Question

A hot tub manufacturer advertises that a water temperature of \(100^{\circ} \mathrm{F}\) can be achieved in 15 minutes or less. A random sample of 25 tubs is selected, and the time necessary to achieve a \(100^{\circ} \mathrm{F}\) temperature is determined for each tub. The sample mean time and sample standard deviation are 17.5 minutes and 2.2 minutes, respectively. Does this information cast doubt on the company's claim? Carry out a test of hypotheses using significance level \(0.05 .\)

Short answer

Carry out the noted calculations to determine the t-value and t-critical value before comparing them to decide whether the null hypothesis can be rejected or not.

Step-by-step solution

Formulate Hypotheses

First, let's set the null hypothesis (\(H_0\)) and the alternative hypothesis (\(H_a\)). The null hypothesis for this scenario will be that the mean time to heat the tub is equal to or less than 15 minutes (\(H_0: \mu \leq 15\)). Correspondingly, the alternative hypothesis will be the possibility that the average heating time is more than 15 minutes (\(H_a: \mu > 15\)).

Calculate Test Statistic

Next, calculate the t-value using the formula: \[t = \frac{\bar{x} - \mu}{\frac{s}{\sqrt n}}\] where \(\bar{x}\) is sample mean (17.5 minutes), \(\mu\) is the population mean (15 minutes), \(s\) is the standard deviation (2.2 minutes), and \(n\) is the number of samples (25).

Determine the t-critical

This is a one-tailed test since the alternative hypothesis is testing if the mean is greater than a certain value, not simply different. With a sample size of 24 degrees of freedom (sample size of 25 - 1) and an alpha level of 0.05, you consult a t-distribution table or use a calculator to find the t-critical value compatible with these conditions. The t-critical value will be used to decide whether the null hypothesis can be rejected.

Compare Test Statistic with t-critical

Now we compare our calculated t-value from Step 2 with the t-critical value obtained in Step 3. If the test statistic is greater than the t-critical value, that allows us to reject the null hypothesis.

Conclusion

Depending on the results of Step 4, make the conclusion. If we found there's enough evidence to reject the null hypothesis, it would mean the average time to heat the tub is statistically significantly more than 15 minutes, bringing doubt upon the manufacturer's claim. If not, there's not enough statistical evidence to disprove the manufacturer's claim.