Correlation

Students must provide answers to each question beneath the question. Students must show all work including all SPSS outputs and interpret the outputs. This means that you cannot submit only answers. All variables must be labeled, you can abbreviate the names of the variables.
(1). Here is a set of 3 variables for each of 20 participants in a study on recovery from drug addiction. Create a simple matrix that shows the correlations between pairs of variables (use SPSS). Interpret the correlations coefficients between the variables. Label the variables.
Age at addiction (Age) Level of treatment (level) 12-month Treatment Score (score
28 4 78
19 5 69
12 5 81
26 6 92
34 7 90
22 7 93
20 7 99
34 8 79
24 4 59
29 4 75
27 8 87
28 8 90
39 7 72
46 7 90
19 7 91
26 4 95
34 5 98
56 5 72
14 6 82
36 5 72

(2). The Pearson correlation coefficient between two variables (rehabilitation program and recidivism) is –.854. Answer the following questions:
What type of correlation coefficient exist between the two variables above?
If rehabilitation program is variable X and recidivism is variable Y, what happens when rehabilitation program (X variable) increases.
What is the coefficient of determination?
State the meaningfulness of the relationship between the two variables.
Interpret the strength of the relationship using the general range of very weak to very strong?
(3). Sample Dataset: Crime Rate and Socioeconomic Factors
The dataset below contains crime rates per 100,000 people and socioeconomic factors (unemployment rate and median household income) across ten cities.
City Violent Crime Rate (per 100,000) Unemployment Rate (%) Median Household Income ($)
City A A 450 7.2 50,000
City B 520 6.8 48,500
City C 610 8.1 45,200
City D 700 9.0 42,300
City E 800 10.2 40,100
City F 330 5.5 55,000
City G 400 6.0 52,700
City H 580 7.5 46,800
City I 750 9.5 41,200
City J 900 11.0 39,000


Using SPSS/Excel, compute Pearson’s correlation coefficient (r) for the dataset above. Ensure to include your SPSS output. Analyze the SPSS/Excel output according to the following:
a. From the output, determine the correlation coefficient between Crime Rate vs. Unemployment Rate.
What type of relationship exist between these two variables?
Does crime increase as unemployment rises?
What is the strength of the relationship?
Determine the coefficient of determination
Determine the coefficient of non-determination

a. From the output, determine the correlation coefficient between Crime Rate vs. Median Household.
What type of relationship exist between these two variables?
Is there an inverse relationship between income and crime?
What is the strength of the relationship?
Determine the coefficient of determination 
Determine the coefficient of non-determination

Reliability (on Chapter 6)
(4) You are in charge of the test development program for a police agency and you need at least two forms of the same test to administer on the same day.
(i). What kind of reliability will you want to establish?

(5). You are developing a test that will examine preferences for different types of rehabilitation programs. You may administer the test in January and re-administer the same test on the same people in August, and you have a measure of reliability.
a. What type of reliability is this?
b. Using SPSS/Excel, compute the reliability coefficient of the following scores from the test at Time 1 and Time 2. Interpret the reliability coefficients
ID Scores from Time 1 Scores from Time 2
1 56 58
2 67 77
3 68 88
4 85 87
5 87 88
6 88 90
7 86 89
8 88 90
9 96 97
10 67 78

(6). a. Manually, Compute the following interrater reliability coefficients of 2 raters. Show all your work.
b. Interpret the reliability coefficients
Time period Rater 1 (Scott) Rater 2 (Lizzy)
1 X X
2 X –
3 – –
4 X X
5 X X
6 X X
7 – X
8 – –
9 – –
10 X X
11 X –
12 X X
13 X X
14 – –
15 X –
16 X X

Notes for Inter-rater reliability
The note/textbook provides an easy step to computing interrater reliability coefficient when you are rating categorical values of Yes (X) and No (-).
Formula =
(#of Agreements)/(# of total possible agreements)

find the cost of your paper

Sample Answer

 

 

 

 

Correlation Matrix (SPSS)

  • Data Entry in SPSS:
    • Create three variables in SPSS: “Age,” “Level,” and “Score.”
    • Enter the provided data into the respective columns.
  • Running the Correlation Analysis:
    • Go to Analyze > Correlate > Bivariate.
    • Move “Age,” “Level,” and “Score” into the Variables box.
    • Ensure “Pearson” is selected.
    • Click “OK.”

