A restaurant that bills its house account monthly is concerned that the average monthly bill exceeds $200 per account. A random sample of twelve accounts is selected, resulting in the sample mean of $220 and a sample standard deviation of $12. The researchers have determined that they should test that the mean bill exceeds $200 at the 5% level of significance.
What is the Null Hypothesis?
What is the Alternate Hypothesis?
How many tail(s) in the test?
What test should the researchers use?
Why do they use this test?
What is the critical value?
If the calculated value for the test statistic is 5.77, then what have the researchers learned with the test?
Are the test results reliable?
n: 12
Mean: 220.00
SD: 12.00
α: 0.05
H0:<=200 H1:>200
Reject H0 if t > 1.796
t=(220.00 - 200.0)/(12.00/sq 12)
t= 20.00 / 3.46
t= 5.77
Reject H0 because t value is greater than 1.796
Null Hypothesis: The average monthly bill for the restaurant's house accounts is $200 or less.
Alternate Hypothesis: The average monthly bill for the restaurant's house accounts exceeds $200.
Number of tails in the test: The test is one-tailed.
Test to be used: The researchers should use a one-sample t-test.
Reason for using this test: The one-sample t-test is appropriate when comparing the mean of a sample to a known or hypothesized population mean.
Critical value: The critical value depends on the level of significance chosen. In this case, the level of significance is 5%, so the critical value is 1.753 (obtained from t-tables with 11 degrees of freedom).
Interpretation of the test statistic: If the calculated value for the test statistic is 5.77, it means that the sample mean of $220 is 5.77 standard deviations away from the hypothesized population mean of $200.
What the researchers have learned: With a calculated test statistic of 5.77, the researchers have learned that the sample mean significantly exceeds the hypothesized population mean of $200.
Reliability of the test results: To determine the reliability of the test results, the p-value needs to be compared to the chosen level of significance. If the p-value is less than the level of significance (5%), then the test results can be considered reliable. However, the p-value is not provided in the given information, so we cannot determine the reliability of the results without further information.
To answer these questions, we need to understand hypothesis testing and the specific details provided in the question.
1. Null Hypothesis: The null hypothesis, denoted as H0, is a statement that assumes no significant difference exists between a set of observations. In this case, the null hypothesis can be written as "The average monthly bill per account is not greater than $200."
2. Alternative Hypothesis: The alternative hypothesis, denoted as Ha, is a statement that contradicts the null hypothesis and suggests that there is a significant difference between the observations. In this case, the alternative hypothesis can be written as "The average monthly bill per account exceeds $200."
3. Number of Tails: Since the alternative hypothesis suggests that the average monthly bill per account is greater than $200, this is a one-tailed test.
4. Test Selection: In this scenario, since we are comparing the mean of a sample to a specific value and have the sample's standard deviation, we should use a t-test for a single sample.
5. Reason for Test Selection: The researchers should use a t-test because we have the sample mean, sample standard deviation, and the population mean we are comparing it to. The t-test allows us to evaluate whether the sample mean is significantly different from the population mean.
6. Critical Value: To determine the critical value for the test, we need to refer to the t-distribution table or use statistical software. Since the significance level is specified as 5%, and the test is one-tailed, we need to find the critical t-value at the 95th percentile with degrees of freedom of 11 (12 - 1).
7. If the calculated value for the test statistic is 5.77, the researchers have learned that the sample mean is significantly greater than $200. The calculated test statistic falls in the rejection region, which means that the null hypothesis can be rejected in favor of the alternative hypothesis.
8. Test Reliability: To determine the reliability of the test results, we compare the calculated test statistic to the critical value. If the calculated value falls in the rejection region (beyond the critical value), we can reject the null hypothesis and conclude that the results are reliable at the specified significance level.
Please note that the actual critical value and conclusion of the test cannot be determined with the provided information.