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: Find the z - score for the 40th percentile

Using a standard normal distribution table (z - table), the z - score corresponding to the 40th percentile (0.4000) is approximately $z=-0.25$.

Step2: Use the 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. We know $\mu = 85$, $\sigma=1.5$ and $z=-0.25$. Rearranging the formula for $x$ gives $x=\mu + z\sigma$.

Step3: Calculate the student's score

Substitute the values into the formula: $x = 85+(-0.25)\times1.5=85 - 0.375 = 84.625$.

for problem 8:

Step1: Determine the z - scores for the middle 60%

The middle 60% leaves 20% in the two tails (10% in each tail). The z - score corresponding to the 10th percentile (0.1000) is approximately $z_1=-1.28$ and the z - score corresponding to the 90th percentile (0.9000) is approximately $z_2 = 1.28$.

Step2: Use the z - score formula for the minimum price

The z - score formula is $z=\frac{x-\mu}{\sigma}$. Rearranging for $x$ gives $x=\mu+z\sigma$. For the minimum price, with $\mu = 8256$, $\sigma = 1150$ and $z=-1.28$, we have $x_1=8256+(-1.28)\times1150=8256-1472 = 6784$.

Step3: Use the z - score formula for the maximum price

Using the same formula $x=\mu + z\sigma$, with $z = 1.28$, $\mu = 8256$ and $\sigma=1150$, we get $x_2=8256 + 1.28\times1150=8256+1472=9728$.

Answer:

84.625