ANOVA, Chi-Square, and Binomial Tests- Statistics Solved Assignment Solution Sample

QUESTION

 

ANOVA, Chi-Square, and Binomial Tests

Complete all of the following problems prior to the Adobe Connect session. Please use the Adobe Connection session to ask any questions you have and to check your answers.

1. Analysis of Variance: A study was performed on five subjects to determine how environmental conditions such as temperature affect pulse. The five subjects were exposed to four different temperatures for 30 minutes before their pulse was taken. The data is recorded in the following table.

Perform an ANOVA test to determine whether or not pulse is affected by the external temperature. Use an α = 0.05. Reference Section 12.3 of Statistics for the Behavioral Sciences for assistance.

Temperature	Pulse (bpm)
40° F	82	81	98	90	75
60° F	68	74	92	88	72
80° F	71	69	90	87	70
100° F	78	75	97	92	73

1. State the null hypothesis.

2. State the alternative hypothesis.

3. Calculate df_between (also known as df_Numerator).

4. Calculate df_within (also known as df_Denominator).

5. Use the values for df_N and df_D you calculated with an α = 0.05 to determine the expected F-ration. (Use Table B.4 “The F Distribution.”)

6. Calculate the total (∑x) for each treatment condition (T).

7. Calculate the sum of all scores in the research study (G = ∑T).

8. Calculate the sum of squares (SS) for each treatment conditions where: .

9. Calculate SS_TOTAL where: .

10. Calculate SS_WITHIN where SS_WITHIN = ∑SS_{inside each treatment}.

11. Calculate SS_BETWEEN where .

12. Calculate df_TOTAL. (Remember that you calculated df_BETWEEN and df_WITHIN in parts c and d.)

13. Calculate MS_BETWEEN where MS_BETWEEN = .

14. Calculate MS_WITHIN where MS_WITHIN = .

15. Calculate the actual F-ration where F = .

16. Based on the values you have calculated in this problem, should you accept or reject the null hypothesis?

17. Based on your conclusion in part p, does temperature affect pulse rate based on this experiment?

2. Chi-Square: Friedman and Roseman (1974) have suggested that personality type is related to heart disease. Specifically, Type A people, who are competitive, driven, pressured, and impatient, are more prone to heart disease. On the other hand, Type B individuals, who are less competitive and more relaxed, are less likely to have heart disease. The data is reported in the following table.

Perform a Chi-square test to determine whether or not there is a relationship between personality and heart disease. Use an alpha value of α = 0.05. Reference Section 17.3 and Demonstration 17.1 of Statistics for the Behavioral Sciences for assistance.

	Heart	Vascular	Hypertension	None
Type A Personality	38	27	43	70
Type B Personality	22	23	17	120

1. 1. State the null hypothesis.

1. 2. State the alternative hypothesis.

1. 3. Calculate the degrees of freedom (df).

1. 4. Locate the critical region (Use Table B.8 “The Chi-Square Distribution.”)

1. 5. Calculate the row and column totals. Record your results in the following table:

	Heart	Vascular	Hypertension	None	fc
Type A Personality	38	29	43	60
Type B Personality	18	22	14	126
fr

1. 6. Calculate n.

1. 7. Calculate the expected frequencies (f_c) and record in the following table:

Personality	Disease
Type A Personality	Heart
	Vascular
	Hypertension
	None
Type B Personality	Heart
	Vascular
	Hypertension
	None

1. 8. Perform the calculations in the following table to calculate: .

Type A: Heart	38
Type A: Vascular	29
Type A: Hypertension	43
Type A: None	60
Type B: Heart	18
Type B: Vascular	22
Type B: Hypertension	14
Type B: None	126

1. 9. Calculate the chi-square statistic: .

1. 10. Based on the chi-square statistic you calculated above and the critical region that you determined earlier, should you accept or reject the null hypothesis?

1. 11. Based on your conclusion in part j, is there a connection between personality type and disease?

3. Binomial Test: A new medication that is designed to decrease anxiety was tested on 87 individuals that have been diagnosed with chronic depression. After the study, 36 of the test subjects reported a noticeable decrease in their anxiety and the other 51 subjects reported that they had noticed no change in their anxiety levels.

Perform a binomial test on this dataset to determine whether or not the drug decreases anxiety in people with chronic depression. Reference Sections 18.1 and 18.2 of Statistics for the Behavioral Sciences for assistance.

1. 1. State the value on n (the total number of subjects).

1. 2. State the value X (the number of subjects who reported a decrease in anxiety).

1. 3. State the null hypothesis.

1. 4. State the alternative hypothesis.

1. 5. Find the critical region using an alpha value of 0.05. (Use Table B.1 “The Unit Normal Table.”)

1. 6. Calculate the z-score. (Use equation 18.1.)

1. 7. Given this result, do you reject the null hypothesis?

1. 8. Based on your conclusions, does this medication help to reduce the anxiety of patients with chronic depression?

 

ANSWER

ANOVA, Chi-Square, and Binomial Tests

Complete all of the following problems prior to the Adobe Connect session. Please use the Adobe Connection session to ask any questions you have and to check your answers.

1. Analysis of Variance: A study was performed on five subjects to determine how environmental conditions such as temperature affect pulse. The five subjects were exposed to four different temperatures for 30 minutes before their pulse was taken. The data is recorded in the following table.

Perform an ANOVA test to determine whether or not pulse is affected by the external temperature. Use an α = 0.05. Reference Section 12.3 of Statistics for the Behavioral Sciences for assistance.

