(1) According to the leadership literature, there are a ...



Notes 9 ANOVA Homework Answers

(1) According to the leadership literature, there are a number of different leadership styles. Listed below are scores obtained from an instrument designed to measure a particular leadership style, which will be referred to as style X. Of interest is whether X differs by school district type in terms of urbanity. A stratified random sample of school principals were selected from three district types (mostly urban, mostly suburban, and mostly rural). The scores on style X range from 100 to 0. The closely the score to 100, the more the respondent conforms to style X, while the closer the score to 0, the less the respondent conforms to style X.

Is there any evidence that X differs among the three district types?

SPSS Results

Descriptive Statistics

Dependent Variable: style_x

|district |Mean |Std. Deviation |N |

|rural |41.1429 |28.86999 |7 |

|suburban |52.6667 |14.19390 |6 |

|urban |74.7143 |17.67026 |7 |

|Total |56.3500 |25.07100 |20 |

Tests of Between-Subjects Effects

Dependent Variable: style_x

|Source |Type III Sum of |df |Mean Square |F |Sig. |

| |Squares | | | | |

|Corrected Model |4060.931(a) |2 |2030.465 |4.380 |.029 |

|Intercept |62779.554 |1 |62779.554 |135.410 |.000 |

|district |4060.931 |2 |2030.465 |4.380 |.029 |

|Error |7881.619 |17 |463.625 | | |

|Total |75449.000 |20 | | | |

|Corrected Total |11942.550 |19 | | | |

a R Squared = .340 (Adjusted R Squared = .262)

Pairwise Comparisons

Dependent Variable: style_x

|(I) district |(J) district |Mean Difference |Std. Error |Sig.(a) |95% Confidence Interval for |

| | |(I-J) | | |Difference(a) |

| | | | |

|Rural |41.14 |28.87 |7 |

|Suburban |52.67 |14.19 |6 |

|Urban |74.71 |17.67 |7 |

|Source |SS |df |MS |F |

|District Type |4060.93 |2 |2030.47 |4.38* |

|Error |7881.62 |17 |463.63 | |

Note: R2 = .34, adj. R2 = .26.

* p < .05

Table 2

Comparisons of Mean Differences in Leadership Style by District Type

|Comparison |Estimated Mean Difference |Standard Error of |Bonferroni Adjusted 95% CI |

| | |Difference | |

|Rural vs. Suburban |-11.52 |11.98 |-43.33, 20.28 |

|Rural vs. Urban |-33.57* |11.51 |-64.13, -3.01 |

|Suburban vs. Urban |-22.05 |11.98 |-53.85, 9.76 |

* p < .05, where p-values are adjusted using the Bonferroni method.

ANOVA results, displayed in Table 1, show that there are statistically significant mean differences in leadership style by district type. The general pattern of means suggest rural principals scored lowest on style X, urban principals scored highest, and suburban principals scored between rural and urban principals. Pairwise comparisons, reported in Table 2, show that the only statistically significant difference in scores occurs between rural and urban principals.

(2) Researchers were interested in learning whether posttest mean scores on a reading proficiency test differed by teacher assessed ratings of students' ability. Of interest is whether teachers are capable of actually distinguishing between students of differing abilities. The scores and ability ratings are given below.

|ABILITY |POSTTEST |ABILITY |POSTTEST |ABILITY |POSTTEST |

|H |92 |M |43 |L |51 |

|L |31 |M |89 |M |30 |

|L |65 |M |77 |L |89 |

|M |12 |H |6 |H |11 |

|H |17 |H |56 |L |32 |

SPSS Results

Descriptive Statistics

Dependent Variable: posttest

|ability |Mean |Std. Deviation |N |

|H |36.4000 |36.81440 |5 |

|L |53.6000 |24.32694 |5 |

|M |50.2000 |32.18229 |5 |

|Total |46.7333 |30.19098 |15 |

Tests of Between-Subjects Effects

Dependent Variable: posttest

|Source |Type III Sum of |df |Mean Square |F |Sig. |

| |Squares | | | | |

|Corrected Model |829.733(a) |2 |414.867 |.417 |.668 |

|Intercept |32760.067 |1 |32760.067 |32.949 |.000 |

|ability |829.733 |2 |414.867 |.417 |.668 |

|Error |11931.200 |12 |994.267 | | |

|Total |45521.000 |15 | | | |

|Corrected Total |12760.933 |14 | | | |

a R Squared = .065 (Adjusted R Squared = -.091)

APA Styled Presentation

Table 3

ANOVA Results and Descriptive Statistics for Leadership Style X by District Type

|Ability Classification |Mean |SD |n |

|High |36.40 |36.81 |5 |

|Low |53.60 |24.33 |5 |

|Middle |50.20 |32.18 |5 |

|Source |SS |df |MS |F |

|Ability |829.73 |2 |414.87 |0.42 |

|Error |11931.20 |12 |994.27 | |

Note: R2 = .07, adj. R2 = -0.09.

* p < .05

Results of the ANOVA show that posttest reading proficiency scores do not differ statistically among the three teacher-generated ability classifications. These lack of differences suggest teachers are unable to validly distinguish students by ability level.

Note that a table of multiple comparisons does not appear since the group means were not statistically different.

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