Question

A certain cell phone advertises a mean battery life of 32 hours. Suppose the battery life of these phones follows a Normal distribution with standard deviation 2.5 hours. If 5 of these phones are randomly selected from the factory, then there is about a 95% probability that the sample mean will fall in which interval?

Find the z-table here.
7.5–56.5 hours
21.0–43.0 hours
27.1–36.9 hours
29.8–34.2 hours
31.0–33.0 hours

Answer 1

29.8–34.2 hours

Explanation:

Empirical Rule:

The Empirical Rule states that, for a normally distributed random variable:

68% of the measures are within 1 standard deviation of the mean.

95% of the measures are within 2 standard deviation of the mean.

99.7% of the measures are within 3 standard deviations of the mean.

Central Limit Theorem

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean
`\mu` and standard deviation
`\sigma`, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
`\mu` and standard deviation
`s = (\sigma)/(√(n))`.

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

Population:

Mean of 32, standard deviation of 2.5.

Sample:

Sample of 5, so
`n = 5, s = (2.5)/(√(5)) = 1.1`

There is about a 95% probability that the sample mean will fall in which interval?

Within 2 standard deviations of the mean. So

32 - 2*1.1 = 32 - 2.2 = 29.8

32 + 2*1.1 = 32 + 2.2 = 34.2 hours