prices are at $2. 72 a gallon with Std. Dev. at $0. 09
National gas prices are at $2.72 a gallon with Std. Dev. at $ How unusual would it be to find a gas station charging $3.00 a gallon?
z = 3.11

3. Chapter 6
(Continued)
THE NORMAL CURVE

4. Andy scored a 650 on the Math portion of the SAT
Andy scored a 650 on the Math portion of the SAT. Mary scored a 25 on the ACT. If the Math SAT has a Normal Distribution, , and the ACT has a Normal Distribution , which student did better and why?
ACT
SAT
Andy = (650 – 500 )/95 = 1.58
Mary = (25 – 21)/3 = 1.33

5. The 68-95-99.7 Rule (Empirical Rule) Chapter 6 (cont.)
In a Normal Distribution with Mean μ and standard deviation σ:
68% of the observations will fall within one σ of the mean.
95% of the observations will fall within two σ’s of the mean.
99.7% of the observations will fall within three σ’s of the mean.
2.35%
.15%
99.7%

6. EXAMPLE: The mean score for the Math portion of the S. A. T
EXAMPLE: The mean score for the Math portion of the S.A.T. test is 500 with a Standard deviation of 95.
FIRST DRAW THE CURVE
N(μ, σ) = N(500, 95)
215
310
405
500
595
690
785
1.) What Proportion of the Population will score between 405 and 595?
2.) What Proportion of the Population will score between 595 and 690?
3.) What Proportion of the Population will score above 785?
4.) What PERCENTILE would a student scoring a 595 be in?
0.68
START HERE
ADD σ TO μ
ADD 2σ TO μ
0.135
0.0015
84 Percentile

7. How do you Start this Problem?
Example: The distribution Gas Prices at stations throughout the Metroplex is approximately normal with Mean $2.72, and Standard Deviation $0.09.
What % of consumers will spend between $2.54 and $2.90 a gallon?
What % of consumers will spend more than $2.90 a gallon?
A price of $2.63 corresponds to what Percentile?
99.7% of consumers will spend between what two prices for a gallon of gas?
How do you Start this Problem?

8. Since this is beyond 2 S.D. = (1.00 - .95) ÷ 2 = 2.5%
First construct and label the Normal Curve.
2.72
2.81
2.90
2.99
MEAN (μ)
ADD σ TO μ
ADD 2σ TO μ
ADD 3σ TO μ
What % of consumers will spend between $2.54 and $2.90 a gallon?
What % of consumers will spend more than $2.90 a gallon?
A price of $2.63 corresponds to what Percentile?
% of consumers will spend between what two prices for a gallon of gas?
95%
Since this is beyond 2 S.D. = ( ) ÷ 2 = 2.5%
16% = ( ) ÷ 2
$ $2.99

9. HW: Page 124: 17, 18, 20, 22, 23
37.2 mpg?

11. This is done with the z-score formula:
Andy scored a 650 on the Math portion of the SAT. Mary scored a 25 on the ACT. If the Math SAT has a Normal Distribution, , and the ACT has a Normal Distribution , which student did better and why?
ACT
SAT
Andy = (650 – 500 )/95 = 1.58
Mary = (25 – 21)/3 = 1.33