Question

examples: percentages to raw data

1. on the most recent test, a student scored in the 40th percentile. the mean of the test scores was an 85 and the standard deviation was 1.5. what was the student’s score?
2. an automobile dealer finds that the average price of a previously owned vehicle is $8,256. he decides to sell cars that will appeal to the middle 60% of the market in terms of price. find the maximum and minimum prices of the cars the dealer will sell. the standard deviation is $1,150 and the variable is normally distributed.

mean (μ)=8,256
std = $1,150
percent = 60%
min = (z, scores+
minimum =
maximum =

Explanation:

Step1: Recall z - score formula

The z - score formula is $z=\frac{x-\mu}{\sigma}$, where $x$ is the raw score, $\mu$ is the mean and $\sigma$ is the standard deviation.

Step2: Solve problem 7

For the 40th percentile, using a standard normal distribution table (z - table), the z - score corresponding to the 40th percentile is approximately $z=-0.25$. Given $\mu = 85$ and $\sigma=1.5$.
Substitute into the z - score formula: $- 0.25=\frac{x - 85}{1.5}$.
Multiply both sides by 1.5: $-0.25\times1.5=x - 85$.
$-0.375=x - 85$.
Add 85 to both sides: $x=85 - 0.375=84.625$.

Step3: Solve problem 8

For the middle 60% of a normal distribution, the area in the two tails is $1 - 0.60 = 0.40$. So the area in each tail is $\frac{0.40}{2}=0.20$.
The z - score corresponding to the left - hand tail area of 0.20 is $z_1\approx - 0.84$ and the z - score corresponding to the right - hand tail area of 0.20 (or the 80th percentile) is $z_2\approx0.84$.
For the minimum price ($x_1$):
Using the z - score formula $z_1=\frac{x_1-\mu}{\sigma}$, with $\mu = 8256$, $\sigma = 1150$ and $z_1=-0.84$.
$-0.84=\frac{x_1 - 8256}{1150}$.
Multiply both sides by 1150: $-0.84\times1150=x_1 - 8256$.
$-966=x_1 - 8256$.
Add 8256 to both sides: $x_1=8256-966 = 7290$.
For the maximum price ($x_2$):
Using the z - score formula $z_2=\frac{x_2-\mu}{\sigma}$, with $\mu = 8256$, $\sigma = 1150$ and $z_2 = 0.84$.
$0.84=\frac{x_2 - 8256}{1150}$.
Multiply both sides by 1150: $0.84\times1150=x_2 - 8256$.
$966=x_2 - 8256$.
Add 8256 to both sides: $x_2=8256 + 966=9222$.

Answer:

1. The student's score is 84.625.
2. The minimum price is $7290$ and the maximum price is $9222$.