Chapter 10.2, Problem 43E

(a)

To determine

To Explain: the two groups' centres are identical or distinct and the spreads are identical or different for both types.

Answer to Problem 43E

Centres: Different

Spread: Similar

Explanation of Solution

The centre lies approximately in the middle of the point most to the left and the point most to the right.

Then we remember that the centre for the upper dot plot is more to the right than to the left and so the centres look distinct.

(b)

To determine

To Explain: for red wine and white wine treatment, build and analyse a 90 percent confidence interval for the difference in mean percent gain in polyphenol levels.

Answer to Problem 43E

(2.6975,7.8539)

Explanation of Solution

Given:

Red wine	3.5	8.1	7.4	4.0	0.7	4.9	8.4	7.0	5.5
White wine	3.1	0.5	-3.8	4.1	-0.6	2.7	1.9	-5.9	0.1

Formula used:

The confidence interval are

  $\begin{array}{l}\left({\overline{x}}_1-{\overline{x}}_2\right)-t_{\alpha /2}\times \sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}\\ \left({\overline{x}}_1-{\overline{x}}_2\right)+t_{\alpha /2}\times \sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}\end{array}$

Calculation:

Mean is the sum of all quantities, divided by the value count.

  $\begin{array}{l}{\overline{x}}_1=5.5\\ {\overline{x}}_2=0.2333\end{array}$

Variance is the number of square deviations separated by n-1 to the norm. The square root of variance is the standard deviation.

  $\begin{array}{l}{s}_1=2.5169\\ s_2=3.2920\end{array}$

Degrees of freedom is

  $df=\min \left(n_1-1,n_2-1\right)=\min \left(9-1,9-1\right)=8$

Find $t^{\ast }$ with $df=8$ and c = 90%

  $t^{\ast }=1.860$

The confidence interval for ${\mu }_1-{\mu }_2$ are:

  $\begin{array}{l}\left({\overline{x}}_1-{\overline{x}}_2\right)-t_{\alpha /2}\times \sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}=\left(5.5-0.2333\right)-1.860\times \sqrt{\frac{{2.5169}^2}{9}+\frac{{3.2920}^2}{9}}=2.6975\\ \left({\overline{x}}_1-{\overline{x}}_2\right)+t_{\alpha /2}\times \sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}=\left(5.5-0.2333\right)+1.860\times \sqrt{\frac{{2.5169}^2}{9}+\frac{{3.2920}^2}{9}}=7.8539\end{array}$

There is confidence of 90 percent that the mean gap is between 2.6975 and 7.8539.

(c)

To determine

To Explain: The Part (b) of the interval indicates the red wine is more powerful than white wine.

Answer to Problem 43E

Yes

Explanation of Solution

Given:

Red wine	3.5	8.1	7.4	4.0	0.7	4.9	8.4	7.0	5.5
White wine	3.1	0.5	-3.8	4.1	-0.6	2.7	1.9	-5.9	0.1

From the result (b): (2.6975,7.8539)

The interval of confidence does not have zero and sits absolutely over zero, this means that the mean differential is more likely to be positive and that the interval indicates that red wine is more successful than white wine.