a dealer finds that the average price of a previously owned vehicle is $8,256. he decides to sell to the…
a dealer finds that the average price of a previously owned vehicle is $8,256. he decides to sell to the middle 60% of the market in terms of price. find the maximum and minimum prices the dealer will sell. the standard deviation is $1,150 and the variable is normally distributed.
Answer
Explanation:
Step1: Determine the z - scores
The middle 60% of a normal distribution leaves 20% in the lower - tail and 20% in the upper - tail. Looking up in the standard normal distribution table, the z - score corresponding to the lower 20% (or 0.20) is approximately $z_1=- 0.84$ and the z - score corresponding to the upper 80% (1 - 0.20) is approximately $z_2 = 0.84$.
Step2: Use the z - score formula
The z - score formula is $z=\frac{x-\mu}{\sigma}$, where $\mu$ is the mean, $\sigma$ is the standard deviation, and $x$ is the value from the data set. We know that $\mu = 8256$ and $\sigma=1150$. For the minimum value $x_1$: [ \begin{align*} -0.84&=\frac{x_1 - 8256}{1150}\ x_1-8256&=-0.84\times1150\ x_1-8256&=-966\ x_1&=8256 - 966\ x_1&=7290 \end{align*} ] For the maximum value $x_2$: [ \begin{align*} 0.84&=\frac{x_2 - 8256}{1150}\ x_2-8256&=0.84\times1150\ x_2-8256&=966\ x_2&=8256 + 966\ x_2&=9222 \end{align*} ]
Answer:
The minimum price is $$7290$ and the maximum price is $$9222$.