Elementary Statistics – One-sample (and paired) t Procedure Examples

Example 1 (One-sample t Confidence Interval)

Suppose that the life spans of a certain light bulb follow an approximate normal distribution. We are interested in estimating the mean life span of all of these light bulbs. We take a random sample of 20 light bulbs, and find the sample mean and sample standard deviation are 1478.3 hours and 521.8 hours, respectively.

1. Find and interpret a 90% confidence interval for the mean life span of the population of light bulbs.

2. Suppose now that we take a random sample of 5,000 light bulbs and find the same sample mean and standard deviation. Find a 90% confidence interval for the mean life span of the population of light bulbs.

Example 2 (One-sample t-test)

How accurate are radon detectors of a type sold to homeowners? To answer this question, researchers placed 14 randomly selected detectors in a chamber that exposed them to 105 picocuries per liter (pCi/l) of radon. The detector readings are shown in the stemplot below. The sample mean and standard deviation are 100.29 pCi/l and 7.15 pCi/l, respectively.

Stem	Leaf
8	9
9	1 4
9	5 6 7 9
10	1 3 4
10	5 6
11	1 3

Leaf unit = 1.0

1. Is it reasonable to use the one-sample t test (based on our rule of thumb for checking the normality condition)?

2. Is there convincing evidence that the mean reading of all detectors of this type differs from the true value of 105?

Example 3 (Paired t-test)

A study was conducted on the effect of a special class designed to improve children’s verbal skills. Each of 41 children took a verbal skills test twice, both before and after a 3-week period in the class. From the sample, the “after score – before score” differences have mean 0.645 and a standard deviation 1.527.

1. Is there evidence that the mean improvement is greater than 0 (check for statistical and practical significance)?

2. If there is evidence, can this improvement be solely attributed to the special class?