Q9 Homework Pearson r

Pearson r -- Homework

Problem 1                   Percent Minority: 10%, 40%, 30%, 20%, 15%, 25%, 33%, 24%, 21% 

                            Burglaries:        1,   7,   6,   3,   4,   5,   6,   3,   2 

A crime study in Cleveland Ohio is aimed at awarding financial aid to minorities as an incentive to refrain from burglaries. In order to justify the expense, they need to establish that burglaries are associated with minority occupancy. The study examined 9 neighborhoods to see if the variables were significantly related. The question to be answered was do neighborhoods with increased minority populations show a pattern of elevated burglaries when examining a representative sample of the city population. The independent variable used in that study was percentage of ethnic minorities. It is theorized that burglaries are more prevalent in areas having high a concentration of minorities. The mean of the independent variable “percent minorities” is 24%.  The mean of the dependent variable “number of burglaries” is  4.111. The correlation between the two variables is Pearson r = .877. Policy Action:  If the relationship is statistically significant, the minority population will get the financial aid money. 

a) Describe the statistical significance of the data; b) the likelihood you would see these results if the null were true; c) the percentage of explained variability; d) your decision on the policy action. 

Theory: As percent Minority increases burglaries will increase. (r ≠ 0)  

Null:  There is not a statistically significant relationship between the number of burglaries and the percentage of ethnic minorities in residence in the neighborhood.(r = 0) 

Results: The data shows support for the theory that burglaries are significantly higher in neighborhoods with higher percentage of minorities. The Null hypothesis is rejected (r(7) = .877, p. < .05). It is unlikely that we would find a relationship this strong if the null is true. The probability of obtaining a Pearson r this size by chance is less than 5 times in 100.

Analysis:  The theory is supported by the data. There is a strong positive correlation between burglaries and percent of minorities and the relationship is statistically significant: r(7) = .877, p. < .05. The effect size of ethnic minorities on burglaries shows that 77% of the variance in burglaries is accounted for by presence of minorities (r2 = .770), leaving 23% of the variance-explained to other factors.

Test Questions:

 a) Describe the statistical significance of the data.  (Is the relationship statistically significant (or) does the presence of minorities significantly impact burglaries (or) are the variables significantly related?)

 Reject the Null Hypothesis. The data shows support for the theory that burglaries are significantly higher in neighborhoods with higher percentage of minorities (r(7) = .877, p. < .05).

 b) Describe the likelihood you would see these results if the null were true  (Is it likely that you would see a relationship this strong if the Null is true (or) If you repeated this experiment over and over, is it likely you would come to a different conclusion?)

 It is unlikely that we would find a relationship this strong if the null is true.

 c) Describe the percentage of explained variability (What is the effect size (or) what is the Magnitude of Effect, (or) How much variability in burglaries is attributable to the presence of minorities?)

 The effect size of ethnic minorities on burglaries shows that 77% of the variance in burglaries is accounted for by presence of minorities (r2 = .770), leaving 23% of the variance-explained to other factors.

 d) What is your decision on the financial aid money? (Will the minority population be awarded the incentive money?)

 Yes, they will receive the money

Problem 2                             Drug Arrests:  3, 8, 4, 3, 2

                                        Burglaries:  2, 7, 3, 5, 1

The same study used drug arrests as a predictor. To see if they could help offset the expenses of the program, it was suggested that they could combine the drug task force and the burglary task force. If the relationship between Drug Arrests and Burglaries is statistically significant, they will combine the two task forces together. Based on the data from 5 neighborhoods, Pearson r is .841. The mean of the independent variable “Drug Arrests” is  = 4.0. The mean of the dependent variable “number of burglaries” is   = 3.6. Policy Action:  If the relationship is significant, the drug task force will combine with the Burglary task force.

Theory: Neighborhoods with more drug activity will have more burglaries.  (r ≠ 0)

 Null Hypothesis: There is not a statistically significant relationship between the number of drug arrests and the number of burglaries occurring in the neighborhood. (r = 0)

 Results: The data does not support the theory that Drug Arrests accompany a significantly higher number of burglaries in the selected areas. The null hypothesis is retained (r(3) = .841, p. > .05). Retain the null because at df = 3, rcrit = .878. If the null is true, the probability of obtaining a Pearson r this size by chance is greater than 5 times in 100, which is unacceptably high.

 Analysis:  The data do not support the theory that drug arrests and burglaries are associated in these neighborhoods as the relationship is not statistically significant. The null hypothesis must be retained until evidence of relationship is demonstrated. 

 Test Questions:

 1.  Is the relationship statistically significant (or) does the presence of drug arrests significantly impact burglaries (or) are the variables significantly related?

 Retain the Null Hypothesis. The data does not show support for the theory that elevated drug arrests are significantly correlated with higher percentage of minorities in neighborhoods (r(3) = .841, p. > .05).

 2.  Is it likely that you would see a relationship this strong if the Null is true (or) If you repeated this experiment over and over, is it likely you would come to a different conclusion?

