Question

The repair cost of a Subaru engine is normally distributed with a mean of $5,850 and a standard deviation of $1,125. Random samples of 20 Subaru engines are drawn from this population and the mean repair cost of each sample is calculated.
Which of the following mean costs would be considered unusual?

A. $6350
B. $6180
C. $5180
D. None of these

Answer 1

Answer:

C. $5180

Step-by-step explanation:

To solve this question, we have to understand the normal probability distribution and the central limit theorem.

Normal probability distribution:

Problems of normally distributed samples can be solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$, the zscore of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

Z-scores lower than -2 or higher than 2 are considered unusual.

Central limit theorem:

The Central Limit Theorem estabilishes that, for a random normally distributed variable X, with mean $\mu$ and standard deviation $\sigma$, the sample means with size n can be approximated to a normal distribution with mean $\mu$ and standard deviation $s = \frac{\sigma}{\sqrt{n}}$

In this problem, we have that:

$\mu = 5850, \sigma = 1125, n = 20, s = \frac{1125}{\sqrt{20}} = 251.56$

Which of the following mean costs would be considered unusual?

We have to find the z-score for each of them

A. $6350

$Z = \frac{X - \mu}{\sigma}$

By the Central Limit Theorem

$Z = \frac{X - \mu}{s}$

$Z = \frac{6350 - 5850}{251.56}$

$Z = 1.99$

Not unusual

B. $6180

$Z = \frac{X - \mu}{s}$

$Z = \frac{6180 - 5850}{251.56}$

$Z = 1.31$

Not unusual

C. $5180

$Z = \frac{X - \mu}{s}$

$Z = \frac{5180 - 5850}{251.56}$

$Z = -2.66$

Unusual, and this is the answer.