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The significance level is a crucial component in hypothesis testing. It represents the probability we are willing to accept for making a Type I error, which is rejecting the null hypothesis when it is actually true. In our problem, we are using a significance level of 1%, denoted as \( \alpha = 0.01 \). This means we are allowing for only a 1% chance of incorrectly concluding that the average length of phone calls has decreased. This low significance level reflects the manager's desire for high confidence in the decision to implement the new policy based on the data at hand.

The test statistic allows us to make statistical comparisons. It is a standardized value that helps determine the difference between the sample statistic and the population parameter if the null hypothesis is true.We calculate the test statistic using the following formula:\[t = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}}\]where:

• \( \bar{x} \) is the sample mean (133 seconds)
• \( \mu \) is the population mean (143 seconds)
• \( s \) is the sample standard deviation (35 seconds)
• \( n \) is the sample size (100 calls)

The test statistic reflects how far the sample mean deviates from the population mean when expressed in terms of the standard error. In our example, the test statistic value calculated is -2.857.

The critical value determines the threshold our test statistic must exceed to reject the null hypothesis. It is found using statistical tables, given the desired significance level and the degrees of freedom in our sample.In this case, with a 1% significance level and 99 degrees of freedom (\( n-1 \)), the critical value is approximately \( t_{critical} = -2.33 \) for a one-tailed test. The critical value acts as a cutoff point: if the test statistic is less than the critical value, we have enough statistical evidence to reject the null hypothesis. Here, because our test statistic of -2.857 is less than -2.33, this supports the manager's claim that the call duration has decreased.

Formulating the null and alternative hypotheses is the first step of hypothesis testing. These hypotheses set the basis for what we are testing.- The **null hypothesis** (\( H_0 \)): asserts that there is no change or effect. In our scenario, it claims that the average call duration remains 143 seconds, i.e., \( \mu = 143 \).- The **alternative hypothesis** (\( H_1 \)): suggests a change or different effect. Here, it states that the average duration has decreased, i.e., \( \mu < 143 \).Understanding these hypotheses guides us through setting up and interpreting the statistical test, as they frame our investigation into whether policy changes have indeed shortened call durations.