3.Correlation: Use the steps of hypothesis testing to evaluate this hypothesis.
a.State the hypotheses.
b.Determine the critical value of your statistic for rejecting the null based on the given alpha level/sample size.
c.Calculate the statistic.
e.Calculate the effect. What does this mean?
The Relationship between Hours Worked and GPA
The relationship between hours studied and GPA:
- What is the predictor variable?
The predictor variable is the variable named “study” denoting the number of hours studied by the student.
- What is the criterion variable?
The criterion variable is the variable named “GPA” denoting the GPA of the student.
- Correlation: Use the steps of hypothesis testing to evaluate this hypothesis.
- State the hypotheses.
The hypothesis states that the GPA scores of the student is affected by the hours studied by the student. The null and alternate hypothesis can be therefore written as:
- Determine the critical value of your statistic for rejecting the null based on the given alpha level/sample size.
The critical value is given by the inverse of t-statistic with n-2 or 27 degrees of freedom and alpha value equal to 0.05. The critical value was determined to be 2.0518.
- Calculate the statistic.
The statistic of interest to test the hypothesis is given by . The Pearson’s correlation coefficient, r, given by , was computed as 0.7539. The data has 29 sample points. Thus the statistic was computed to have the value 5.9628.
- Based on this statistic alone, would you reject or accept the null hypothesis?
The test suggests rejection of null hypothesis at 5% level of significance since the statistic has value greater than the critical value.
The value of is 0.4316, which is the effect that the predictor variable has on the criterion variable. It denotes the proportion of variation in criterion variable as explained by the predictor variable.
- Regression: Address the following items
- Calculate the slope of the regression line.
The slope, given by , where is the standard deviation of the predictor X and is the sum of squared deviation from mean for the criterion variable was found to be 0.1024.
- Calculate the intercept of the regression line.
The intercept a given by was found to be 1.632 where is the mean of the criterion variable and is the mean of predictor.
- Insert the calculated values for the slope and intercept into the equation for the regression line.
The regression line was thus obtained as:
- Determine the predicted GPA for students who study:
- 4
Putting X as 4 in the equation specified in part c, the predicted GPA was found to be 2.042.
- 8
Putting X as 8 in the equation specified in part c, the predicted GPA was found to be 2.4519.
Putting X as 12 in the equation specified in part c, the predicted GPA was found to be 2.862.
- 16
Putting X as 16 in the equation specified in part c, the predicted GPA was found to be 3.272
- 20
Putting X as 20 in the equation specified in part c, the predicted GPA was found to be 3.681.
- Determine the sum of squares for the error in predictions (standard error of the estimate).
The sum of squares for error in predictions was found to be 7.282
- Determine the regression sum of squares.
The regression sum of squares was found to be 9.59
The relationship between hours worked and GPA:
- What is the predictor variable?
The predictor variable is the variable named “work” denoting the number of hours worked by the student.
- What is the criterion variable?
The criterion variable is the variable named “GPA” denoting the GPA of the student
- Correlation: Use the steps of hypothesis testing to evaluate this hypothesis.
- State the hypotheses.
The hypothesis states that the GPA scores of the student are affected by the hours worked by the student. The null and alternate hypothesis can be therefore written as:
The Relationship between Hours Worked and Hours Studied:
H0 : ρ=0 against H1: ρ ≠0
- Determine the critical value of your statistic for rejecting the null based on the given alpha level/sample size.
The critical value is given by the inverse of t-statistic with n-2 or 27 degrees of freedom and alpha value equal to 0.05. The critical value was determined to be 2.0518.
- Calculate the statistic.
The statistic of interest to test the hypothesis is given by. The Pearson’s correlation coefficient, r, given as, was computed as -0.3426. The data has 29 sample points. Thus the statistic was computed to have the value 1.8949.
- Based on this statistic alone, would you reject or accept the null hypothesis?
The critical value was found to be greater than the value of the statistic. Therefore the test fails to reject the null hypothesis at 5% level of significance.
- Calculate the effect. What does this mean?
The value of is 0.882621, which is the effect that the predictor variable has on the criterion variable. It denotes the proportion of variation in criterion variable as explained by the predictor variable.
- Regression: Address the following items
- Calculate the slope of the regression line.