Temperature	Pulse (bpm)
40° F	82	81	98	90	75
60° F	68	74	92	88	72
80° F	71	69	90	87	70
100° F	78	75	97	92	73

1. State the null hypothesis.

H₀: μ₁ = μ₂ = μ₃ … = μ_k

2. State the alternative hypothesis.

H₁: Means are not all equal.

3. Calculate df_between (also known as df_Numerator).

1.7

4. Calculate df_within (also known as df_Denominator).

2.3

5. Use the values for df_N and df_D you calculated with an α = 0.05 to determine the expected F-ration. (Use Table B.4 “The F Distribution.”)

1.3

6. Calculate the total (∑x) for each treatment condition (T).

76

7. Calculate the sum of all scores in the research study (G = ∑T).

65

8. Calculate the sum of squares (SS) for each treatment conditions where: .

44

9. Calculate SS_TOTAL where: .

1843

10. Calculate SS_WITHIN where SS_WITHIN = ∑SS_{inside each treatment}.

1213

11. Calculate SS_BETWEEN where .

554

12. Calculate df_TOTAL. (Remember that you calculated df_BETWEEN and df_WITHIN in parts c and d.)

4

13. Calculate MS_BETWEEN where MS_BETWEEN = .

3.3

14. Calculate MS_WITHIN where MS_WITHIN = .

3.1

15. Calculate the actual F-ration where F = .

1.1

16. Based on the values you have calculated in this problem, should you accept or reject the null hypothesis?

reject

17. Based on your conclusion in part p, does temperature affect pulse rate based on this experiment?

yes

2. Chi-Square: Friedman and Roseman (1974) have suggested that personality type is related to heart disease. Specifically, Type A people, who are competitive, driven, pressured, and impatient, are more prone to heart disease. On the other hand, Type B individuals, who are less competitive and more relaxed, are less likely to have heart disease. The data is reported in the following table.

Perform a Chi-square test to determine whether or not there is a relationship between personality and heart disease. Use an alpha value of α = 0.05. Reference Section 17.3 and Demonstration 17.1 of Statistics for the Behavioral Sciences for assistance.

	Heart	Vascular	Hypertension	None
Type A Personality	38	27	43	70
Type B Personality	22	23	17	120

1. 1. State the null hypothesis.

H0: Personality and heart disease are independent.

1. 2. State the alternative hypothesis.

2. H1: Personality and heart disease are not independent.

1. 1. Calculate the degrees of freedom (df).

3

1. 2. Locate the critical region (Use Table B.8 “The Chi-Square Distribution.”)

1. 3. Calculate the row and column totals. Record your results in the following table:

	Heart	Vascular	Hypertension	None	fc
Type A Personality	38	29	43	60	170
Type B Personality	18	22	14	126	180
fr	56	51	57	186	350

1. 4. Calculate n.

350

1. 5. Calculate the expected frequencies (f_c) and record in the following table:

Personality	Disease
Type A Personality	Heart	27.2
	Vascular	24.78
	Hypertension	27.68
	None	90.34
Type B Personality	Heart	28.8
	Vascular	26.22
	Hypertension	29.31
	None	95.65

1. 6. Perform the calculations in the following table to calculate: .

Type A: Heart	38	27.2	10.8	116.64	4.288235294
Type A: Vascular	29	24.78	4.22	17.8084	0.71866021
Type A: Hypertension	43	27.68	15.32	234.7024	8.479132948
Type A: None	60	90.34	-30.34	920.5156	10.1894576
Type B: Heart	18	28.8	-10.8	116.64	4.05
Type B: Vascular	22	26.22	-4.22	17.8084	0.679191457
Type B: Hypertension	14	29.31	-15.31	234.3961	7.997137496
Type B: None	126	95.65	30.35	921.1225	9.630135912

1. 7. Calculate the chi-square statistic: .

46.03195092

1. 8. Based on the chi-square statistic you calculated above and the critical region that you determined earlier, should you accept or reject the null hypothesis?

We have statistically significant evidence at α=0.05 to show that H₀ is false.

1. 9. Based on your conclusion in part j, is there a connection between personality type and disease?

In the χ² goodness-of-fit test, we conclude that either the distribution specified in H₀ is false (when we reject H₀) or that we do not have sufficient evidence to show that the distribution specified in H₀ is false 

3. Binomial Test: A new medication that is designed to decrease anxiety was tested on 87 individuals that have been diagnosed with chronic depression. After the study, 36 of the test subjects reported a noticeable decrease in their anxiety and the other 51 subjects reported that they had noticed no change in their anxiety levels.

Perform a binomial test on this dataset to determine whether or not the drug decreases anxiety in people with chronic depression. Reference Sections 18.1 and 18.2 of Statistics for the Behavioral Sciences for assistance.

1. 1. State the value on n (the total number of subjects).

87

1. 2. State the value X (the number of subjects who reported a decrease in anxiety).

36

1. 3. State the null hypothesis.

the drug has no effect on people.

1. 4. State the alternative hypothesis.

the drug has effect on people.

1. 5. Find the critical region using an alpha value of 0.05. (Use Table B.1 “The Unit Normal Table.”)

1. 6. Calculate the z-score. (Use equation 18.1.)

1.645 (taking from the left )

1. 7. Given this result, do you reject the null hypothesis?

accept

1. 8. Based on your conclusions, does this medication help to reduce the anxiety of patients with chronic depression?

yes