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Name of Database: R7031education.sav

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Question 1: Is there a relationship among the variables measuring different aspects of parent satisfaction?

Step 1: The null and alternate hypothesis

For this research question, there is one hypothesis for each pair of variables. Is there an association between-

1. Safety and Education

2. Safety and Social
6. Education and Social

3. Safety and Physical
7. Education and Physical
10. Social and Physical

4. Safety and Overall Satisfaction in January
8. Education and Overall Satisfaction in January
11. Social and Overall Satisfaction in January
13. Physical and Overall Satisfaction in January

5. Safety and Overall Satisfaction in June
9. Education and Overall Satisfaction in June
12. Social and Overall Satisfaction in June
14. Physical and Overall Satisfaction in June
15. Overall Satisfaction in January and Overall Satisfaction in June

The notation for the hypothesis of each correlation to be tested is:

Step 2: Level of significance, ?

The level of significance, ? selected is .05.

Step 3: The data collection

In January 08, One hundred elementary school parents of a county school district were surveyed regarding their satisfaction with a free after-school daycare program. The parents filled out the survey on after the end of the fall semester (in January 08). In June 08, the parents were telephoned and re-surveyed and were asked to rate their overall satisfaction again (database R7031education.sav).

Step 4: The test statistic and p-value

Checking first, the assumptions have been met for correlation.

Normal Distribution: Examining the measures of central tendency (see the cells in yellow) and variability, it can be seen that none of the variables have violated this assumption: the means, medians, and modes are “close,” and the indicators of skewness and kurtosis (cells in blue) are well within the normal range (close to 0.00), see Table 1.

Table 1

Descriptive Statistics

Safety of Children
Educational Activities
Social Activities
Physical Activities
Overall Satisfaction in January
Overall Satisfaction in June
N
Valid
50
50
50
50
50
50

Missing
0
0
0
0
0
0
Mean
4.340
5.340
2.880
2.660
3.5800
4.700
Std. Error of Mean
.2367
.1632
.1266
.1199
.19397
.1348
Median
4.000
5.000
3.000
3.000
3.5000
5.000
Mode
4.0
5.0
3.0
3.0
3.00
4.0
Std. Deviation
1.6734
1.1537
.8953
.8478
1.37158
.9530
Skewness
-.376
-.048
-.112
.101
.320
.206
Std. Error of Skewness
.337
.337
.337
.337
.337
.337
Kurtosis
-.287
.321
-.221
.372
.217
-.546
Std. Error of Kurtosis
.662
.662
.662
.662
.662
.662
Minimum
1.0
3.0
1.0
1.0
1.00
3.0
Maximum
7.0
8.0
5.0
5.0
7.00
7.0

Next, look at the scatterplots to see if there is enough of a linear relationship.

The relationships between Safety of Children and Overall Satisfaction in June looks like a linear, strong positive correlation.

The relationships between Educational Activities and Overall Satisfaction in June looks like a linear, moderately strong positive correlation.

Table 2

Correlation Matrix

Safety of Children
Educational Activities
Social Activities
Physical Activities
Overall Satisfaction in January
Overall Satisfaction in June
Safety of Children
Pearson Correlation
1
.193
.436(**)
.112
.206
.628(**)

Sig. (2-tailed)

.180
.002
.439
.152
.000

N
50
50
50
50
50
50
Educational Activities
Pearson Correlation
.193
1
.317(*)
.726(**)
.195
.484(**)

Sig. (2-tailed)
.180

.025
.000
.174
.000

N
50
50
50
50
50
50
Social Activities
Pearson Correlation
.436(**)
.317(*)
1
.133
.340(*)
.555(**)

Sig. (2-tailed)
.002
.025

.356
.016
.000

N
50
50
50
50
50
50
Physical Activities
Pearson Correlation
.112
.726(**)
.133
1
.173
.326(*)

Sig. (2-tailed)
.439
.000
.356

.229
.021

N
50
50
50
50
50
50
Overall Satisfaction in January
Pearson Correlation
.206
.195
.340(*)
.173
1
.479(**)

Sig. (2-tailed)
.152
.174
.016
.229

.000

N
50
50
50
50
50
50
Overall Satisfaction in June
Pearson Correlation
.628(**)
.484(**)
.555(**)
.326(*)
.479(**)
1

Sig. (2-tailed)
.000
.000
.000
.021
.000

N
50
50
50
50
50
50
**  Correlation is significant at the 0.01 level (2-tailed).

*  Correlation is significant at the 0.05 level (2-tailed).

Step 5: Retain or reject the null hypothesis.

The strongest positive correlation is between Educational Activities and Physical Activities, r =.726, p <.001.