 Since the null was retained, It is likely that we would find a relationship this strong if the null is true.

 3.  What is the effect size (or) what is the Magnitude of Effect, (or) How much variability in burglaries is attributable to drug arrests?

 There is no effect size of drug arrests on burglaries. The study shows that 0% of the variance in burglaries is accounted for by presence of drug arrests (r2 = 0), leaving 100% of the variance-explained to other factors.

 4.  Will the two task forces be combined?

 No way Jose. The two units will not combine.

Problem 3

 Police officers are typically rated on a "Composite Officer Proficiency" (COP) scale that is designed to reflect job capability. A component of job capability is ability to miss fewer targets on the marksmanship range. Female and male officers who have higher COP scores are theorized to miss fewer targets. A police chief created a rating team to study the relationship. She wanted to determine if COP ratings could be used to predict who will miss the least number of targets. Theory: AS COP scores go up, targets missed go down.

Policy Action: If the relationship is significant, the agency will remodel the firing range. 

(Pearson r = -.806, N = 6). 

X = COP Score:           5, 7, 4, 6, 2, 2           = 4.33

Y = Targets Missed:      4, 2, 5, 3, 7, 4           = 4.167

Theory: Officers with higher COP scores will miss fewer targets. (r ≠ 0) 

  Null: There is not a statistically significant relationship between scores on the Composite Officer Proficiency scale and the number of targets missed by the officer. (r = 0)

  Results: r(4) = -.806, p. > .05. Retain the null because at df = 4, rcrit = .811. If the null is true, the probability of obtaining a Pearson r this size by chance is greater than 5 times in 100, which is unacceptably high.

  Analysis:  The theory is unsupported by the data collected in this study. Although the relationship approaches statistical significance, there is not a statistically significant relationship between COP scores and marksmanship ability at the .05 level. The null hypothesis must be assumed true. Policy Action: The agency will not remodel the firing range.

 Test Questions:

 1.  Is the relationship statistically significant (or) do higher cop scores indicate better performance on the shooting range (fewer targets missed) (or) are the variables significantly related?

 Retain the Null Hypothesis. The data does not show support for the theory that elevated cop scores are significantly correlated with fewer targets missed (r(4) = -.806, p. > .05).

 2.  Is it likely that you would see a relationship this strong if the Null is true (or) If you repeated this experiment over and over, is it likely you would come to a different conclusion?

 Since the null was retained, It is likely that we would find a relationship this strong if the null is true.

 3.  What is the effect size (or) what is the Magnitude of Effect, (or) How much variability in targets missed is attributable to COP scores?

 There is no effect size of cop scores on shooting skills. The study shows that 0% of the variance in targets missed is accounted for by cop scores (r2 = 0), leaving 100% of the variance-explained to other factors. 

 4.  Will the firing range be remodeled?

 The agency will not remodel the firing range.

Problem 4 Answer: 

  Theory: Persons with higher education have higher income. (r ≠ 0)

  Null: There is not a statistically significant relationship between total family income and education as measured by the highest grade completed. (r = 0)

  Results: r(1430) = .303, p. = .000. Reject the null because, if the null is true, at df = 1431 the probability of obtaining a Pearson r this size by chance is less than 1 in 1000.  

  Analysis: Examination of the data show that there is a statistically significant relationship between education and income r(1430) = .303, p. = .000, as was predicted by the theory. However, the relationship is weak (r2 = .091) with only 9.1% of the variance in income attributable to education, leaving 90.9% to other factors.

Problem 5 Answer:  Problem 5 Answer:  

  Theory: Persons with higher education marry later in life. (+r   0)

  Null: There is not a statistically significant relationship between age first married and education as measured by the highest grade completed. (r = 0)

  Results: r(1158) = .291, p. = .000. The null is rejected, because if the null is true this is a very rare event. At df = 1158, the probability of obtaining a Pearson r this size by chance is less than 1 in 1000.

  Analysis: The data show some support for the theory. As predicted, there is a statistically significant relationship between education andeducation as measured by the highest grade completed. r(1158) = .291, p. = .000. However, although statistically significant, the relationship is weak (r2 = .084) with only 8.4% of the variance in age wed being accounted for through knowledge of education, leaving 91.6% to other factors. Therefore these results should be interpreted with caution.

  Theory: Persons with higher education marry later in life. (+r   0)

  Null: There is not a statistically significant relationship between age first married and education as measured by the highest grade completed. (r = 0)

  Results: r(1158) = .291, p. = .000. The null is rejected, because if the null is true this is a very rare event. At df = 1158, the probability of obtaining a Pearson r this size by chance is less than 1 in 1000.

  Analysis: The data show some support for the theory. As predicted, there is a statistically significant relationship between age married andeducation as measured by the highest grade completed. r(1158) = .291, p. = .000. However, although statistically significant, the relationship is weak (r2 = .084) with only 8.4% of the variance in age wed being accounted for through knowledge of education, leaving 91.6% to other factors. Therefore these results should be interpreted with caution.

