Concept explainers
(b)
The null and alternative hypotheses for the consumption of proteins.
(b)
Answer to Problem 72E
Solution: The null and alternative hypotheses are formulated as
Explanation of Solution
Therefore, the hypotheses are formulated as:
In the above hypotheses
To find: The null and alternative hypotheses for the consumption of carbohydrates.
Answer to Problem 72E
Solution: The null and alternative hypotheses are formulated as
Explanation of Solution
Calculation: The significance test is to compare the two groups in terms of consumption of carbohydrates. Hence, the hypothesis is that the consumption of carbohydrates in the two groups is the same against the alternative that the consumption of carbohydrates in the two groups is not the same.
Therefore, the hypotheses are formulated as
In the above hypothesis
(c)
To find: The test statistic with the degrees of freedom and the p-value for consumption of proteins.
(c)
Answer to Problem 72E
Solution: The t – test statistic is obtained as 2.47 and p – value as 0.0144 for 199 degrees of freedom for consumption of proteins.
Explanation of Solution
Calculation: The two-sample t – test statistic for the hypothesis formulated in part (a) is defined as:
Where,
The difference of means is considered as 0 in the null hypothesis. Substitute the provided values in the above defined formula to compute the two sample t statistic. So,
The p-value for the provided one–sided test is calculated as
So, the degrees of freedom are 199. The Excel function to determine the p- value from t-test statistic is displayed in the attached screenshot,
Therefore, the p-value is obtained as 0.0144.
To explain: The conclusion of the performed significance test.
Answer to Problem 72E
Solution: There is significant difference between the early eaters and late eaters in terms of consumption of proteins.
Explanation of Solution
To find: The test statistic with the degrees of freedom and the p-value for the consumption of carbohydrates.
Answer to Problem 72E
Solution: The t – test statistic is obtained as 0.29 and p – value as 0.772 for 199 degrees of freedom for carbohydrates consumption.
Explanation of Solution
Calculation: The two-sample t – test statistic for the hypothesis formulated in part (a) is defined as:
Where,
The difference of true means is considered as 0 according to the null hypothesis. Substitute the provided values in the above defined formula to compute the two sample t statistic. So,
The p-value for the provided one-sided test is calculated as
So, the degrees of freedom are 199. The Excel function to determine the p- value from t-test statistic is displayed in the attached screenshot,
Therefore, the p-value is obtained as 0.772.
The conclusion of the performed significance test.
Answer to Problem 72E
Solution: There is no significant difference between the early eaters and late eaters in terms of consumption of carbohydrates.
Explanation of Solution
(d)
To find: A 95% confidence interval for the difference of means between early eaters and late eaters in terms of consumption of proteins.
(d)
Answer to Problem 72E
Solution: A required 95% confidence interval is
Explanation of Solution
Calculation: The formula for confidence interval for the difference between the means is defined as:
where
According to the Table D provided in the Appendix, the critical value for a two-tailed test at 95% confidence level is 1.962 for 199 degrees of freedom. Substitute the provided values in the above-defined formula to determine the 95% confidence interval for the difference between the early eaters and late eaters in terms of consumption of proteins. So,
Therefore, the 95% confidence interval for the difference between the means is obtained as
To explain: The comparison of information from the obtained confidence interval with the information given by the significance test.
Answer to Problem 72E
Solution: The information provided by both confidence interval and the significance test shows that there is a significant difference between the two means of proteins.
Explanation of Solution
To find: A 95% confidence interval for the difference of means between early eaters and late eaters in terms of consumption of carbohydrates.
Answer to Problem 72E
Solution: A required confidence interval is
Explanation of Solution
Calculation: The formula for confidence interval for the difference between the means is defined as:
where
According to the Table D provided in the Appendix, the critical value for a two-tailed test at 95% confidence level is 1.962 for 199 degrees of freedom. Substitute the provided values in the above defined formula to determine the 95% confidence interval for the difference between the early eaters and late eaters in terms of consumption of carbohydrates. So,
Interpretation: Therefore, the 95% confidence interval for the difference between the means is obtained as
To explain: The comparison of information from obtained confidence interval with the information given by the significance test.
Answer to Problem 72E
Solution: The information provided by both confidence interval and the significance test shows that there is no significant difference between the two means of carbohydrates.
Explanation of Solution
The obtained confidence interval
To explain: A short summary on the results of the performed significance test and obtained confidence intervals for three dietary compositions.
Answer to Problem 72E
Solution: From the calculated value of the test statistic from the previous exercise for the consumption of fats, the null hypothesis is not rejected and it is concluded that there is no significant difference between the consumption of fats. The obtained confidence interval
Explanation of Solution
From the calculated value of the test statistic for the consumption of proteins, the null hypothesis is rejected and it is concluded that there is a significant difference between the consumption of proteins. The obtained confidence interval
From the calculated value of the test statistic for the consumption of carbohydrates, the null hypothesis is not rejected and it is concluded that there is no significant difference between the consumption of carbohydrates. The obtained confidence interval
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Chapter 7 Solutions
Introduction to the Practice of Statistics
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