Final Answer:
At the 0.050 significance level, the decision rule is to reject the null hypothesis if the sample mean waiting time at Farmer Jack's is less than 7.225 minutes.
Step-by-step explanation:
Statistical hypothesis testing is employed to determine if there is a significant difference between the local Farmer Jack's waiting time and the national average reported by the magazine. The null hypothesis (H₀) assumes no difference, stating that the local mean waiting time is equal to or greater than the national average. The alternative hypothesis (H₁) posits that the local mean waiting time is less than the national average.
To set up the decision rule, a critical value is determined based on the significance level (α), which is 0.050 in this case. For a one-tailed test, as we are only interested in whether the local waiting time is less than the national average, we find the critical z-value. Consulting a standard normal distribution table, a z-value of -1.645 corresponds to the 0.050 significance level.
Next, the critical value is calculated by multiplying the standard deviation of the sample mean by the critical z-value and adding it to the national average waiting time. The result, 7.225 minutes, becomes the threshold. If the sample mean waiting time at Farmer Jack's is less than this threshold, we reject the null hypothesis.
In conclusion, this decision rule allows us to determine if the waiting time at Farmer Jack's is statistically less than the national average, providing a robust method for drawing meaningful conclusions from the sample data.
The complete question is:
"A national grocer's magazine reports the typical shopper spends 75 minutes in line waiting to check out. A sample of 25 shoppers at the local Farmer Jack's showed a mean of 6.5 minutes with a standard deviation of 2.2 minutes. Is the waiting time at the local Farmer Jack's less than that reported in the national magazine? Use the 0.050 significance level. What is the decision rule? (Negative amounts should be indicated by a minus sign. Round your answer to 3 decimal places.)"