Final answer:
To test the claim that the standard deviation of points scored in NFL games is less than 13 points, we will perform a hypothesis test using a 0.05 level of significance. The null hypothesis is that the standard deviation is greater than or equal to 13, while the alternative hypothesis is that the standard deviation is less than 13. We will calculate the test statistic, determine the rejection region, and make a decision based on the critical value and the calculated test statistic.
Step-by-step explanation:
In this question, we are testing the claim that the standard deviation of points scored in NFL games is less than 13 points, at a 0.05 level of significance. To do this, we will perform a hypothesis test.
Step 1: State the Hypotheses
The null hypothesis, denoted by H0, is that the standard deviation is greater than or equal to 13. The alternative hypothesis, denoted by Ha, is that the standard deviation is less than 13.
H0: σ >= 13
Ha: σ < 13
Step 2: Test Statistic
We will use the chi-square distribution to test the hypotheses. The test statistic is calculated as (n-1)s^2/σ0^2, where n is the sample size, s is the sample standard deviation, and σ0 is the hypothesized standard deviation.
Step 3: Determine the Rejection Region
We will reject the null hypothesis if the test statistic falls in the critical region, which is determined using the chi-square distribution and the level of significance. In this case, we will use a one-tailed test because we are testing for a specific direction (less than 13 points).
Step 4: Calculate the Test Statistic
Using the given sample data, we can calculate the test statistic as (15-1)(32.7^2)/(13^2), where 32.7 is the sample standard deviation calculated from the given scores.
Step 5: Make a Decision
Using a chi-square distribution table or a calculator, we can find the critical value for our level of significance of 0.05 and degrees of freedom of 14. If the calculated test statistic is less than the critical value, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
Step 6: Conclusion
Based on the calculated test statistic and the critical value, we make a conclusion about whether to reject or fail to reject the null hypothesis. If we reject the null hypothesis, it means that there is evidence to support the claim that the standard deviation of points scored in NFL games is less than 13 points.