Problem 6 Answer

  Theory: Inmates with longer sentences should be able to read better. (r ≠ 0)

   Null: There is not a statistically significant relationship between reading achievement scores as measured by scores on the Test of Adult Basic Education (TABE) and the length of sentence measured in months.  (r = 0)

   Results: r(68) = .065, p. = .594. Retain the null because the probability of getting a Pearson Correlation coefficient this size by chance from a random sample is greater than the acceptable minimum of less than 5% (p. < .05). Therefore the risk is too high to assume that the relationship is statistically significant.

   Analysis: The theory is not supported by the data as the relationship between sentence length and reading achievement is not statistically significant.

 Test Questions:

 1.  Is the relationship statistically significant? (or) Do longer sentences result in higher TABE scores? (or) Are the variables significantly related?

 Retain the Null Hypothesis. The data does not show support for the theory that longer sentences are significantly correlated with higher TABE scores (r(68) = .065, p. = .594.).

 2.  Is it likely that you would see a relationship this strong if the Null is true? (or) If you repeated this experiment over and over, is it likely you would come to a different conclusion?

 Since the null was retained, It is likely that we would find a relationship this strong if the null is true.

 3.  What is the effect size (or) what is the Magnitude of Effect, (or) How much variability in burglaries is attributable to drug arrests?

 There is no effect size of sentence length on TABE scores. The study shows that 0% of the variance in TABE Scores is accounted for by sentence lenght (r2 = 0), leaving 100% of the variance-explained to other factors. 

Problem 7. A researcher is studying the respiratory rate of cardiac patients. To tests the patient’s baseline responses, she conducts a test of their respiratory system. Naturally she expects that increased emotional stimulation will cause a rise in respiratory rate.  The data-set consists of simulated scores showing the relationship between emotional stimulation and respiratory rate. Obtain the requested statistics, test for significance, provide the null, results, and analysis, and if appropriate, describe how much respiratory rate is attributable to the emotional stimulation and how much is attributable to other factors. (Pearson r = .919,

 N = 4)

Emotional Stimulation:  9, 8, 4, 6      Respiratory Rate:  8, 6, 1, 6

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Problem 7 Answer

  Theory: Increased emotional stimulation will result in elevated respiratory rate. (+r  0)

  Null: There is not a statistically significant relationship between respiratory rate of cardiac patients and the amount of emotional stimulation they receive.  (r = 0)

  Results: r(2) = .919, p. >.05. Retain the null because the probability of getting a Pearson Correlation coefficient this size by chance from a random sample is greater than the acceptable minimum of less than 5% (p. < .05). Therefore the risk is too high to assume that the relationship is statistically significant.

  Analysis: Theory is not supported, there is not a statistically significant relationship between the two variables. Since Pearson r is not significant, MOE is not appropriate.

Problem 8. A neighborhood watch committee has noticed the presence of more drug dealers in neighborhoods over the past 24 months. (Pearson r = .989, N = 14)

 A real estate agent spoke before the committee and announced that real estate prices have declined in some neighborhoods during that same period.  The agent believes that drugs are driving real estate prices down. 

Problem 8 Answer:

  Theory: Increased drug activity is associated with falling real estate prices and home values. (r ≠ 0) 

  Null: There is not a statistically significant relationship between the number of drug arrests within a neighborhood and the mean sales price of a home in that neighborhood. (r = 0)

  Results: Reject, r(12) = -.989, p. < .05. Reject the null because, if the null is true, at df = 12 the probability of obtaining a Pearson r this size by chance is less than 1 in 1000.  

  Analysis: Examination of the data show that there is a statistically significant relationship between drug activity and real estate prices as was predicted by the theory: r(12) = -.919, p. < .05. The relationship is very strong (r2 = .845) with 84.5% of the variance in real estate prices attributable to drug activity, leaving only 15.5% to other factors.

Problem 9. The Chief of Police believes that bigger officers are better at running down and capturing fleeing felons. She asks a research question: Are heavier officers more efficient at capturing fleeing felons? A study is conducted to determine the weight of each officer in each precinct over the past 10 years, and how many fleeing felons were actually caught in those precincts out of the total number of felonies reported. 

Problem 9 Answer:

  Theory: Increased officer weight is associated with higher number of fleeing felon captures. (r ≠ 0)

  Null: There is not a statistically significant relationship between officer weight and the number of fleeing felons captured. (r = 0)

  Results: Reject, r(9) = .807, p. < .05. The null hypothesis is rejected.  If the null is true, at df = 9 the probability of obtaining a Pearson r this size by chance is less than 3 in 1000.  

  Analysis: Examination of the data show that there is a statistically significant relationship between officer weight and the number of felons apprehended. It was theorized that higher officer weight would be associated with more captures. This theory was supported by the data, as higher officer weight is strongly associated with more felon captures: r(9) = .807, p. < .05. The relationship is very strong (r2 = .651) with 65.1% of the variance in felon arrests attributable to officer weight, leaving 34.9% to other factors.