Concept explainers
To check: Whether there is sufficient evidence to conclude a difference in means.
To perform: The appropriate test to find out where the difference in means if there is sufficient evidence to conclude a difference in means
Answer to Problem 15CQ
Yes, there is sufficient evidence to conclude a difference in means.
There is significant difference between the means “Asia and Europe” and “Asia and Africa”.
Explanation of Solution
Given info:
The table shows the particulate matter for prominent cities of three continents. The level of significance is 0.05.
Calculation:
The hypotheses are given below:
Null hypothesis:
Alternative hypothesis:
Here, at least one mean is different from the others is tested. Hence, the claim is that, at least one mean is different from the others.
The level of significance is 0.05. The number of samples k is 3, the sample sizes
The degrees of freedom are
Where
Substitute 3 for k in
Substitute 11 for N and 3 for k in
Critical value:
The critical F-value is obtained using the Table H: The F-Distribution with the level of significance
Procedure:
- Locate 8 in the degrees of freedom, denominator row of the Table H.
- Obtain the value in the corresponding degrees of freedom, numerator column below 2.
That is, the critical value is 4.46.
Rejection region:
The null hypothesis would be rejected if
Software procedure:
Step-by-step procedure to obtain thetest statistic using the MINITAB software:
- Choose Stat > ANOVA > One-Way.
- In Response, enter the Gasoline prices.
- In Factor, enter the Factor.
- Click OK.
Output using the MINITAB software is given below:
From the MINITAB output, the test value F is 6.65.
Conclusion:
From the results, the test value is 6.65.
Here, the F-statistic value is greater than the critical value.
That is,
Thus, it can be concluding that, the null hypothesis is rejected.
Hence, the result concludes that, there is sufficient evidence to conclude a difference in means.
Consider,
Step-by-step procedure to obtain the test mean and standard deviation using the MINITAB software:
- Choose Stat > Basic Statistics > Display
Descriptive Statistics . - In Variables enter the columns Asia, Europe and Africa.
- Choose option statistics, and select Mean, Variance and N total.
- Click OK.
Output using the MINITAB software is given below:
The sample sizes
The means are
The sample variances are
Here, the samples of sizes of three states are not equal. So, the test used here is Scheffe test.
Tukey test:
Critical value:
The formula for critical value F1 for the Scheffe test is,
Here, the critical value of F test is 4.46.
Substitute 4.46 for critical value is of F and 2 for k-1 in
Comparison of the means:
The formula for finding
That is,
Comparison between the means
The hypotheses are given below:
Null hypothesis:
Alternative hypothesis:
Rejection region:
The null hypothesis would be rejected if absolute value greater than the critical value.
The formula for comparing the means
Substitute 74.0 and 35.50 for
Thus, the value of
Conclusion:
The value of
Here, the value of
That is,
Thus, the null hypothesis is rejected.
Hence, there is significant difference between the means
Comparison between the means
The hypotheses are given below:
Null hypothesis:
Alternative hypothesis:
Rejection region:
The null hypothesis would be rejected if absolute value greater than the critical value.
The formula for comparing the means
Substitute 74.0 and 30.67 for
Thus, the value of
Conclusion:
The value of
Here, the value of
That is,
Thus, the null hypothesis is rejected.
Hence, there is significant difference between the means
Comparison between the means
The hypotheses are given below:
Null hypothesis:
Alternative hypothesis:
Rejection region:
The null hypothesis would be rejected if absolute value greater than the critical value.
The formula for comparing the means
Substitute 35.50 and 30.67 for
Thus, the value of
Conclusion:
The value of
Here, the value of
That is,
Thus, the null hypothesis is not rejected.
Hence, there is no significant difference between the means
Justification:
Here, there is significant difference between the means
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