**Answer:**

0.0091 = 0.91% probability that the sample mean would be less than 48,101 miles in a sample of 281 tires if the manager is correct

**Step-by-step explanation:**

To solve this question, we need 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 and standard deviation , the zscore of a measure X is given by:

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.

**Central limit theorem:**

The Central Limit Theorem estabilishes that, for a random variable X, with mean and standard deviation , the sample means with size n of at least 30 can be approximated to a normal distribution with mean and standard deviation

**In this problem, we have that:**

**What is the probability that the sample mean would be less than 48,101 miles in a sample of 281 tires if the manager is correct?**

This is the pvalue of Z when X = 48101. So

By the Central Limit Theorem

has a pvalue of 0.0091

0.0091 = 0.91% probability that the sample mean would be less than 48,101 miles in a sample of 281 tires if the manager is correct