Question

1. data on dating a student wonders if tall women tend to date taller men than do short women. she measures herself, her dormitory roommate, and the women in the adjoining dorm room. then she measures the next man each woman dates. here are the data (heights in inches):

women (x) 66 64 66 65 70 69

men (y) 79 68 70 68 71 65

(a) make a scatterplot of these data. describe what you see.

(b) find the correlation r step by step, using the formula on page 166. explain how your value for r matches your description in part (a).

1. more hot dogs refer to exercise 19.

(a) explain why it isn’t correct to say that the correlation is 0.87 mg/cal.

(b) what would happen to the correlation if the variables were reversed on the scatterplot? explain your reasoning.

(c) what would happen to the correlation if sodium was measured in grams instead of milligrams? explain your reasoning.

1. correlation in... both high school english teachers, students think that rob is a harder grader, so rob and marie decide to grade the same 10 essays and see how their scores compare. the correlation is r = 0.99, but rob’s scores are always lower than marie’s. draw a possible scatterplot that illustrates this situation.

1. limitations of correlation a carpenter sells hand - made wooden benches at a craft fair every week. over the past year, the carpenter has varied the price of the benches from $80 to $120 and recorded the average weekly profit he made at each selling price. the prices of the bench and the corresponding average profits are shown in the table.

price $80 $90 $100 $110 $120

average profit $2400 $2800 $3000 $2800 $2400

(a) make a scatterplot to show the relationship between price and profit.

(b) the correlation for these data is r = 0. explain how this can be true even though there is a strong relationship.

Explanation:

Let's solve part (b) of problem 22, which is about finding the correlation \( r \) step - by - step. The formula for the correlation coefficient \( r \) is:

\[
r=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sqrt{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}\sum_{i = 1}^{n}(y_{i}-\bar{y})^{2}}}
\]

Step 1: Calculate the means of \( x \) (women's heights) and \( y \) (men's heights)

First, we find the number of data points \( n = 6 \).

For \( x \) values: \( x=\{66,64,66,65,70,69\} \)
\[
\bar{x}=\frac{66 + 64+66 + 65+70 + 69}{6}=\frac{390}{6}=65
\]

For \( y \) values: \( y = \{79,68,70,68,74,65\} \)
\[
\bar{y}=\frac{79+68 + 70+68+74+65}{6}=\frac{424}{6}\approx70.67
\]

Step 2: Calculate \( (x_{i}-\bar{x}) \), \( (y_{i}-\bar{y}) \), \( (x_{i}-\bar{x})(y_{i}-\bar{y}) \), \( (x_{i}-\bar{x})^{2} \) and \( (y_{i}-\bar{y})^{2} \) for each data point

\( x_i \)	\( y_i \)	\( x_i-\bar{x} \)	\( y_i - \bar{y}\)	\( (x_i-\bar{x})(y_i - \bar{y}) \)	\( (x_i-\bar{x})^2 \)	\( (y_i - \bar{y})^2 \)
64	68	\( 64 - 65=- 1\)	\( 68 - 70.67=-2.67\)	\( (-1)\times(-2.67) = 2.67\)	\( (-1)^2 = 1\)	\( (-2.67)^2\approx7.13\)
66	70	\( 66 - 65 = 1\)	\( 70 - 70.67=-0.67\)	\( 1\times(-0.67)=- 0.67\)	\( 1^2 = 1\)	\( (-0.67)^2\approx0.45\)
65	68	\( 65 - 65=0\)	\( 68 - 70.67=-2.67\)	\( 0\times(-2.67) = 0\)	\( 0^2=0\)	\( (-2.67)^2\approx7.13\)
70	74	\( 70 - 65 = 5\)	\( 74 - 70.67 = 3.33\)	\( 5\times3.33=16.65\)	\( 5^2 = 25\)	\( 3.33^2\approx11.09\)
69	65	\( 69 - 65 = 4\)	\( 65 - 70.67=-5.67\)	\( 4\times(-5.67)=-22.68\)	\( 4^2 = 16\)	\( (-5.67)^2\approx32.15\)

Step 3: Calculate the sums

• Sum of \( (x_{i}-\bar{x})(y_{i}-\bar{y}) \):

\[
\sum(x_{i}-\bar{x})(y_{i}-\bar{y})=8.33 + 2.67-0.67 + 0+16.65-22.68=4.2
\]

• Sum of \( (x_{i}-\bar{x})^{2} \):

\[
\sum(x_{i}-\bar{x})^{2}=1 + 1+1 + 0+25+16 = 44
\]

• Sum of \( (y_{i}-\bar{y})^{2} \):

\[
\sum(y_{i}-\bar{y})^{2}=69.4+7.13 + 0.45+7.13+11.09+32.15=127.35
\]

Step 4: Calculate the correlation coefficient \( r \)

First, calculate the denominator:
\[
\sqrt{\sum(x_{i}-\bar{x})^{2}\sum(y_{i}-\bar{y})^{2}}=\sqrt{44\times127.35}=\sqrt{5603.4}\approx74.85
\]

Then, calculate \( r \):
\[
r=\frac{4.2}{74.85}\approx0.056
\]

The small positive value of \( r \) (close to 0) indicates a very weak positive linear relationship between the height of women and the height of the men they date, which matches the description from the scatterplot (where the points are somewhat scattered with a very slight upward trend).

Answer:

The correlation coefficient \( r\approx0.056 \)