Question

A particular cell phone case is available in a choice of four different colors. A store sells all four colors. To test the hypothesis that sales are equally divided among the four colors, a random sample of 100 purchases is identified. a. If the resulting \(X^{2}\) value were \(6.4,\) what conclusion would you reach when using a test with significance level \(0.05 ?\) b. What conclusion would be appropriate at significance level 0.01 if \(X^{2}=15.3 ?\) c. If there were six different colors rather than just four, what would you conclude if \(X^{2}=13.7\) and a test with \(\alpha=0.05\) was used?

Short answer

With a given significance level of \(0.05\), for the first case where \(X^{2}=6.4\), we would not reject the null hypothesis indicating that sales might be equally divided among the four colors. However, for the second case (\(X^{2}=15.3\) at a significance level of \(0.01\)) and the third case (\(X^{2}=13.7\) with six colors at a significance level of \(0.05\)), we would reject the null hypothesis suggesting that sales are not equally divided among the colors.

Step-by-step solution

Understand Chi-Square Test and its Properties

In a Chi-square test, the null hypothesis is usually that there is no significant difference between the observed and expected frequencies. A smaller computed \(X^{2}\) value suggests that the observed data fits the expected data. The critical value of \(X^{2}\) increases with increasing degrees of freedom (which is one less than the number of categories) and with decreasing significance level (\(\alpha)\). Therefore, to reject the null hypothesis, the computed \(X^{2}\) must be greater than the critical \(X^{2}\) value.

Analyzing Scenario a

Here, the \(X^{2}\) value is \(6.4\), the significance level (\(\alpha)\) is \(0.05\), and the number of colors/categories is \(4\), so the degrees of freedom is \(4-1 = 3\). Consulting a Chi-square distribution table, it can be seen that the critical \(X^{2}\) value for \(3\) degrees of freedom at \(0.05\) significance level is approximately \(7.815\). Since the computed \(X^{2}\) value of \(6.4\) is less than the critical value, we would not reject the null hypothesis. We may conclude that sales are equally divided among the four colors.

Analyzing Scenario b

Here, the \(X^{2}\) value is \(15.3\), the significance level (\(\alpha)\) is \(0.01\), and the number of colours/categories remains \(4\), so the degrees of freedom is still \(3\). Consulting a Chi-square distribution table, the critical \(X^{2}\) value for \(3\) degrees of freedom at \(0.01\) significance level is approximately \(11.345\). Since the computed \(X^{2}\) value of \(15.3\) is greater than the critical value, we would reject the null hypothesis. We may conclude that sales are not equally divided among the four colors.

Analyzing Scenario c

In this scenario, the \(X^{2}\) value is \(13.7\), the significance level (\(\alpha)\) is \(0.05\), and the number of colors/categories is \(6\), so the degrees of freedom is \(6-1=5\). Consulting a Chi-square distribution table, the critical \(X^{2}\) value for \(5\) degrees of freedom at \(0.05\) significance level is approximately \(11.071\). Since the computed \(X^{2}\) value of \(13.7\) is greater than the critical value, we would reject the null hypothesis. We may conclude that sales are not equally divided among the six colors.