The weakest positive correlation is between Safety of Children and Physical Activities, r = .112, p =.439.

The null hypothesis will be rejected for highlighted (yellow) pair of observations as shown in Table 2.

Step 6: The Risk of Type I and Type II Error assessment

Type I Error

The correlation squared ( ) is a measure of effect size. The correlation between Educational Activities and Physical Activities, r =.726, p <.001, is significant, and has a strong effect size (  = .527).

Type II Error

The biggest risk in associational research is the risk of unreliable measures and the consequences unreliability has on the strength and direction of the correlation coefficient. Survey research is notoriously at risk for Type II error because of unreliability. The sample size equal to 50 is less than the general rule of thumb for correlation studies that is at least 100 cases to make sure we have sufficient power and sample satisfy assumptions of normality and linearity.

Step 7: APA format result

The null hypothesis for the correlations is rejected for the statistically significant pairs that are:

·         Safety of Children and Social Activities

·         Safety of Children and Overall Satisfaction in June

·         Educational Activities and Social Activities

·         Educational Activities and Physical Activities

·         Educational Activities and Overall Satisfaction in June

·         Social Activities and Overall Satisfaction in January

·         Social Activities and Overall Satisfaction in June

·         Physical Activities and Overall Satisfaction in June

·         Overall Satisfaction in January and Overall Satisfaction in June

The strongest positive correlation is between Educational Activities and Physical Activities, r =.726, p <.001.

The weakest positive correlation is between Safety of Children and Physical Activities, r = .112, p =.439.

Question 2: Does Safety Impact Overall Satisfaction in June?

Step 1: The null and alternate hypothesis

There are two research questions and two hypotheses.

Does Safety predict Overall Satisfaction in June?

How good a predictor is safety?

Step 2: Level of significance, ?

The level of significance, ? selected is .05.

Step 3: The data collection

In January 08, One hundred elementary school parents of a county school district were surveyed regarding their satisfaction with a free after-school daycare program. The parents filled out the survey on after the end of the fall semester (in January 08). In June 08, the parents were telephoned and re-surveyed and were asked to rate their overall satisfaction again (database R7031education.sav).

Step 4: The test statistic and p-value

Checking first, the assumptions have been met.

Normal Distribution: Examining the measures of central tendency (see the cells in yellow) and variability, it can be seen that none of the variables have violated this assumption: the means, medians, and modes are “close,” and the indicators of skewness and kurtosis (cells in blue) are well within the normal range (close to 0.00), see Table 3.

Table 3

Descriptive Statistics

Safety of Children
Overall Satisfaction in June
N
Valid
50
50
Missing0
0
Mean
4.340
4.700
Median
4.000
5.000
Mode
4.0
4.0
Std. Deviation
1.6734
.9530
Skewness
-.376
.206
Std. Error of Skewness
.337
.337
Kurtosis
-.287
-.546
Std. Error of Kurtosis
.662
.662

Linearity: look at the scatterplot to see if there is enough of a linear relationship.

The relationships between Safety of Children and Overall Satisfaction in June looks like a linear, strong positive correlation.

Homoscedasticity (the Distribution of Residuals): Examining the plot of the residuals as compared to the predicted values, the lack of a pattern (i.e., the dots are scattered randomly) supports the assumption of homoscedasticity (equal variance of error across values of the independent variables).

= .395, or 39.5% of the variance is explained—60.5% is not (See Table 4).

Table 4

Model Summary

Model
R
R Square
Std. Error of the Estimate
1
.628(a)
.395
.382
.7490
a  Predictors: (Constant), Safety of Children

The F-test (the ratio of regression to residual variation) is F(1, 48) = 31.314, p < .001. The model is statistically significant (See Table 5).

Table 5

ANOVA(b)

Model

Sum of Squares
df
Mean Square
F
Sig.
1
Regression
17.569
1
17.569
31.314
.000(a)
Residual26.931
48
.561

Total44.500
49

a  Predictors: (Constant), Safety of Children

b  Dependent Variable: Overall Satisfaction in June

Last, examining the regression coefficient. In the case of one predictor, ? (Beta) is the same as the correlation coefficient, 0.628 (See Table 6). Therefore, this indicates a strong positive impact, that is statistically significant, at t = 5.596, p < .001.

Table 6

Coefficients(a)

Model

Unstandardized Coefficients
Standardized Coefficients
t
Sig.
BStd. Error
Beta

1
(Constant)

3.147

.297

10.595

.000

Safety of Children.358
.064
.628
5.596
.000
a  Dependent Variable: Overall Satisfaction in June

The formula for the best fitting line is

Overall Satisfaction in June = 3.147 + 0.358(Safety of Children)

Step 5: Retain or reject the null hypothesis.

