The police reported that the average driver exceeds the 65 miles per hour speed limit by more than 5 miles per hour on the busiest street in a small town. They know that the standard deviation of everybodyls speed is 2.83. A sample of 60 randomly selected cars were clocked by airplane radar. The average speed was 69.3 miles per hour. Determine if the average speed is greater than what the police department reported Which function in your calculator should you use to solve the problem?
Answer
Final Answer: We cannot conclude that the average speed is greater than what the police department reported. The p-value is \(\boxed{0.9723151286488966}\), which is greater than the usual significance level of 0.05.
Step 1 ：This is a hypothesis testing problem. We are given the sample mean, the population standard deviation, and the sample size. We are asked to test if the average speed is greater than what the police department reported (70 mph).
Step 2 ：We can use a one-sample z-test to solve this problem. The null hypothesis is that the average speed is equal to 70 mph, and the alternative hypothesis is that the average speed is greater than 70 mph.
Step 3 ：We can calculate the z-score using the formula: \(z = \frac{{\bar{x} - \mu}}{{\sigma/\sqrt{n}}}\) where \(\bar{x}\) is the sample mean, \(\mu\) is the population mean, \(\sigma\) is the population standard deviation, and \(n\) is the sample size.
Step 4 ：Then, we can find the p-value associated with this z-score. If the p-value is less than the significance level (usually 0.05), we reject the null hypothesis and conclude that the average speed is greater than 70 mph.
Step 5 ：Given values: sample mean (\(\bar{x}\)) = 69.3, population mean (\(\mu\)) = 70, population standard deviation (\(\sigma\)) = 2.83, sample size (\(n\)) = 60.
Step 6 ：Calculate z-score: \(z = \frac{{69.3 - 70}}{{2.83/\sqrt{60}}} = -1.9159634928234648\)
Step 7 ：Calculate p-value: p-value = 0.9723151286488966
Step 8 ：The p-value is greater than 0.05, which means we fail to reject the null hypothesis. Therefore, we cannot conclude that the average speed is greater than 70 mph based on this sample.
Step 9 ：Final Answer: We cannot conclude that the average speed is greater than what the police department reported. The p-value is \(\boxed{0.9723151286488966}\), which is greater than the usual significance level of 0.05.
