Question

Confidence interval Assume that we will use the sample data from Exercise 1 “Video Games” with a 0.05 significance level in a test of the claim that the population mean is greater than 90 sec. If we want to construct a confidence interval to be used for testing the claim, what confidence level should be used for the confidence interval? If the confidence interval is found to be 21.1 sec < $\mu$< 191.4 sec, what should we conclude about the claim?

Short answer

The 90% confidence interval should be used to test the claim at a 0.05 level of significance.

It can be concluded that there is insufficient evidence to support the claim that the population mean is greater than 90 seconds.

Step-by-step solution

Given information

Refer to Exercise 1 for the duration times of alcohol use for 12 different video games. The claim tested at the 0.05 level of significance indicates that the population mean is greater than 90 sec.

The confidence interval is $21.1\text{sec}<\mu <191.4\text{sec}$.

State the hypotheses

For true population mean $\mu$, the hypotheses are formulated as follows:

$$\begin{gathered} H_0:\mu =90\sec (\text{null}\text{hypothesis}) \\ {H}_1:\mu >90\sec (\text{alternative}\text{hypothesis}\text{and}\text{original}\text{claim}) \end{gathered}$$

Hence, this is a right-tailed test or a one-tailed test.

Determine the confidence level

The relationship between significance level and confidence level is stated below.

		Confidence level
		Two-tailed test	One-tailed test
Significance level	0.01	99%	98%
	0.05	95%	90%
	0.10	90%	80%

For a one-tailed test with a significance level $\alpha =0.05$, a 90% confidence interval would be used.

State the decision using confidence interval

The 90% confidence interval is 21.1 sec to 191.4 sec.

The hypothesized value of 90 sec lies within the confidence interval. Thus, there is insufficient evidence to reject the null hypothesis.

Thus, it is concluded that we fail to reject the null hypothesis.

Therefore, it is concluded that there is insufficient evidence to support the claim that the population mean is greater than 90 seconds.