Question

College Graduation Rates. Data from the College Results Online website compared the 2011 graduation rate and median SAT score for 92 similar-sized public universities and colleges in the United States. The scatterplot below shows the relationship between these two variables along with the least squares fit. Round all calculated results to 4 decimal places.

1. The relationship between median SAT score and graduation rate is  ? positive negative ,  ? weak strong , and  ? linear non-linear .

2. The explanatory variable is  ? graduation rate median SAT college year  and the response variable is  ? graduation rate median SAT college year .

The summary statistics for graduation rate and median SAT score are listed below. The correlation between graduation rate and median SAT score is 0.663.

Median SAT score: mean = 1030.3, standard deviation = 80.7
Graduation rate: mean = 49.6, standard deviation = 14.3

3. The equation of the regression line is y =  +  x

4. Complete the following sentence to interpret the slope of the regression line:

An increase of  in Median SAT score corresponds to a/an  ? decrease increase  of  in Graduation Rate.

5. The recorded median SAT score for Northern Michigan University is 1026. Use the regression equation to estimate the graduation rate for Northern Michigan University.

6. The recorded graduation rate for Northern Michigan University is 46.3. Complete the following sentence.

The residual for Northern Michigan University is  . This means the graduation rate at Northern Michigan University is
A. lower than
B. higher than
C. the same as
the rate predicted by the regression model.

7. Stanford University (an elite private university in California not included in this data set) has a median SAT score of 1455. Would it be appropriate to use this linear model to predict the graduation rate for Stanford?

A. Yes, because 1455 is a reasonable median SAT score for an elite university.
B. No, because 1455 is beyond the range of the data used to build the regression model.
C. No, because 99.495% is too large to be a reasonable graduation rate, even for an elite university.

Answer 1

Answer:

1. The relationship between the median SAT score and graduation rate is positive, strong and linear.

Positive because the slope is increasing, strong because points are on or close to the line and linear because it is a straight line.

2. The explanatory variable is Median SAT score and response variable is the graduation rate.

The response variable is the focus of a question in a study or experiment. An explanatory variable is one that explains changes in that variable. It can be anything that might affect the response variable.

3. Correlation coefficient r = 0.663

Median SAT score mean = 1030.3 and the standard deviation is 80.7

Graduation rate score mean = 49.6 and the standard deviation is 14.3

The least square regression equation can be represented as follows

y = a + b*x

where a is the intercept and b is the slope

where the slope is calculated using the following formula

and the intercept is calculated as follows

The formula for calculating correlation coefficient is

Thus we can calculate covariance of X and Y as

The equation of the regression line is

Y = -70.8045 + 0.117*x

4. An increase of 1 in median SAT score corresponds to an increase of 0.117 in Graduation rate.

5. The recorded median SAT score for the northern Michigan university is 1026.

Y = -70.8045 + 0.117*1026 = 49.23

The estimated graduation rate is = 49.23

6. The recorded graduation rate for the nrothern michigan university is 46.3.

Residual = recorded- predicted = 46.3 - 49.23 = -2.93

The residual for the northern university is -2.93. This means the graduation rate at northern Michigan university is lower than the predicted by the regression model.

7. No, because 1455 is beyond the range of the data used to build the regression model.

Regression analysis theory indicates that the safest place to obtain interpolation is in the middle of the range of the x values.  It is less secure at the ends of the range.  One should be cautious with extrapolation because the results become more and more unreliable very quickly as one goes further away from the range of the x values.

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