The average amount of time it takes for couples to further communicate with each other after their first date has ended is 3.22 days. Is this average different for blind dates? A researcher interviewed 51 couples who had recently been on blind dates and found that they averaged 3.6 days to communicate with each other after the date was over. Their standard deviation was 0.962 days. What can be concluded at the the $\alpha=0.05$ level of significance? a. For this study, we should use t-test for a population mean b. The null and alternative hypotheses would be: $0^{\infty}$ $0^{8}$ $\sigma^{\circ}$ $0^{\circ}$ $0^{\circ}$ $\infty^{\circ}$ c. The test statistic $t t^{2}=$ (please show your answer to 3 decimal places.)

Step 1 ：First, we need to identify the null and alternative hypotheses. The null hypothesis is that the average time it takes for couples to communicate after a blind date is the same as the overall average, which is 3.22 days. The alternative hypothesis is that the average time is different from 3.22 days. In mathematical terms, we can express these as: $H_0: \mu = 3.22$ and $H_1: \mu \neq 3.22$.

Step 2 ：Next, we need to calculate the test statistic. We can use the formula for the t-statistic, which is $t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}}$, where $\bar{x}$ is the sample mean, $\mu_0$ is the population mean under the null hypothesis, $s$ is the sample standard deviation, and $n$ is the sample size.

Step 3 ：Substituting the given values into the formula, we get $t = \frac{3.6 - 3.22}{0.962/\sqrt{51}}$.

Step 4 ：Calculating the above expression, we get $t \approx 2.726$.

Step 5 ：Now, we need to find the critical value for a two-tailed t-test with $\alpha = 0.05$ and degrees of freedom $df = n - 1 = 51 - 1 = 50$. Using a t-distribution table or calculator, we find that the critical value is approximately $\pm 2.009$.

Step 6 ：Since the calculated t-value is greater than the critical value, we reject the null hypothesis. This means that there is significant evidence at the $\alpha = 0.05$ level to suggest that the average time it takes for couples to communicate after a blind date is different from the overall average of 3.22 days.

Step 7 ：Therefore, the final answer is $t \approx 2.726$, and we reject the null hypothesis.