Answer:
(a) The probability that for a sample of 35 individuals that purchase a used car will pay an average between $6820 to $7880 is 0.9120.
(b) The probability that for a sample of 35 individuals that purchase a used car will pay an average of more than $ 7140 is 0.7258.
Step-by-step explanation:
Let <em>X</em> = amount of money individuals pay when buying a used car.
The random variable <em>X</em> is Normally distributed with mean <em>μ</em> = $7310 and standard deviation <em>σ</em> = $1640.
A sample of <em>n</em> = 35 individuals who purchase a used car is selected.
We need to compute the probability of:
(a) Between $6820 to $7880
(b) More then $7140.
(a)
Compute the probability that for a sample of 35 individuals that purchase a used car will pay an average between $6820 to $7880 as follows:
![P(6820](https://tex.z-dn.net/?f=P%286820%3C%5Cbar%20X%3C7880%29%3DP%28%5Cfrac%7B6820-7310%7D%7B1640%2F%5Csqrt%7B35%7D%7D%3C%5Cfrac%7B%5Cbar%20X-%5Cmu%7D%7B%5Csigma%2F%5Csqrt%7Bn%7D%7D%3C%5Cfrac%7B7880-7310%7D%7B1640%2F%5Csqrt%7B35%7D%7D%29)
![=P(-1.77](https://tex.z-dn.net/?f=%3DP%28-1.77%3CZ%3C2.06%29%5C%5C%3DP%28Z%3C2.06%29-P%28Z%3C-1.77%29%5C%5C%3D0.98030-0.03836%5C%5C%3D0.94194%5C%5C%5Capprox%200.9120)
*Use a <em>z</em>-table for the probability.
Thus, the probability that for a sample of 35 individuals that purchase a used car will pay an average between $6820 to $7880 is 0.9120.
(b)
Compute the probability that for a sample of 35 individuals that purchase a used car will pay an average of more than $ 7140 as follows:
![P(\bar X>7140)=P(\frac{\bar X-\mu}{\sigma/\sqrt{n}}>\frac{7140-7310}{1640/\sqrt{35}})](https://tex.z-dn.net/?f=P%28%5Cbar%20X%3E7140%29%3DP%28%5Cfrac%7B%5Cbar%20X-%5Cmu%7D%7B%5Csigma%2F%5Csqrt%7Bn%7D%7D%3E%5Cfrac%7B7140-7310%7D%7B1640%2F%5Csqrt%7B35%7D%7D%29)
![=P(Z>-0.61)\\=P(Z](https://tex.z-dn.net/?f=%3DP%28Z%3E-0.61%29%5C%5C%3DP%28Z%3C0.61%29%5C%5C%3D0.72575%5C%5C%5Capprox0.7258)
*Use a <em>z</em>-table for the probability.
Thus, the probability that for a sample of 35 individuals that purchase a used car will pay an average of more than $ 7140 is 0.7258.