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

QUESTION

ANOVA, Chi-Square, and Binomial Tests

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.
1. State the alternative hypothesis.
1. Calculate dfbetween (also known as dfNumerator).
1. Calculate dfwithin (also known as dfDenominator).
1. Use the values for dfN and dfD you calculated with an α = 0.05 to determine the expected F-ration. (Use Table B.4 “The F Distribution.”)
1. Calculate the total (∑x) for each treatment condition (T).
1. Calculate the sum of all scores in the research study (G = ∑T).
1. Calculate the sum of squares (SS) for each treatment conditions where: .
1. Calculate SSTOTAL where: .
1. Calculate SSWITHIN where SSWITHIN = ∑SSinside each treatment.
1. Calculate SSBETWEEN where .
1. Calculate dfTOTAL. (Remember that you calculated dfBETWEEN and dfWITHIN in parts c and d.)
1. Calculate MSBETWEEN where MSBETWEEN = .
1. Calculate MSWITHIN where MSWITHIN = .
1. Calculate the actual F-ration where F = .
1. Based on the values you have calculated in this problem, should you accept or reject the null hypothesis?
1. Based on your conclusion in part p, does temperature affect pulse rate based on this experiment?
1. 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. State the null hypothesis.
1. State the alternative hypothesis.
1. Calculate the degrees of freedom (df).
1. Locate the critical region (Use Table B.8 “The Chi-Square Distribution.”)
1. 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. Calculate n.
1. Calculate the expected frequencies (fc) and record in the following table:
 Personality Disease Type A Personality Heart Vascular Hypertension None Type B Personality Heart Vascular Hypertension None
1. 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. Calculate the chi-square statistic: .
1. 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. Based on your conclusion in part j, is there a connection between personality type and disease?
1. 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. State the value on n (the total number of subjects).
1. State the value X (the number of subjects who reported a decrease in anxiety).
1. State the null hypothesis.
1. State the alternative hypothesis.
1. Find the critical region using an alpha value of 0.05. (Use Table B.1 “The Unit Normal Table.”)
1. Calculate the z-score. (Use equation 18.1.)
1. Given this result, do you reject the null hypothesis?
1. Based on your conclusions, does this medication help to reduce the anxiety of patients with chronic depression?

ANOVA, Chi-Square, and Binomial Tests

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 … = μk

1. State the alternative hypothesis.

H1: Means are not all equal.

1. Calculate dfbetween (also known as dfNumerator).

1.7

1. Calculate dfwithin (also known as dfDenominator).

2.3

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

1.3

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

76

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

65

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

44

1. Calculate SSTOTAL where: .

1843

1. Calculate SSWITHIN where SSWITHIN = ∑SSinside each treatment.

1213

1. Calculate SSBETWEEN where .

554

1. Calculate dfTOTAL. (Remember that you calculated dfBETWEEN and dfWITHIN in parts c and d.)

4

1. Calculate MSBETWEEN where MSBETWEEN = .

3.3

1. Calculate MSWITHIN where MSWITHIN = .

3.1

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

1.1

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

reject

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

yes

1. 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. State the null hypothesis.

H0: Personality and heart disease are independent.

1. State the alternative hypothesis.

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

1. Calculate the degrees of freedom (df).

3

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

1. 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. Calculate n.

350

1. Calculate the expected frequencies (fc) 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. 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. Calculate the chi-square statistic: .

46.03195092

1. 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. 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

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

87

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

36

1. State the null hypothesis.

the drug has no effect on people.

1. State the alternative hypothesis.

the drug has effect on people.

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

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

1.645 (taking from the left )

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

accept

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

yes

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