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.
$\left(A\right)$ Use $\alpha =0.10$ & the critical value method to test if the true mean daily sales are higher than \$1200.

• A. Because test stat. $\left(z\right)$ is INSIDE rejection region, we REJECT $H_0$. There is ENOUGH evidence to conclude that μ = 1200.
• B. Because test stat. $\left(z\right)$ is INSIDE rejection region, we REJECT $H_0$. There is ENOUGH evidence to conclude that μ > 1200.
• C. Because test stat. $\left(z\right)$ is OUTSIDE rejection region, we FAIL TO REJECT $H_0$. There is NOT ENOUGH evidence to conclude that μ = 1200.
• D. Because test stat. $\left(z\right)$ is OUTSIDE rejection region, we FAIL TO REJECT $H_0$. There is NOT ENOUGH evidence to conclude that μ > 1200.

Answer: (B)

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 \(t\)-distribution table depending on whether the population standard deviation is known and the sample size.
4. Calculate the test statistic using the sample data. For example, if using a \(Z\)-test, the formula is: \[\text{Test Statistic} = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}}\] 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) to decide whether to reject or fail to reject the null hypothesis. If the test statistic falls into the critical region, reject \(H_0\); otherwise, do not reject \(H_0\).