BeanZone Coffee is testing a new brand of espresso beans that the supplier claims will increase average customer satisfaction, which is currently 7.8 (out of 10). A sample of 35 stores using the new beans reports an average satisfaction score of 8.1 & standard deviation 0.5.
$\left(A\right)$ Use $\alpha =0.10$ to test if the true mean satisfaction score increased.

• A. Because $P$-value < $\alpha =$ 0.1, we REJECT $H_0$, there is ENOUGH evidence to suggest μ = 7.8.
• B. Because $P$-value < $\alpha =$ 0.1, we REJECT $H_0$, there is ENOUGH evidence to suggest μ > 7.8.
• C. Because $P$-value > $\alpha =$ 0.1, we FAIL TO REJECT $H_0$, there is NOT ENOUGH evidence to suggest μ > 7.8.
• D. Because $P$-value > $\alpha =$ 0.1, we FAIL TO REJECT $H_0$, there is NOT ENOUGH evidence to suggest μ = 7.8.

Answer: (B)

Verified step by step guidance

1. Step 1: Define the null and alternative hypotheses based on the claim. Since the supplier claims the average satisfaction score has increased from 7.8, set the null hypothesis as \(H_0: \mu = 7.8\) and the alternative hypothesis as \(H_a: \mu > 7.8\) (a right-tailed test).
2. Step 2: Identify the sample statistics and parameters given: sample mean \(\overline{x} = 8.1\), sample size \(n = 35\), sample standard deviation \(s = 0.5\), and significance level \(\alpha = 0.10\).
3. Step 3: Calculate the test statistic using the formula for a one-sample t-test (since population standard deviation is unknown): \[ t = \frac{\overline{x} - \mu_0}{s / \sqrt{n}} \] where \(\mu_0 = 7.8\) is the hypothesized population mean.
4. Step 4: Determine the critical value from the t-distribution with \(n - 1 = 34\) degrees of freedom at the \(\alpha = 0.10\) significance level for a right-tailed test. Alternatively, find the p-value corresponding to the calculated t-statistic.
5. Step 5: Compare the p-value to \(\alpha\): if the p-value is less than \(\alpha\), reject \(H_0\); otherwise, fail to reject \(H_0\). Then, interpret the result in the context of the problem to conclude whether there is enough evidence to support that the true mean satisfaction score has increased.