There are two research questions and two hypotheses.

Does Safety predict Overall Satisfaction in June?

= .395, and the F-test (the ratio of regression to residual variation is F(1, 48) = 31.314, p < .001. The model is statistically significant. Therefore, reject the null hypothesis.

How good a predictor is safety?

The standardized coefficient ? = .628, indicate a strong positive impact, t = 5.596, p < .001. Therefore, reject the null hypothesis.

Step 6: The Risk of Type I and Type II Error assessment

Type I Error

is a measure of effect size. = .395 indicates only 39.5% of the variance is explained, 60.5% is unexplained. While it is statistically significant, this should be interpreted cautiously.  is statistically significant, and has a moderately strong effect size. In addition, all of the assumptions have been met, suggesting that the risk of Type I error has been minimized.

Survey research is notoriously at risk for Type II error because of unreliability. The sample size equal to 50 is less than the general rule of thumb for correlation studies that is at least 100 cases to make sure we have sufficient power and sample satisfy assumptions of normality and linearity.

Step 7: APA format result

Safety is a strong predictor of Overall Satisfaction in June, and the model is statistically significant, = .395, F(1, 48) = 31.314, p < .001. However, 60.5% of the variance remains unexplained. The standardized coefficient ? = .628, indicates a strong positive impact, t = 5.596, p < .001.

Question 3: Does Safety and Education Impact Overall Satisfaction in June?

Step 1: The null and alternate hypothesis

There are two research questions and two hypotheses.

Does the combination of Safety and Education predict Overall Satisfaction in June?

Which independent variables are the best predictors?

Step 2: Level of significance, ?

The level of significance, ? selected is .05.

Step 3: The data collection

In January 08, One hundred elementary school parents of a county school district were surveyed regarding their satisfaction with a free after-school daycare program. The parents filled out the survey on after the end of the fall semester (in January 08). In June 08, the parents were telephoned and re-surveyed and were asked to rate their overall satisfaction again (database R7031education.sav).

Step 4: The test statistic and p-value

Checking first, the assumptions have been met.

Normal Distribution: Examining the measures of central tendency (see the cells in yellow) and variability, it can be seen that none of the variables have violated this assumption: the means, medians, and modes are “close,” and the indicators of skewness and kurtosis (cells in blue) are well within the normal range (close to 0.00), see Table 7.

Table 7

Descriptive Statistics

Safety of Children
Educational Activities
Overall Satisfaction in June
N
Valid
50
50
50

Missing
0
0
0
Mean
4.340
5.340
4.700
Std. Error of Mean
.2367
.1632
.1348
Median
4.000
5.000
5.000
Mode
4.0
5.0
4.0
Std. Deviation
1.6734
1.1537
.9530
Skewness
-.376
-.048
.206
Std. Error of Skewness
.337
.337
.337
Kurtosis
-.287
.321
-.546
Std. Error of Kurtosis
.662
.662
.662
Minimum
1.0
3.0
3.0
Maximum
7.0
8.0
7.0

Linearity: looking at the scatterplots to see if there is enough of a linear relationship.

The relationships between Safety of Children and Overall Satisfaction in June looks like a linear, strong positive correlation.

The relationships between Educational Activities and Overall Satisfaction in June looks like a linear, moderately strong positive correlation.

Homoscedasticity (the Distribution of Residuals): Examining the plot of the residuals as compared to the predicted values, the lack of a pattern (i.e., the dots are scattered randomly) supports the assumption of homoscedasticity (equal variance of error across values of the independent variables).

Normality of the Residual: The straight line of the P-P plot is the benchmark. This is how the data would appear if they were “multivariate normal.” Multivariate refers to the fact that there are more than one variable and cannot just look at “the distribution” to see if it is normally distributed. The pattern of dots closely fit the line. Some of the dots are only slightly off the line. Therefore, the assumption of normality is met.

Table 8

Correlation Matrix

Safety of Children
Educational Activities
Overall Satisfaction in June
Safety of Children
Pearson Correlation
1
.193
.628
Sig. (2-tailed)
.180
.000
N50
50
50
Educational Activities
Pearson Correlation
.193
1
.484
Sig. (2-tailed).180

.000
N50
50
50
Overall Satisfaction in June
Pearson Correlation
.628
.484
1
Sig. (2-tailed).000
.000

N50
50
50

Multicolinearity: It refers to the correlation among the IV or predictor variables. Examining the correlation matrix it can be seen that the two independent variables (Safety of Children and Educational Activities) have a weak positive correlation, r = .193, p =.0180 (see Table 8). Concerns for multicolinearity should arise only when correlations are >.70 (+/-). Therefore, this assumption is also met as well.

