A recent study of food portion sizes reported that over a 17-year period, the average size of a soft drink consumed by Americans aged 2 years and older increased from 13.1 ounces (oz) to 19.9 oz. The authors state that the difference is statistically significant with P < 0.01.
Explain what additional information you would need to compute a confidence interval for the increase, and outline the procedure that you would use for the computations. (Select all that apply.)
- Degrees of freedom and a more accurate P-value could be used to find the confidence interval. In this case we could determine t, then calculate SED and t*.
- Sample sizes and a more accurate P-value could be used to find the confidence interval. In this case we could determine standard deviations and the confidence interval in the usual way.
- Sample sizes and standard deviations could be used to find the confidence interval. In this case we could find the interval in the usual way.
- Standard deviations and degrees of freedom could be used to find the confidence interval. In this case we could now find the P-value, which could be used to find SED.
- t and degrees of freedom could be used to find the confidence interval. In this case we could compute SED and use degrees of freedom to find t*.
Do you think that a confidence interval would provide useful additional information? Explain why or why not.
- Yes, the confidence interval could give us useful information about the average size of soft drinks.
- Yes, the confidence interval could give us useful information about the magnitude of the difference.
- No, the confidence interval could give us no more useful information because the P-value already tells us that the interval does not contain 0.
- Yes, the confidence interval could give us useful information about the variability between sample participants in the study.
- No, the confidence interval could give us no more useful information because it cannot tell us the sample size in the study.
A friend has performed a significance test of the null hypothesis that two means are equal. His report states that the null hypothesis is rejected in favor of the alternative that the first mean is larger than the second. In a presentation on his work, he notes that the first sample mean was larger than the second mean and this is why he chose this particular one-sided alternative.
(a) Explain what is wrong with your friend's procedure and why.
- We should never choose a one-sided alternative.
- We should only choose a one-sided alternative if we have some reason to expect a specific directional outcome before looking at the sample results.
- The null hypothesis in this case should have been that the first mean is larger than the second.
- The first mean can never be larger than the second mean; this indicates a mistake was made during statistical analysis.
- The null hypothesis in this case should have been that the two means were not equal.
(b) Suppose he reported t = 1.50 with a P-value of 0.08. What is the correct P-value that he should report?
A study of iron deficiency among infants compared samples of infants following different feeding regimens. One group contained breast-fed infants, while the children in another group were fed a standard baby formula without any iron supplements. Here are summary results on blood hemoglobin levels at 12 months of age.
Group | n | x̄ | s
--- | --- | --- | ---
Breast-fed | 22 | 13.3 | 1.6
Formula | 18 | 12.7 | 1.7
(a) Is there significant evidence that the mean hemoglobin level is higher among breast-fed babies? State H0 and Ha.
- H0: μbreast-fed > μformula; Ha: μbreast-fed = μformula
- H0: μbreast-fed ≠ μformula; Ha: μbreast-fed < μformula
- H0: μbreast-fed = μformula; Ha: μbreast-fed > μformula
- H0: μbreast-fed < μformula; Ha: μbreast-fed = μformula
Carry out a t test. Give the P-value. (Use α = 0.01. Use μbreast-fed − μformula. Round your value for t to three decimal places, and round your P-value to four decimal places.)
What is your conclusion?
- Reject the null hypothesis. There is significant evidence that the mean hemoglobin level is higher among breast-fed babies.
- Fail to reject the null hypothesis. There is significant evidence that the mean hemoglobin level is higher among breast-fed babies.
- Reject the null hypothesis. There is not significant evidence that the mean hemoglobin level is higher among breast-fed babies.
- Fail to reject the null hypothesis. There is not significant evidence that the mean hemoglobin level is higher among breast-fed babies.
(b) Give a 95% confidence interval for the mean difference in hemoglobin level between the two populations of infants. (Round your answers to three decimal places.)