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Chapter 15: Problem 1

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

Expert verified
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

01

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.
02

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.
03

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.
04

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.

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Most popular questions from this chapter

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