The slope, given by , where is the standard deviation of the predictor X and is the standard deviation of the criterion variable was found to -0.026.
- Calculate the intercept of the regression line.
The intercept a given by was found to be 3.3269 where is the mean of the criterion variable and is the mean of predictor.
- Insert the calculated values for the slope and intercept into the equation for the regression line.
- Determine the predicted GPA for students who work:
- 28
Putting X as 28 in the equation specified in part c, the predicted GPA was found to be 2.598
- 24
Putting X as 24 in the equation specified in part c, the predicted GPA was found to be 2.703.
Putting X as 16 in the equation specified in part c, the predicted GPA was found to be 2.9108.
- 12
Putting X as 12 in the equation specified in part c, the predicted GPA was found to be 3.0148.
- 8
Putting X as 8 in the equation specified in part c, the predicted GPA was found to be 3.1188.
- Determine the sum of squares for the error in predictions (standard error of the estimate).
The sum of squares error was computed as 14.892.
- Determine the regression sum of squares.
The regression sum of squares was found to be 1.9805.
The relationship between hours worked and hours studied:
- What is the predictor variable?
The predictor variable is the variable named “work” denoting the number of hours worked by the student.
- What is the criterion variable?
The criterion variable is the variable named “study” denoting number of hours studied by the student
- Correlation: Use the steps of hypothesis testing to evaluate this hypothesis.
- State the hypotheses.
The hypothesis states that the hours studied by the student are affected by the hours worked by the student. The null and alternate hypothesis can be therefore written as:
H0 : ρ=0 against H1: ρ ≠0
- Determine the critical value of your statistic for rejecting the null based on the given alpha level/sample size.
The critical value is given by the inverse of t-statistic with n-2 or 27 degrees of freedom and alpha value equal to 0.05. The critical value was determined to be 2.0518.
- Calculate the statistic.
The Relationship between Hours Worked and Hours Studied: High Conscientiousness
The statistic of interest to test the hypothesis is given by . The Pearson’s correlation coefficient, r, given by , was computed as –0.235. The data has 29 sample points. Thus the statistic was computed to have the value 1.2563.
- Based on this statistic alone, would you reject or accept the null hypothesis?
The critical value was found to be greater than the value of the statistic. Therefore the test fails to reject the null hypothesis at 5% level of significance. The null is therefore accepted.
- Calculate the effect. What does this mean?
The value of is 0.94476, which is the effect that the predictor variable has on the criterion variable. It denotes the proportion of variation in criterion variable as explained by the predictor variable.
The relationship between hours worked and hours studied for students with high conscientiousness:
- What is the predictor variable?
The predictor variable is the variable named “study” denoting the number of hours studied by the students whose conscientiousness was reportedly high.
- What is the criterion variable?
The criterion variable is the variable named “work” denoting the number of hours studied by the students whose conscientiousness was reportedly high.
The Pearson’s correlation coefficient, r, given by was found to be 0.1739.
- Calculate the effect. What does this mean?
The value of is 0.94476, which is the effect that the predictor variable has on the criterion variable. It denotes the proportion of variation in criterion variable as explained by the predictor variable.
The relationship between hours worked and hours studied for students with low conscientiousness:
- What is the predictor variable?
The predictor variable is the variable named “study” denoting the number of hours studied by the students whose conscientiousness was reportedly low.
- What is the criterion variable?
The criterion variable is the variable named “work” denoting the number of hours studied by the students whose conscientiousness was reportedly low.
Calculate the statistic.
The Pearson’s correlation coefficient, r, given by was found to be -0.6220.
- Calculate the effect. What does this mean?
The value of is 0.61303, which is the effect that the predictor variable has on the criterion variable. It denotes the proportion of variation in criterion variable as explained by the predictor variable.
It can thus be seen that a student who invests more hours to studying has more GPA scores that those who invest lesser hours. However, not enough evidence was found that change in length of working hours would correspond with change in GPA scores. The data failed to provide any evidence to support the conjecture that longer working hours corresponded with shorter lengths time invested to studying. Again, it was seen that students with high conscientiousness showed positive association between study hours and work hours while those with low conscientiousness depicted negative association. Hence it is concluded that amount of time invested to studying does in fact affect GPA score, however, provided that the student is conscientious, work would not negatively impact his or her studies and hence GPA scores.
