Question

1. Alice thinks that the more often you read or watch the news, the more likely you are to vote. She asks five people how many days per week they read or watch the news (X) and how likely they are to vote on a scale from 1 to 10 (Y).
a. (14 points) Calculate the correlation between the two variables:
X Y
3 2
2 1
1 3
5 8
6 9

b. (2 points) How much of the variability in likelihood to vote is explained by frequency of reading or watching the news?

2. Over the years, Coach Bob has developed a formula to predict how many of the country’s top 100 football recruits will sign with his university based on how many games his team won that year. The formula is:
Y = 2(X) – 14
a. (6 points) How many top recruits would be predicted for seasons that ended with:
12 wins?
10 wins?
7 wins?
b. (2 points) What is the predictor variable? What is the criterion variable?
c. (2 points) What is the slope? What is the y-intercept?
d. (2 points) Is the correlation between games won and number of top recruits signed positive or negative? How do we know?

3. (6 points) Calculate the standard error for the following samples. The population standard deviation for each sample is 15.
a. N = 4
b. N = 16
c. N = 225

4. a. (3 points) If we know the population mean is 50 and the standard error of the mean is .5, what is the z-score for a sample mean of 51? What is the likelihood of getting a sample mean of 51 or more?   

b. (3 points) If we know the population mean is 20 and the standard error of the mean is 5, what is the z-score for a sample mean of 105? What is the likelihood of getting a sample mean of 105 or higher？

Answer 1

Solution:

Question 1)

Given:

X: Number of days per week they read or watch the news

Y: how likely they are to vote on a scale from 1 to 10

X Y
3 2
2 1
1 3
5 8
6 9

Part a) Calculate the correlation between the two variables:

Formula for correlation :

where

Thus we need to make following table:

X	Y	X^2	Y^2	XY
3	2	9	4	6
2	1	4	1	2
1	3	1	9	3
5	8	25	64	40
6	9	36	81	54

Thus

Thus

Thus the correlation between the two variables is r = 0.8860

Part b) How much of the variability in likelihood to vote is explained by frequency of reading or watching the news?

Coefficient of determination is the amount of variation explained in dependent variable by variation is independent variable.

Thus we need to find :

Coefficient of determination = r²

Coefficient of determination = 0.8860²

Coefficient of determination = 0.7849

Coefficient of determination = 78.49%

Thus about 78.49% of the variability in likelihood to vote is explained by frequency of reading or watching the news.