Statistics for the Behavioral Sciences
Statistics for the Behavioral Sciences
3rd Edition
ISBN: 9781506386256
Author: Gregory J. Privitera
Publisher: SAGE Publications, Inc
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Chapter 17, Problem 21CAP

1.

To determine

Find the Chi-square test for independence at 0.05 significance level.

Check whether to retain or reject the null hypothesis test by using the chi-square test for independence.

1.

Expert Solution
Check Mark

Answer to Problem 21CAP

The value of test statistic is 6.482.

The decision is to reject the null hypothesis.

The cravings for cocaine and relapse are related.

Explanation of Solution

Calculation:

The given information is that, a study was conducted by the tests of cravings for

cocaine and relapse. The summarized table is,

RelapseTotals
YesNo
CravingYes201030
No81725
Totals2827N=55

Table 1

The formula for total participants is,

N=i=1nfoi

In the formula, fo denotes the observed frequencies for each category.

The formula for expected frequency is,

fe=row total×column totalN

In the formula, N denotes the total participants.

Test statistic:

The formula for chi-square test statistic is,

χobt2=(fofe)2fe

In the formula, fo denotes the observed frequency and fe denotes the expected frequency.

The formula for degrees of freedom for chi-square independence test is,

df=(k11)(k21)

In the formula, k1 denotes the number of levels of first categorical variable and k2 denotes the number of levels of second categorical variable.

Null hypothesis:

H0: The cravings for cocaine and relapse are independent.

Alternative hypothesis:

H1: The cravings for cocaine and relapse are related.

Expected frequency:

Substitute the corresponding values of roe totals, column totals and total frequencies for each level of categorical variable in the expected frequency formula.

Expected frequencyRelapseTotals
YesNo
CravingYes30×2855=15.330×2755=14.730
No25×2855=12.725×2755=12.325
Totals2827N=55

Table 2

Substitute, k1=2,k2=2 in degrees of freedom formula for chi-square independence test.

df=(21)(21)=1×1=1

Critical value:

The given significance level is α=0.05 and the degrees of freedom are 1.

From the Appendix C: Table C.7 Critical values for Chi-square:

  • Locate the value 1 in the degrees of freedom (df) column.
  • Locate the 0.05 in level of significance row.
  • The intersecting value that corresponds to the 1 with level of significance 0.05 is 3.84.

The critical value for df=1 at a 0.05 level of significance is 3.84.

Test statistic value:

Substitute the corresponding values of observed and expected frequencies for each level of categorical variable in the test statistic formula.

χobt2=(2015.3)215.3+(1014.7)214.7+(812.7)212.7+(1712.3)212.3=22.0915.3+22.0914.7+22.0912.7+22.0912.3=1.444+1.503+1.739+1.796=6.482

Hence, the value of test statistic is 6.482.

Conclusion:

The test statistic value is 6.482.

The critical value is 3.84.

The test statistic value is greater than the critical value.

The null hypothesis is rejected.

The cravings for cocaine and relapse are related.

2.

To determine

Find the effect size using ϕ.

Find the effect size using Cramer’s V.

2.

Expert Solution
Check Mark

Answer to Problem 21CAP

The effect size using ϕ is 0.34.

The effect size using Cramer’s V is 0.34.

Explanation of Solution

Calculation:

The test statistic is 6.482, and value of N is 55.

The formula for Phi coefficient is,

ϕ=χ2N

In the formula, χ2 denotes the calculated test statistic, N denotes the total participants.

The formula for effect size using Cramer’s V is,

V=χ2N×dfSmaller

In the formula, χ2 denotes the calculated test statistic, N denotes the total participants and dfSmaller denotes the smaller value for two sets of degrees of freedom, that is smaller of (k11) and (k21).

Substitute, N=55, and χ2=6.482 in effect size formula using Phi coefficient.

ϕ=6.48255=0.1179=0.34

Hence, effect size using ϕ is 0.34.

For 2×2 chi-square, the degrees of freedom are 1(=21), and 1(=21).

The smaller degrees of freedom are 1.

Substitute, N=55, χ2=6.482, and dfsmaller=1 in effect size formula using Cramer’s V.

V=6.48255×1=6.48255=0.1179=0.34

Hence, effect size using Cramer’s V is 0.34.

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Chi Square test; Author: Vectors Academy;https://www.youtube.com/watch?v=f53nXHoMXx4;License: Standard YouTube License, CC-BY