Question

In Exercises 12.101-12.106, tase either the critical-value approach or the P-value approach to perform a chi-square homogeneity test, provided the conditions for using the test are met.
12.101 Self-Concept and Sightedness. Self-concept can be defined as the general view of oneself in terms of personal value and capabilities. A study of whether visual impairment affects self-concept was reported in the article "An Exploration into Self Concept: A Comrarative Analysis between the Adolescents Who Are Sighted and
Elind in India" (British Journal of Visual Impairment, Vol. 30, No. 1, of sighted and blind Indian adolescents gave the following data on self-concept.

a. At the 5% significance level, do the data provide sufficient evidence to conclude that a difference exists in self-concept distributions between sighted and blind Indian adolescents?
b. Repeat part (a) at the1%significance level.

Short answer

Part (a)Pearson Chi-Square=6.589,DF=2,P−Value=0.037
Likelihood Ratio Chi-Square=6.701,DF=2,p-Value=0.035
From the MINITAB output, the value of the chi-square statistic is 6.589.

The value of p$=0.037$

Part (b) The data do not provide sufficient evidence to conclude that a difference exists in selfconcept distributions between sighted and blind Indian adolescents at the$1\%$significance level.

Step-by-step solution

Part (a) Step 1. Given information

Given sighted and blind indian adolescents gave the following data on self-respect

Part (a) Step 2. Check whether or not the data provide sufficient evidence to conclude that a difference exists in self-concept distributions between sighted and blind Indian adolescents. 

Step 1:
The test hypotheses are given below:
Null hypothesis:
H₀ : There is no difference exists in self-concept distributions between sighted and blind Indian adolescents.
Alternative hypothesis:
H₁ : There is difference exists in self-concept distributions between sighted and blind Indian adolescents.
Step 2: Decide the level of significance.
Here, the level of significance is, $\alpha =0.05$.

Step 3:
Find the expected frequency and test statistic.
MINITAB procedure:
Step 1: Choose Stat > Tables > Chi-Square test for association.
Step 2: In Columns containing the table, enter the column of High, Moderate and Low.
Step 3: In Rows, select Sightedness.
Step 4: Under Statistics, select Chi-square test, Display counts in each cell, Display marginal counts and expected cell counts

Step 5. Click on OK

Part (a) Step 3. Finding MATLAB Count 

Chi-Square Test for Association: SIGHTEDNESS, Worksheet columns
Rows: SIGHTEDNESS Columns: Worksheet columns
High Moderate Low All

Sighted 13 73 14 100

10.00 70.63 19.38

0.9000 0.0799 1.4911

Blind 3 40 17 60

6.00 42.38 11.63

1.5000 0.1331 2.4852

All 16 113 31 160
Cell Contents: Count

Expected count
Contribution to Chi-square
Pearson Chi-Square=6.589,DF=2,P−Value=0.037
Likelihood Ratio Chi-Square=6.701,DF=2,p-Value=0.035
From the MINITAB output, the value of the chi-square statistic is 6.589.

Part  (a) Step 4. Finding the value of p

From the MINITAB output,the p-value is$0.037$

Part (b) Step 1. Given information

Given sighted and blind indian adolescents gave the following data on self-respect

Part (b) Step 2. Repeat part (a) at the 1% significance level.

If P-value$\le \alpha$, then reject the null hypothesis.
Here, theP-value is lesser than the level of significance.
That is, P-value$(=0.037)<\alpha (=0.05)$.
Therefore, the null hypothesis is rejected at$5\%$ level.
Thus, the results are statistically significant at$5\%$ level of significance.
Step 6:
Interpretation:
Thus, the data provide sufficient evidence to conclude that a difference exists in self-concept distributions between sighted and blind Indian adolescents at the significance level.

Repeat Part (a) at$1\%$ significance level.
Decide the significance level .
Here, the level of significance is$\alpha =0.01$,,
p-value approach:
From Part (a)., the value of the chi-square statistic is 6.589 and theP-value is 0.037

Here, the P-value is greater than the level of significance.
That is, P-value$(=0.037)>\alpha (=0.01)$.
Therefore, the null hypothesis is not rejected at$1\%$ level.
Thus, the results are not statistically significant at $1\%$level of significance.

Interpretation:
Thus, the data do not provide sufficient evidence to conclude that a difference exists in selfconcept distributions between sighted and blind Indian adolescents at the 1% significance level.