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The p-value is an essential element of hypothesis testing in statistics. It represents the probability of observing a test statistic as extreme as, or more extreme than, the one computed from your sample data, assuming the null hypothesis is true. In simpler terms, it helps to measure the strength of evidence against the null hypothesis. For our exercise, the null hypothesis is that the average time adults spend on chores is 14 hours. If the p-value calculated from the z-score is less than the predefined significance level (in this case, \( \alpha = 0.01 \)), we reject the null hypothesis. This implies that there is strong evidence to support the alternative hypothesis stating that adults spend more than 14 hours on chores.

The critical-value approach involves determining a threshold value or 'critical value' from a z-table, which corresponds to the significance level \( \alpha \). This critical value tells us where the "extreme" tail of the distribution begins. In our context, for a one-sided test with \( \alpha=0.01 \), we locate the critical z-value that marks the cut-off for the extreme 1% of the data distribution. If the calculated test statistic (z-score) falls beyond this region, this is a basis for rejecting the null hypothesis. This method provides a specific point of reference to decide whether our observed statistic is significantly different from what was expected under the null hypothesis.

To begin hypothesis testing, we compute the z-score. The z-score indicates how many standard deviations an element is from the mean. For our problem,

• \( \bar{x} = 14.65 \text{ hours} \) - sample mean,
• \( \mu = 14 \text{ hours} \) - population mean under the null hypothesis,
• \( \sigma = 3 \text{ hours} \) - population standard deviation,
• \( n = 200 \) - sample size.

Applying these values to the formula \( Z = \frac{\bar{x} - \mu}{\sigma/\sqrt{n}} \), we compute the z-score. This calculation facilitates comparison with both the p-value and critical-value approaches, helping to determine whether the observed data significantly deviates from what was expected.

The alternative hypothesis is a core part of any hypothesis testing process. It proposes what we are testing to prove or find evidence for.
In this exercise, the alternative hypothesis is that adults spend more than 14 hours on chores during a weekend. This vision contrasts with the null hypothesis (which states the average is 14 hours). We set the stage for statistical testing using either the p-value or critical-value approaches to determine which hypothesis is supported by the sample data. The alternative hypothesis is accepted if the test results show significant deviation—that is, if the sample data's mean is significantly higher than 14 hours.