Answer: The average number of days the movie runs in the southern districts is significantly different from 100 at a 2% significance level.
Explanation:
Null hypothesis (H0): The average number of days the movie runs in the southern districts is equal to 100.
Alternative hypothesis (H1): The average number of days the movie runs in the southern districts is not equal to 100.
Next, we need to calculate the test statistic. The formula for the one-sample t-test is:
t = (sample mean - population mean) / (sample standard deviation / √sample size)
Given that the sample mean is 86, the population mean is 100, the sample standard deviation is 8, and the sample size is 80, we can plug these values into the formula to calculate the test statistic:
t = (86 - 100) / (8 / √80)
After performing the calculations, we find that the test statistic is approximately -5.92.
To determine the critical value for a 2% significance level, we need to consult the t-distribution table or use statistical software. Assuming a two-tailed test, the critical value for a 2% significance level with a sample size of 80 is approximately ±2.617.
Since the absolute value of our test statistic (-5.92) exceeds the critical value (2.617), we can reject the null hypothesis. Therefore, we have sufficient evidence to conclude that the average number of days the movie runs in the southern districts is significantly different from 100 at a 2% significance level.