AP StatisticsHW: p95 22 – 24, 27 (skip part c)

Obj: to understand and use a z-score and standard normal distribution

Do Now:The mean monthly cost of gas is $125 with a standard

deviation of $10. The distribution of the gas bills is approximately normal.

a) What percentage of homes have a monthly bill of more than $115?

b) Less than $115?c) What bill amount represents the top 16 percent?d) What bill amount represents the top 84%e) How many standard devations above the mean is a bill of

$150?

C2 D4

Z-Score (standardized value)

• Allows us to identify the position of a data value relative to the μ and σ of its set of data values.

z = x – μ

σ

• Ex: If x = 13.75, μ = 10, and σ = 2.5, then the

z-score = 13.75 – 10

2.5

This means that 13.75 is 1.5 standard deviations above the mean of 10

• Ex: If x = 100, μ = 120, and σ = 15, calculate the z-score and tell what it means.

• If a variable x, which takes on the values

x = {x1, x2, …, xn}has a normal distribution N(μ, σ) and we change every data value into its standardized score (z-score), this new variable z takes on the values

z = {z1, z2, …, zn}and has the normal distribution N(0, 1) which we call the standard normal distribution

Standard Normal Dist’n

• Ex: A student scores 625 on the math section of the SAT and a 28 on the math section of the ACT. She can only report one score to her college. If the SAT summary statistics include μ = 490 and σ = 100 and the ACT summary statistics include μ = 21 and σ = 6, which score should she report?

• Ex: For data with a distribution N(0,1) calculate the following percentages:

a) % of data values between -1 and 1.b) % of data values less than 1.c) % of data values greater than -1.d) % of data values less than 2.

• For what data values are 99.85% of the scores lower?

• Do p.95 #19, 20