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Nationally, patients who go to the emergency room wait an average of 4 hours to be admitted into the hospital. Do patients at rural hospitals have a lower waiting time? The 14 randomly selected patients who went to the emergency room at rural hospitals waited an average of 2.6 hours to be admitted into the hospital. The standard deviation for these 14 patients was 2.3 hours. What can be concluded at the the $\alpha=$ 0.01 level of significance level of significance?
a. For this study, we should use Select an answer
b. The null and alternative hypotheses would be:
$H_{0}: ? \vee$ Select an answer $\vee$
$H_{1}: ? \vee$ Select an answer $\vee$
c. The test statistic ? $?$ (please show your answer to 3 decimal places.)
d. The $\mathrm{p}$-value $=\quad$ (Please show your answer to 4 decimal places.)
e. The $p$-value is ? $\vee \alpha$
f. Based on this, we should Select an answer $\vee$ the null hypothesis.
g. Thus, the final conclusion is that ...
The data suggest the population mean is not significantly lower than 4 at $\alpha=0.01$, so there is statistically insignificant evidence to conclude that the population mean waiting time to be admitted into the hospital from the emergency room for patients at rural hospitals is equal to 4 hours.
The data suggest that the population mean awaiting time to be admitted into the hospital from the emergency room for patients at rural hospitals is not significantly lower than 4 hours at $\alpha=0.01$, so there is statistically insignificant evidence to conclude that the population mean waiting time to be admitted into the hospital from the emergency room for patients at rural hospitals is lower than 4 hours.
The data suggest the populaton mean is significantly lower than 4 at $\alpha=0.01$, so there is statistically significant evidence to conclude that the population mean waiting time to be admitted into the hospital from the emergency room for patients at rural hospitals is lower than 4 hours.
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Thus, the final conclusion is that there is not enough evidence to conclude that the average waiting time for patients at rural hospitals is less than the national average of 4 hours.
Step 1 ：We are given that the national average waiting time for patients to be admitted into the hospital is 4 hours. We want to test if the average waiting time for patients at rural hospitals is less than this national average. We have a sample of 14 patients from rural hospitals with an average waiting time of 2.6 hours and a standard deviation of 2.3 hours. We will perform a hypothesis test at the 0.01 level of significance.
Step 2 ：For this study, we should use a one-sample t-test because we are comparing the sample mean to a known population mean.
Step 3 ：The null hypothesis is that the average waiting time for patients at rural hospitals is equal to or greater than the national average, \(H_{0}: \mu \geq 4\). The alternative hypothesis is that the average waiting time is less than the national average, \(H_{1}: \mu < 4\).
Step 4 ：We calculate the test statistic using the formula for a one-sample t-test: \(t = \frac{\bar{x} - \mu}{s/\sqrt{n}}\). Substituting the given values, we get \(t = \frac{2.6 - 4}{2.3/\sqrt{14}} \approx -2.278\).
Step 5 ：We calculate the p-value using the t-distribution with \(n - 1 = 14 - 1 = 13\) degrees of freedom. The p-value is the probability of observing a test statistic as extreme as -2.278 or more extreme, given that the null hypothesis is true. Using a t-distribution table or calculator, we find that the p-value is approximately 0.0202.
Step 6 ：We compare the p-value to the level of significance. Since the p-value (0.0202) is greater than the level of significance (0.01), we fail to reject the null hypothesis.
Step 7 ：Thus, the final conclusion is that there is not enough evidence to conclude that the average waiting time for patients at rural hospitals is less than the national average of 4 hours.
