In: Statistics and Probability
4. (22 pts) A psychologist conducts a study to determine the relationship between religion and self-esteem. Ten eighth graders are randomly selected for the study. Each individual undergoes two tests, one measuring self-esteem and the other religious involvement. For the self-esteem test, a higher score indicates higher self-esteem; for the test measuring religious involvement, a higher the score indicates higher religious involvement. The self-esteem test has a range from 1 to 10 and the religious involvement test ranges from 0 to 50. Or the purposes of this question, assume both tests are well standardized and of interval scaling. The following data are collected (see numbers to the left): |
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Answer the following:
A. (2 pts) If a relationship exists such that the more religiously involved one is, the higher actual self-esteem is, would you expect r computed on the provided values to be negative or positive?
B. (4 pts) Using SPSS, JASP, or the online calculator, compute the Pearson correlation coefficient for these data. Report the value of r. What does the direction of the effect (the positive or negative sign of r) tell you about whether higher self-esteem is associated with higher religious involvement or lower religious involvement? (Choose one.)
C. (2 pts) In SPSS, use religious involvement (the IV) to predict self-esteem (the DV). What proportion of variability in self-esteem is explained by variability in religious involvement?
D. (4 pts) Write the equation to predict self-esteem from religious involvement. Make the equation as detailed as you can (variable names, number for the coefficients).
E. (2 pts) Use the equation to predict the self-esteem of a person who had a religious involvement score of 27. Show your work.
F. (2 pts) For the Pearson correlation between religious involvement and self-esteem, state the null and alternative hypotheses.
G. (2 pts) Using Table E, report the degrees of freedom and the critical value for the Pearson r at an alpha of .05 (two-tailed).
H. (2 pts) What is your decision about the null hypothesis for the Pearson correlation (reject or fail to reject)? Also state the basis for your decision.
I. (2 pts) In psychology, a Pearson correlation of -.388 is usually considered a fairly substantial effect. If you failed to reject the null with this large a correlation, what could a possible explanation be? Write a sentence or two answering this question.
ΣX | ΣY | Σ(x-x̅)² | Σ(y-ȳ)² | Σ(x-x̅)(y-ȳ) | |
total sum | 230 | 48 | 1860 | 19.6 | -74.00 |
mean | 23.00 | 4.80 | SSxx | SSyy | SSxy |
correlation coefficient , r = Sxy/√(Sx.Sy) = -0.3876
negative
...............
sample size , n = 10
here, x̅ = Σx / n= 23.00 ,
ȳ = Σy/n = 4.80
SSxx = Σ(x-x̅)² = 1860.0000
SSxy= Σ(x-x̅)(y-ȳ) = -74.0
estimated slope , ß1 = SSxy/SSxx = -74.0
/ 1860.000 = -0.0398
intercept, ß0 = y̅-ß1* x̄ =
5.7151
so, regression line is Ŷ =
5.715 + -0.040 *x
...................
R² = (Sxy)²/(Sx.Sy) = 0.1502
15.02% of variation is explained by x
...............
Predicted Y at X= 27 is
Ŷ = 5.71505 +
-0.039785 * 27 =
4.64
...............
correlation hypothesis test
Ho: ρ = 0
Ha: ρ ╪ 0
n= 10
correlation , r= -0.3876
t-test statistic = r*√(n-2)/√(1-r²) =
-1.189
DF=n-2 = 8
p-value = 0.2685
you did not provide alpha (for 0.05 , 0.01 , 0.1)
Decison: P value > α, So, Do not reject
Ho
..................
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