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The sample mean is a fundamental concept in statistics, representing the average of a set of observations. In this exercise, the sample mean is used to estimate the average time shoppers spend at checkout lines. Calculating the sample mean involves summing up all observed waiting times and then dividing by the number of observations, or \[ \bar{x} = \frac{\sum_{i=1}^{n}x_i}{n} \]Where:

• \(\bar{x}\) is the sample mean,
• \(x_i\) are individual sample observations,
• \(n\) is the sample size.

In our exercise, 120 shoppers were sampled, and the average waiting time calculated was 8.4 minutes. It serves as an estimate of the true population mean, which helps us understand the central tendency of our sampled data. This value becomes a key player when performing statistical tests, like hypothesis testing, as it gives us an initial insight into whether or not the waiting times align with expected values.

Confidence intervals provide a range of values, derived from the sample mean, within which we expect the population mean to fall. In essence, it's like stating that we are "confident" the true mean might lie in this interval if we were to take many samples. This is a fantastic way to account for sampling variability.To compute a confidence interval for a population mean, we use the formula:\[ \bar{x} \pm z_{\frac{\alpha}{2}} \times \frac{\sigma}{\sqrt{n}} \]Where:

• \(\bar{x}\) is the sample mean,
• \(z_{\frac{\alpha}{2}}\) is the critical value from the normal distribution for our confidence level (e.g., 1.96 for 95%),
• \(\sigma\) is the population standard deviation,
• \(n\) is the sample size.

The confidence interval calculated in the exercise was (7.827, 8.973), meaning we are 95% sure that the actual mean waiting time is between these two values. It's crucial to note that if a confidence interval includes the hypothesized mean (8 minutes in this case), it supports the notion that there's no significant difference from the expected value.

The p-value is a critical concept in hypothesis testing, providing a measure of the probability that the observed results (or more extreme) could occur under the null hypothesis. A smaller p-value indicates that, assuming the null hypothesis is true, the observed data is less likely.In hypothesis testing, we compare the p-value to a predefined significance level \(\alpha\) (commonly 0.05). If the p-value is less than \(\alpha\), it suggests that the data provides enough evidence to reject the null hypothesis.For the exercise, the calculated p-value was approximately 0.172. Given \(\alpha = 0.05\), we find \(p > \alpha\), meaning we do not reject the null hypothesis. This suggests that there isn't strong enough evidence to claim that the mean waiting time significantly differs from the assumed value (8 minutes). Thus, the p-value plays a key role in decision-making processes for statistical hypotheses.