The following tables are generated from SPSS. The first table contains the multiple R and the R^{2} values. The multiple R is 0.638, indicating that the correlation between the actual y values and the predicted y values using the new regression equation is 0.638. The R^{2} is 0.407, indicating that 40.7% of the variance in months to program completion can be explained by knowing the student’s number of earned academic degrees at enrollment.

The second table contains the ANOVA table. As presented in Exercises 18 and 33, the ANOVA is usually performed to test for differences between group means. However, ANOVA can also be performed for regression, where the null hypothesis is that “knowing the value of x explains no information about y”. This table indicates that knowing the value of x explains a significant amount of variance in y. The contents of the ANOVA table are rarely reported in published manuscripts, because the significance of each predictor is presented in the last SPSS table titled “Coefficients” (see below).

The third table contains the b and a values, standardized beta (β), t, and exact p value. The a is listed in the first row, next to the label “Constant.” The β is listed in the second row, next to the name of the predictor. The remaining information that is important to extract when interpreting regression results can be found in the second row. The standardized beta (β) is −0.638. This value has limits just like a Pearson r, meaning that the standardized β cannot be lower than −1.00 or higher than 1.00. The t value is −3.516, and the exact p value is 0.002.

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The following interpretation is written as it might appear in a research article, formatted according to APA guidelines (APA, 2010). Simple linear regression was performed with number of earned academic degrees as the predictor and months to program completion as the dependent variable. The student’s number of degrees significantly predicted months to completion among students in an RN to BSN program, β = −0.638, p = 0.002, and R^{2} = 40.7%. Higher numbers of earned academic degrees significantly predicted shorter program completion time.

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1. If you have access to SPSS, compute the Shapiro-Wilk test of normality for months to completion (as demonstrated in Exercise 26). If you do not have access to SPSS, plot the frequency distributions by hand. What do the results indicate?

2. State the null hypothesis for the example where number of degrees was used to predict time to BSN program completion.

3. In the formula y = bx + a, what does “b” represent?

4. In the formula y = bx + a, what does “a” represent?

5. Using the new regression equation, ŷ = −2.9x + 16.25, compute the predicted months to program completion if a student’s number of earned degrees is 0. Show your calculations.

6. Using the new regression equation, ŷ = −2.9x + 16.25, compute the predicted months to program completion if a student’s number of earned degrees is 2. Show your calculations.

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7. What was the correlation between the actual y values and the predicted y values using the new regression equation in the example?

8. What was the exact likelihood of obtaining a t value at least as extreme as or as close to the one that was actually observed, assuming that the null hypothesis is true?

9. How much variance in months to completion is explained by knowing the student’s number of earned degrees?

10. How would you characterize the magnitude of the R^{2} in the example? Provide a rationale for your answer.

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1. The Shapiro-Wilk p value for months to RN to BSN program completion was 0.16, indicating that the frequency distribution did not significantly deviate from normality. Moreover, visual inspection of the frequency distribution indicates that months to completion is approximately normally distributed. See SPSS output below for the histograms of the distribution:

2. The null hypothesis is: “The number of earned academic degrees does not predict the number of months until completion of an RN to BSN program.”

3. In the formula y = bx + a, “b” represents the slope of the regression line.

4. In the formula y = bx + a, “a” represents the y-intercept, or the point at which the regression line intersects the y-axis.

5. The predicted months to program completion if a student’s number of academic degrees is 0 is calculated as: ŷ = −2.9(0) + 16.25 = 16.25 months.

6. The predicted months to program completion if a student’s number of academic degrees is 2 is calculated as: ŷ = −2.9(2) + 16.25 = 10.45 months.

7. The correlation between the actual y values and the predicted y values using the new regression equation in the example, also known as the multiple R, is 0.638.

8. The exact likelihood of obtaining a t value at least as extreme as or as close to the one that was actually observed, assuming that the null hypothesis is true, was 0.2%. This value was obtained by looking at the SPSS output table titled “Coefficients” in the last value of the column labeled “Sig.”

9. 40.7% of the variance in months to completion is explained by knowing the student’s number of earned academic degrees at enrollment.

10. The magnitude of the R^{2} in this example, 0.407, would be considered a large effect according to the effect size tables in Exercises 24 and 25.

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Using the example from Mancini and colleagues (2014), students enrolled in an RN to BSN program were assessed for demographics at enrollment. The predictor in this example is age at program enrollment, and the dependent variable was number of months it took for the student to complete the RN to BSN program. The null hypothesis is: “Student age at enrollment does not predict the number of months until completion of an RN to BSN program.” The data are presented in Table 29-2. A simulated subset of 20 students was randomly selected for this example so that the computations would be small and manageable.

TABLE 29-2

AGE AT ENROLLMENT AND MONTHS TO COMPLETION IN AN RN TO BSN PROGRAM

Student ID | x | y | x^{2} |
xy |

(Student Age) | (Months to Completion) | |||

1 | 23 | 17 | 529 | 391 |

2 | 24 | 9 | 576 | 216 |

3 | 24 | 17 | 576 | 408 |

4 | 26 | 9 | 676 | 234 |

5 | 31 | 16 | 961 | 496 |

6 | 31 | 11 | 961 | 341 |

7 | 32 | 15 | 1,024 | 480 |

8 | 33 | 12 | 1,089 | 396 |

9 | 33 | 15 | 1,089 | 495 |

10 | 34 | 12 | 1,156 | 408 |

11 | 34 | 14 | 1,156 | 476 |

12 | 35 | 10 | 1,225 | 350 |

13 | 35 | 17 | 1,225 | 595 |

14 | 39 | 20 | 1,521 | 780 |

15 | 40 | 9 | 1,600 | 360 |

16 | 42 | 12 | 1,764 | 504 |

17 | 42 | 14 | 1,764 | 588 |

18 | 44 | 10 | 1,936 | 440 |

19 | 51 | 17 | 2,601 | 867 |

20 | 24 | 11 | 576 | 264 |

sum Σ | 677 | 267 | 24,005 | 9,089 |

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