A coffee shop owner believes the shop’s average daily sales are \$1,200, with a known population standard deviation of \$50. A manager claims it’s higher because of their new initiatives & collects a sample of 25 days and finds an average daily sales of \$1,230.
$\left(B\right)$ The owner’s policy is to reward store managers with a yearly bonus for increased sales. Should the owner give this manager with the bonus?

• A. Yes, the manager was able to increase sales.
• B. No, there was no increase in sales.
• C. More info is required.

Answer: (A)

Verified step by step guidance

1. Identify the null hypothesis (\(H_0\)) and the alternative hypothesis (\(H_a\)) based on the context of the problem. Typically, \(H_0\) represents the status quo or no effect, and \(H_a\) represents the claim you want to test.
2. Determine the significance level (\(\alpha\)), which is the probability of rejecting the null hypothesis when it is actually true. Common values are 0.05, 0.01, or 0.10.
3. Find the critical value(s) corresponding to the chosen significance level and the type of test (left-tailed, right-tailed, or two-tailed). Use the standard normal distribution (\(Z\)) or the \(t\)-distribution depending on the sample size and whether the population standard deviation is known.
4. Calculate the test statistic using the sample data. For example, if using a \(Z\)-test, the formula is: \[ Z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}} \] where \(\bar{x}\) is the sample mean, \(\mu_0\) is the hypothesized population mean, \(\sigma\) is the population standard deviation, and \(n\) is the sample size.
5. Compare the test statistic to the critical value(s). If the test statistic falls into the rejection region (beyond the critical value), reject the null hypothesis; otherwise, fail to reject it.