= .729, or 72.9% of the variance is explained—27.1% is not (See Table 9).

Table 9

Model Summary(b)

Model
R
R Square
Std. Error of the Estimate
1
.729(a)
.532
.512
.6657
a  Predictors: (Constant), Educational Activities, Safety of Children

b  Dependent Variable: Overall Satisfaction in June

The F-test (the ratio of regression to residual variation) is F(2, 47) = 26.713, p < .001. The model is statistically significant (See Table 10).

Table 10

ANOVA(b)

Model

Sum of Squares
df
Mean Square
F
Sig.
1
Regression
23.674
2
11.837
26.713
.000(a)
Residual20.826
47
.443

Total44.500
49

a  Predictors: (Constant), Educational Activities, Safety of Children

b  Dependent Variable: Overall Satisfaction in June

Safety of Children indicates a strong positive impact, that is statistically significant, at ? = .556, t = 5.464, p < .001.

Educational Activities indicates a strong positive impact, that is statistically significant, at ? = .377, t = 3.712, p = .001.

Table 11

Coefficients(a)

Model

Unstandardized Coefficients
Standardized Coefficients
t
Sig.
BStd. Error
Beta

1
(Constant)

1.662

.479

3.467

.001

Safety of Children.316
.058
.556
5.464
.000
Educational Activities.312
.084
.377
3.712
.001
a  Dependent Variable: Overall Satisfaction in June

The formula for the best fitting line is

Overall Satisfaction in June = 1.662 + 0.316(Safety) + 0.312(Education)

Step 5: Retain or reject the null hypothesis.

Does the combination of Safety and Education predict Overall Satisfaction in June?

= .729, and the F-test (the ratio of regression to residual variation is F(2, 47) = 26.713, p < .001. The model is statistically significant. Therefore, reject the null hypothesis.

Which independent variables are the best predictors?

The standardized coefficient ? = .556 for Safety of children, indicate a strong positive impact, t = 5.464, p < .001. Therefore, reject the null hypothesis.

The standardized coefficient ? = .377 for Educational Activities, indicate a strong positive impact, t = 3.712, p = .001. Therefore, reject the null hypothesis.

Both predictors Safety of Children and Educational Activities significantly predict Overall Satisfaction in June. Since the value of Beta for Safety of Children is higher, this indicates that Safety of Children is slightly more important.

Step 6: The Risk of Type I and Type II Error assessment

Type I Error

is a measure of effect size. = .729 indicates 72.9% of the variance is explained (strong effect size), and only 27.1% is unexplained.  is statistically significant, and has a strong effect size. In addition, all of the assumptions have been met, suggesting that the risk of Type I error has been minimized.

Survey research is notoriously at risk for Type II error because of unreliability. The sample size equal to 50 is less than the general rule of thumb for correlation studies that is at least 100 cases to make sure we have sufficient power and sample satisfy assumptions of normality and linearity.

Step 7: APA format result

The results of this study indicate a statistically significant model. Both Safety of Children and Education Activities significantly predict Overall Satisfaction in June.

Safety and Education are strong predictor of Overall Satisfaction in June, and the model is statistically significant, = .729, F(2, 47) = 26.713, p < .001. However, 27.1% of the variance remains unexplained. The standardized coefficient ? = .556 for Safety of Children , indicates a strong positive impact, t = 5.464, p < .001. The standardized coefficient ? = .377 for Educational Activities, indicates a strong positive impact, t = 3712, p = .001.

CONCLUSION

Overall Satisfaction in June has significant association with Safety of Children, Educational Activities, Social Activities, Physical Activities and Overall Satisfaction in January. One important thing is that Overall Satisfaction in January has not significant association with Safety of Children, Educational Activities, Social Activities, and Physical Activities, and this is the area they need to improve.

Using only Safety of Children, 39.5% of the variation in Overall Satisfaction in June is explained. Using variables Safety of Children and Educational Activities, 72.9% of the variation in Overall Satisfaction in June is explained. Therefore, adding a second variable to the regression equation increases prediction of parent satisfaction in June.

Reference:

Methods and Analysis of Quantitative Research (January, 2009).  Argosy University

Online Course. E7031. Instructor: Michael Marrapodi.  Retrieved March 11, 2009, from http://www.myeclassonline.com/ec/crs/default.learn?CourseID=3282843&CPURL=www.myeclassonline.com&Survey=1&47=4157986&ClientNodeID=404511&coursenav=0&bhcp=1