Question

A psychologist wanted to assess the relationship between depression and self-esteem. She believes that as depression increases, self-esteem decreases. She asked a group of 25 participants to complete both Beck’s Depression Inventory (Beck, Ward, Mendelson, Mock, & Erbaugh, 1961) and Rosenberg’s Self-Esteem Scale (1965). The data for each participant is as follows:

Depression (X)	Self-Esteem (Y)
60	11
27	34
15	33
30	24
56	14
52	16
40	23
49	33
24	27
38	26
48	17
58	10
22	39
51	18
41	22
56	17
49	16
54	13
52	12
41	23
35	28
20	35
55	15
16	32
28	20

Descriptive Statistics

For Depression:

1. What is the scale of measurement of the variable?
2. Is the distribution normally distributed, positively skewed, or negatively skewed?
3. Based on your answers two the first two questions, what would be the best statistics to use to describe this variable (central tendency and variability)?
4. Why would these be the correct/best choice?
5. What are the values of these statistics (the appropriate statistics ONLY – not all)?

For Self-Esteem:

6. What is the scale of measurement of the variable?
7. Is the distribution normally distributed, positively skewed, or negatively skewed?
8. Based on your answers two the first two questions, what would be the best statistics to use to describe this variable (central tendency and variability)?
9. Why would these be the correct/best choice?
10. What are the values of these statistics (the appropriate statistics ONLY – not all)?

Correlation

1. What is the predictor variable?
2. What is the criterion variable?
3. Correlation: Use the steps of hypothesis testing to evaluate this hypothesis.
   1. State the hypotheses.
   2. Determine the critical value of your statistic for rejecting the null based on the given alpha level/sample size.
   3. What is the value of the statistic.
   4. Based on this statistic alone, would you reject or accept the null hypothesis?
4. What is the effect size? What does this mean?

Regression

5. What is the slope of the regression line?
6. What is the intercept of the regression line.
7. Insert the calculated values for the slope and intercept into the regression line equation and enter that here.
8. Determine the predicted self-esteem scores for participants who have the following levels of depression:
   1. 28: Y’ =
   2. 42: Y’ =
   3. 54: Y’ =

Answer 1

For dipression

i) The scale of measurement of the variable: Quantitative data (numerical values are provided).

ii) The data is normally distributed. We plot normal probability in Minitab.

iii) Since we have numerical measurements we can calculate mean and variance (standard deviation).

iv) Since the data is numerical and normally distributed, the mean and variance are the correct measures to calculate.

v) Mean=sum of all observations/total no of observation=1017/25=40.68

Variance=(sum of all squared observations/total no of observation)-(mean)^2=196.2176

For Self-esteem

i) The scale of measurement of the variable: Quantitative data (numerical values are provided).

ii) The data is normally distributed. We plot normal probability in Minitab.

iii) Since we have numerical measurements we can calculate mean and variance (standard deviation).

iv) Since the data is numerical and normally distributed, the mean and variance are the correct measures to calculate.

v) Mean=sum of all observations/total no of observation=558/25=22.32

Variance=(sum of all squared observations/total no of observation)-(mean)^2=69.0176

Correlation

i) Predictor variable: Self-esteem

ii) Criterion variable: Dipression

	X^2	Y^2	XY
	3600	121	660
	729	1156	918
	225	1089	495
	900	576	720
	3136	196	784
	2704	256	832
	1600	529	920
	2401	1089	1617
	576	729	648
	1444	676	988
	2304	289	816
	3364	100	580
	484	1521	858
	2601	324	918
	1681	484	902
	3136	289	952
	2401	256	784
	2916	169	702
	2704	144	624
	1681	529	943
	1225	784	980
	400	1225	700
	3025	225	825
	256	1024	512
	784	400	560
Total	46277	14180	20238

	X	Y
Mean	40.68	22.32
Varince	196.2176	69.0176
Covariance	-98.4576
Correlation	-0.84606

Null hypothesis H₀: ρ = 0
Alternative hypothesis H_A: ρ ≠ 0 or H_A: ρ < 0 or H_A: ρ > 0

Test statistic: $t=r\sqrt{n-1}/\sqrt{(1-r^2))}$ , where r is correlation and n is total no of observation

t=-14.5849

|t|=14.5849

p-value=0.0000

Since the p-value< 5%.

We may reject H0.

Regression

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.846058
R Square	0.715815
Adjusted R Square	0.703459
Standard Error	4.617286
Observations	25
ANOVA
	df	SS	MS	F	Significance F
Regression	1	1235.096	1235.096	57.93314	9.97E-08
Residual	23	490.3445	21.31933
Total	24	1725.44
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	42.73231	2.836353	15.06593	2.09E-13	36.86487	48.59976	36.86487	48.59976
X Variable 1	-0.50178	0.065925	-7.61138	9.97E-08	-0.63815	-0.3654	-0.63815	-0.3654

Here intercept: b0=42.73231

slope: b1= -0.50178

a) x=28: Y=28.68254

b) x=42: Y=21.6576

c) x=54: Y=15.6363