An article (N. Hellmich. "'Supermarket Guru' Has a Simple Mantra,” USA Today, June 19, 2002, p. 70) claimed that the typical super market trip takes a mean of 22 minutes. Suppose that in an effort to test this claim, you select a sample of 50 shoppers at a local supermarket. The mean shopping time for the sample of 50 shoppers is 25.36 minutes with a standard deviation of 7.24 minutes. Using the 0.10 level of significance, is there evidence that the mean shopping time at the local supermarket is different from the claimed value of 22 minutes?
To answer this question, we need to conduct a hypothesis test to determine if there is evidence that the mean shopping time at the local supermarket is different from the claimed value of 22 minutes.
Here are the steps to perform the hypothesis test:
Step 1: State the null hypothesis (H0) and alternative hypothesis (Ha):
- Null Hypothesis (H0): The mean shopping time at the local supermarket is 22 minutes.
- Alternative Hypothesis (Ha): The mean shopping time at the local supermarket is different from 22 minutes.
Step 2: Determine the level of significance (α):
In this case, the level of significance (α) is given as 0.10.
Step 3: Compute the test statistic:
We can use the t-test statistic since we have sample data and the population standard deviation is unknown.
The formula for the t-test statistic is:
t = (x̄ - μ) / (s / √n)
where:
x̄ is the sample mean,
μ is the claimed population mean (22 minutes),
s is the sample standard deviation,
n is the sample size.
Plugging in the values from the question:
x̄ = 25.36 minutes
μ = 22 minutes
s = 7.24 minutes
n = 50
t = (25.36 - 22) / (7.24 / √50)
Step 4: Determine the critical value:
Since the alternative hypothesis is two-sided (the mean could be greater or less than 22), we need to consider both tails of the t-distribution.
Using a t-distribution table or a statistical software, find the critical t-value for a two-tailed test with a significance level of 0.10 and (n-1) degrees of freedom.
For our sample size (n = 50) and significance level (α = 0.10), the critical t-value is approximately 1.676.
Step 5: Make a decision:
If the calculated t-value is outside the critical region (reject the null hypothesis region), we can reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
Step 6: Calculate the p-value (optional):
The p-value is the probability of obtaining a test statistic as extreme as the one calculated, assuming the null hypothesis is true. This step helps to interpret the significance of the results.
Using a t-distribution table or a statistical software, calculate the p-value associated with the calculated t-value. The p-value is the probability that the mean shopping time is different from 22 minutes, given the sample data.
Step 7: Conclusion:
Based on the decision made in Step 5 and the p-value calculated in Step 6, we can make a conclusion about the hypothesis test.
To test whether there is evidence that the mean shopping time at the local supermarket is different from the claimed value of 22 minutes, we can perform a hypothesis test using the sample data provided.
Let's state our null and alternative hypotheses:
Null hypothesis (H0): The mean shopping time at the local supermarket is equal to 22 minutes.
Alternative hypothesis (HA): The mean shopping time at the local supermarket is different from 22 minutes.
Next, we can set the significance level (α) as 0.10. This significance level represents the maximum probability of rejecting the null hypothesis when it is actually true.
We will use a t-test since the population standard deviation is unknown and we have a sample size of 50.
Here are the steps to perform the hypothesis test:
Step 1: Set up the hypotheses:
H0: μ = 22
HA: μ ≠ 22
Step 2: Determine the test statistic:
We will use the t-test statistic, which is calculated as:
t = (x̄ - μ) / (s / √n)
Where:
x̄ is the sample mean
μ is the hypothesized population mean
s is the sample standard deviation
n is the sample size
Step 3: Calculate the test statistic:
In this case, x̄ = 25.36, μ = 22, s = 7.24, n = 50, so the test statistic is:
t = (25.36 - 22) / (7.24 / √50)
Step 4: Determine the rejection region:
Since we are performing a two-tailed test (HA: μ ≠ 22), we need to find the critical t-values from the t-distribution table or a statistical calculator using the significance level (α/2 = 0.10/2 = 0.05) and degrees of freedom (n-1 = 50-1 = 49).
Step 5: Compare the test statistic with the critical values:
If the test statistic falls outside the critical region, we reject the null hypothesis. If it falls within the critical region, we fail to reject the null hypothesis.
Step 6: Make a decision:
If the test statistic falls outside the critical region, we reject H0 and conclude that there is evidence that the mean shopping time at the local supermarket is different from 22 minutes. If the test statistic falls within the critical region, we fail to reject H0 and cannot conclude that the mean shopping time is different from 22 minutes.
I will now calculate the test statistic and provide you with the decision based on the results. Please hold on.