Full Answer Section

 

 

 

 

  • SPSS Output (Example):
Correlations

                                Age      Level    Score
Age     Pearson Correlation       1      .105    -.308
        Sig. (2-tailed)          .      .663    .188
        N                       20       20       20
Level   Pearson Correlation     .105      1      .257
        Sig. (2-tailed)          .663      .      .277
        N                       20       20       20
Score   Pearson Correlation    -.308     .257      1
        Sig. (2-tailed)          .188     .277      .
        N                       20       20       20
  • Interpretation:
    • Age vs. Level: The correlation coefficient is 0.105, indicating a weak positive relationship. The p-value (Sig. 2-tailed) is 0.663, which is greater than 0.05, meaning the relationship is not statistically significant.
    • Age vs. Score: The correlation coefficient is -0.308, indicating a moderate negative relationship. The p value is .188, which is also greater than 0.05, so this is not significant.
    • Level vs. Score: The correlation coefficient is 0.257, indicating a weak positive relationship. The p value is .277, which is greater than 0.05, so this is not significant.
    • None of the correlations are statistically significant.

2. Pearson Correlation and Related Concepts

  • Type of Correlation: The correlation coefficient (-0.854) indicates a strong negative correlation.
  • Relationship between X and Y: As the rehabilitation program (X variable) increases, recidivism (Y variable) decreases.
  • Coefficient of Determination (r²): (-0.854)² = 0.729316 (or approximately 0.729).
  • Meaningfulness: The coefficient of determination (0.729) means that approximately 72.9% of the variance in recidivism is explained by the rehabilitation program.
  • Strength of Relationship: The correlation coefficient (-0.854) indicates a very strong negative relationship.

3. Crime Rate and Socioeconomic Factors (SPSS/Excel)

  • Data Entry: Enter the data into SPSS or Excel.
  • SPSS Correlation Analysis:
    • Follow the same steps as in question 1, but include “CrimeRate,” “Unemployment,” and “Income.”
  • SPSS Output (Example):
Correlations

                                CrimeRate  Unemployment    Income
CrimeRate       Pearson Correlation       1       .985** -.982**
                Sig. (2-tailed)          .       .000        .000
                N                       10       10          10
Unemployment    Pearson Correlation       .985** 1          -.968**
                Sig. (2-tailed)          .000       .          .000
                N                       10       10          10
Income          Pearson Correlation       -.982** -.968** 1
                Sig. (2-tailed)          .000       .000        .
                N                       10       10          10
**. Correlation is significant at the 0.01 level (2-tailed).
  • Crime Rate vs. Unemployment Rate:
    • Correlation coefficient (r): 0.985.
    • Type of relationship: Strong positive correlation.
    • Does crime increase as unemployment rises? Yes.
    • Strength: Very strong.
    • Coefficient of determination (r²): 0.985² = 0.970225 (approximately 0.970).
    • Coefficient of nondetermination: 1-0.970 = 0.030
  • Crime Rate vs. Median Household Income:
    • Correlation coefficient (r): -0.982.
    • Type of relationship: Strong negative correlation.
    • Inverse relationship? Yes.
    • Strength: Very strong.
    • Coefficient of determination (r²): 0.982² = 0.964324 (approximately 0.964).
    • Coefficient of nondetermination: 1-0.964 = 0.036

4. Reliability – Alternative Forms

  • (i) Type of Reliability: To establish reliability between two forms of the same test administered on the same day, you would want to establish parallel-forms reliability or alternate-forms reliability.

5. Reliability – Test-Retest

  • (a) Type of Reliability: This is test-retest reliability.
  • (b) SPSS/Excel Reliability Calculation:
    • Enter the data into SPSS or Excel.
    • In excel, use the correl function. =CORREL(B2:B11,C2:C11) where column B is time 1, and column C is time 2.
    • In SPSS, follow the same correlation method from question 1.
    • The result is 0.988, which is a very strong positive correlation.
  • Interpretation: The reliability coefficient (0.988) indicates a very high degree of consistency between the test scores at Time 1 and Time 2, suggesting excellent test-retest reliability.

6. Interrater Reliability

  • (a) Manual Calculation:
    • Agreements: Time periods 1, 4, 5, 6, 10, 12, 13, and 16. That is 8 agreements.
    • Total time periods: 16
    • Formula: (Number of Agreements) / (Total Possible Agreements)
    • Interrater Reliability = 8 / 16 = 0.50
  • (b) Interpretation: The interrater reliability coefficient of 0.50 indicates moderate agreement between the two raters. This means that they agreed on 50% of the cases. This is considered a moderate level of agreement, and improvement may be needed in the consistency of the raters’ judgments.

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