Final answer:
To determine whether there is enough evidence to support the director's feelings, a hypothesis test is performed. The test concludes that there is not enough evidence to support the director's feelings.
Step-by-step explanation:
To determine whether there is enough evidence to support the director's feelings, we can perform a hypothesis test with the critical value method.
Step 1: State the null and alternative hypotheses. Null hypothesis (H0): The mean number of operations performed by surgeons is equal to the national average of 211. Alternative hypothesis (Ha): The mean number of operations performed by surgeons is less than 211.
Step 2: Set the significance level. The significance level (alpha) is given as 0.10.
Step 3: Calculate the test statistic. The test statistic is calculated using the formula: z = (sample mean - population mean) / (standard deviation / sqrt(sample size)). In this case, the sample mean is 209.5, the population mean is 211, the standard deviation is 3.8, and the sample size is 15. Calculating the test statistic, we get z = (209.5 - 211) / (3.8 / sqrt(15)) = -0.197.
Step 4: Determine the critical value. Since the alternative hypothesis is one-tailed (less than), we need to find the critical value for a left-tailed test at a significance level of 0.10. Looking up the critical value in the Z-table, we find the critical value to be -1.282.
Step 5: Compare the test statistic with the critical value. Since the test statistic (-0.197) is greater than the critical value (-1.282), we fail to reject the null hypothesis.
Step 6: Interpret the results. There is not enough evidence to support the director's feelings that the surgeons at the hospital perform fewer operations per year than the national average of 211.