Bighorn sheep are found in the mountainous western United States.  In the following data, x = the age of bighorn sheep in years and y = the percentage of that age group that die during the course of a year (mortality rate).  In the table below, when x = 5, y = 20 meaning that 20% of the sheep aged between 5 and 6 years old die.  A random sample of sheep provided the following information:

∑ x = 15 ; ∑ y = 86.9 ; ∑ x 2 = 55 ; ∑ y 2 = 1544.73 ; ∑ x y = 273.4

Find the correlation coefficient (r) for this data.

A study was done to look at the relationship between number of vacation days employees take each year and the number of sick days they take each year. The results of the survey are shown below.

1. Find the correlation coefficient:  r=r=   Round to 2 decimal places.
2. The null and alternative hypotheses for correlation are:
   H0:H0: ? ρ r μ  == 0
   H1:H1: ? r ρ μ   ≠≠ 0    
   The p-value is:    (Round to four decimal places)
3. Use a level of significance of α=0.05α=0.05 to state the conclusion of the hypothesis test in the context of the study.
   • There is statistically significant evidence to conclude that an employee who takes more vacation days will take more sick days than an employee who takes fewer vacation days.
   • There is statistically insignificant evidence to conclude that there is a correlation between the number of vacation days taken and the number of sick days taken. Thus, the use of the regression line is not appropriate.
   • There is statistically significant evidence to conclude that an employee who takes more vacation days will take fewer sick days than an employee who takes fewer vacation days .
   • There is statistically significant evidence to conclude that there is a correlation between the number of vacation days taken and the number of sick days taken. Thus, the regression line is useful.
4.  r2r2 = (Round to two decimal places)  
5.  Interpret r2r2 :  
   • 77% of all employees will take the average number of sick days.
   • Given any group with a fixed number of vacation days taken, 77% of all of those employees will take the predicted number of sick days.
   • There is a large variation in the number of sick days employees take, but if you only look at employees who take a fixed number of vacation days, this variation on average is reduced by 77%.
   • There is a 77% chance that the regression line will be a good predictor for the number of sick days taken based on the number of vacation days taken.
6. The equation of the linear regression line is:   
   ˆyy^ = + xx   (Please show your answers to two decimal places)  
7. Use the model to predict the number of sick days taken for an employee who took 5 vacation days this year.
   Sick Days = (Please round your answer to the nearest whole number.)  
8. Interpret the slope of the regression line in the context of the question:  
   • For every additional vacation day taken, employees tend to take on average 0.73 fewer sick days.
   • The slope has no practical meaning since a negative number cannot occur with vacation days and sick days.
   • As x goes up, y goes down.
9. Interpret the y-intercept in the context of the question:
   • If an employee takes no vacation days, then that employee will take 8 sick days.
   • The average number of sick days is predicted to be 8.
   • The y-intercept has no practical meaning for this study.
   • The best prediction for an employee who doesn’t take any vacation days is that the employee will take 8 sick days.

Run a regression analysis on the following bivariate set of data with y as the response variable.

Find the correlation coefficient and report it accurate to three decimal places.
r =

What proportion of the variation in y can be explained by the variation in the values of x? Report answer as a percentage accurate to one decimal place.  (If the answer is 0.84471, then it would be 84.5%…you would enter 84.5 without the percent symbol.)
r² = %

Based on the data, calculate the regression line (each value to three decimal places)

y = x +

Predict what value (on average) for the response variable will be obtained from a value of 57.8 as the explanatory variable. Use a significance level of α=0.05 to assess the strength of the linear correlation.

What is the predicted response value?  (Report answer accurate to one decimal place.)
y =