Question

The average age of the population of licensed drivers in a county is µ = 42.6, σ = 12, and the distribution is approximately normal. A county police officer was interested in whether the average age of drivers receiving parking tickets differed from the average age of the driving population. She obtained a sample of N = 25 drivers receiving parking tickets. The average age of these drivers was M = 40.5. Perform the appropriate hypothesis test to determine whether this group differs from the population of drivers in the county.

a. What are the hypotheses in formal statistical notation?
b. Draw the distribution/rejection region(s)/critical value(s).
c. Compute the test statistic and show where it falls on your distribution in part b.
d. Make decision & Communicate results.

Answer 1

a. The null hypothesis (H0) is that the average age of drivers receiving parking tickets is equal to the average age of the driving population. The alternative hypothesis (Ha) is that the average age of drivers receiving parking tickets differs from the average age of the driving population. b. To perform the hypothesis test, we need to determine the critical region based on the significance level (alpha). c. The test statistic is calculated by finding the standard error (SE) of the sample mean and the test statistic (t-score). d. We compare the test statistic to the critical values from the t-distribution to make a decision and communicate the results.

Step-by-step explanation:

a. Hypotheses in formal statistical notation:

Null hypothesis (H0): The average age of drivers receiving parking tickets is equal to the average age of the driving population.

Alternative hypothesis (Ha): The average age of drivers receiving parking tickets differs from the average age of the driving population.

b. Distribution/rejection region(s)/critical value(s):

To perform the hypothesis test, we need to determine the critical region based on the significance level (alpha). Let's assume alpha = 0.05 (or 5%). Since we are testing whether the group differs from the population, this is a two-tailed test. We'll need to find the critical values for the upper and lower tails.

c. Test statistic and where it falls:

To compute the test statistic, we need to calculate the standard error (SE) of the sample mean. The standard error can be calculated using the formula: SE = σ / √N, where σ is the population standard deviation and N is the sample size. In this case, σ = 12 and N = 25, so SE = 12 / √25 = 2.4.

Next, we calculate the test statistic (t-score) using the formula: t = (M - µ) / SE, where M is the sample mean and µ is the population mean. In this case, M = 40.5 and µ = 42.6, so t = (40.5 - 42.6) / 2.4 = -0.875.

d. Decision and results:

Now we compare the absolute value of the test statistic (|t|) to the critical value(s) from the t-distribution. Using a significance level of 0.05 (two-tailed), we find the critical value(s) at the corresponding degrees of freedom (df = N - 1 = 25 - 1 = 24).

If |t| > the critical value(s), we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.

In this case, the critical values at df = 24 and alpha = 0.05 are ±2.064.

Since |-0.875| < 2.064, we fail to reject the null hypothesis. Therefore, we do not have sufficient evidence to conclude that the average age of drivers receiving parking tickets differs from the average age of the driving population.