Q. 45 Observational studies suggest th... [FREE SOLUTION]

91影视

The Practice of Statistics for AP Examination

Daren Starnes, Josh Tabor

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 837 pages

ISBN: 9781319113339

Chapter 10: Q. 45 (page 666)

Observational studies suggest that moderate use of alcohol by adults reduces heart attacks and that red wine may have special benefits. One reason may be that red wine contains polyphenols, substances that do good things to cholesterol in the blood and so may reduce the risk of heart attacks. In an experiment, healthy men were assigned at random to drink half a bottle of either red or white wine each day for two weeks. The level of polyphenols in their blood was measured before and after the $2 -$ week period. Here are the percent changes in polyphenols for the subjects in each group:
a. A dot-plot of the data is shown, along with summary statistics. Write a few sentences comparing the distributions.
b. Construct and interpret a $90 %$ confidence interval for the difference in true mean percent change in polyphenol levels for healthy men like the ones in this study when drinking red wine versus white wine.

Short Answer

Expert verified

Part a. There appear to be no outliers in either distribution because there are no dots that appear to be unusually far from the other dots in the dot plot.

Part b. There is $90 %$ confidence that the mean percent change in polyphenol levels for healthy men when drinking red wine is between $(2.6977, 7.8363)$ lower than the mean percent change in polyphenol levels for healthy men when drinking white wine.

Step by step solution

Part a. Step 1. Explanation

A dot plot of the data is shown in the question along with the summary statistics. Thus, on comparing the two distribution we see that:

Shape: the two distributions are slightly skewed to the left because most of the dots lie near the right of the dot plot.

Center: The center for the red wine distribution is higher than the center for the white wine because most of the dots in the dot plot of red wine lie to the right of most of the dots in the dot plot of white wine and the mean and the median are both greater for the red wine.

Spread: The distribution of white wine is little bit more variable than the distribution of red wine because the dot plot of the white wine is little wider than the dot plot of the red wine and the standard deviation of the white wine is little higher than the standard deviation of red wine.

Unusual features: There appear to be no outliers in either distribution because there are no dots that appear to be unusually far from the other dots in the dot plot.

Part b. Step 1. Given information

Given:

${\overset{炉}{x}}_{1} = 5.500 {\overset{炉}{x}}_{2} = 0.233 n_{1} = 9 n_{2} = 9 s_{1} = 2.517 s_{2} = 3.292 c = 0.90$

Part b. Step 2. Calculation

Now we will be calculating the t-value for this we need to find out the degree of freedom. Thus, the degree of freedom will be:

$d f = m i n (n_{1} - 1, n_{2} - 1) = m i n (9 - 1, 9 - 1) = 8$

Then the t-value will be as:

$t_{伪 / 2} = 1.860$

Thus the confidence interval be:

$({\overset{炉}{x}}_{1} - {\overset{炉}{x}}_{2}) - t_{伪 / 2} 脳 \sqrt{\frac{s_{1}^{2}}{n_{1}} + \frac{s_{2}^{2}}{n_{2}}} = (5.500 - 0.233) - 1.860 脳 \sqrt{\frac{2 . 517^{2}}{9} + \frac{3 . 292^{2}}{9}} = 5.267 - 2.5693 = 2.6977 ({\overset{炉}{x}}_{1} - {\overset{炉}{x}}_{2}) + t_{伪 / 2} + \sqrt{\frac{s_{1}^{2}}{n_{1}} + \frac{s_{2}^{2}}{n_{2}}} = (5.500 - 0.233) + 1.860 脳 \sqrt{\frac{2 . 517^{2}}{9} + \frac{3 . 292^{2}}{9}} = 5.267 + 2.5693 = 7.8363$

Thus, we conclude that there is $90 %$ confidence that the mean percent change in polyphenol levels for healthy men when drinking red wine is between $(2.6977, 7.8363)$ lower than the mean percent change in polyphenol levels for healthy men when drinking white wine.