Question

In a study reported in the Flurry Blog on Oct. 29, 2012, the mean age of tablet users was 34 yrs, with a standard deviation of 15 years. Assuming a normal distribution, what is the approximate probability of picking a random sample of 40 tablet users with a mean age between 31 and 35 yrs?

Answer 1

55.90% probability of picking a random sample of 40 tablet users with a mean age between 31 and 35 yrs

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
`\mu` and standard deviation
`\sigma`, the zscore of a measure X is given by:

`Z = (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.

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.

In this problem, we have that:

`\mu = 34, \sigma = 15, n = 40, s = (15)/(√(40)) = 2.37`

Assuming a normal distribution, what is the approximate probability of picking a random sample of 40 tablet users with a mean age between 31 and 35 yrs?

This is the pvalue of Z when X = 35 subtracted by the pvalue of Z when X = 31.

X = 35

`Z = (X - \mu)/(\sigma)`

By the Central Limit Theorem

`Z = (X - \mu)/(s)`

`Z = (35 - 34)/(2.37)`

`Z = 0.42`

`Z = 0.42` has a pvalue of 0.6628

X = 31

`Z = (X - \mu)/(s)`

`Z = (31 - 34)/(2.37)`

`Z = -1.26`

`Z = -1.26` has a pvalue of 0.1038

0.6628 - 0.1038 = 0.5590

55.90% probability of picking a random sample of 40 tablet users with a mean age between 31 and 35 yrs