A computer battery manufacturer claims their batteries will hold a charge for an average of 15 hours, with standard deviation 3 hours. A random sample of 36 batteries is drawn. Each of the batteries is tested to see how long it holds a charge. Let X = sample mean of the 36 measured times. (a) If the claim is true, what is P(X < 14)? (b) How many batteries must be tested so that the P(X < 14) = 0.01?

a) P(X < 14) = 0.0228.

b) 49 batteries must be tested.

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 are 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 normally distributed random variable X, with mean and standard deviation , the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean and standard deviation .

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

In this problem, we have that:

(a) If the claim is true, what is P(X < 14)?

This is the pvalue of Z when X = 14. So

By the Central Limit Theorem

has a pvalue of 0.0228

So

P(X < 14) = 0.0228.

(b) How many batteries must be tested so that the P(X < 14) = 0.01?

Now we want to find the size of the sample for which X = 14 is in the first percentile, that is, Z when X = 14 has a pvalue of 0.01. So it is Z = -2.327.

First we have to find the standard deviation of the sample in this case.

We know that

So

Rounding up

49 batteries must be tested.

the answer is 4.8 units on e2020.