Question

1. suppose that we have a 6 - sided die. we assume that the die is unbiased (upon rolling the die, each outcome is equally likely). an experiment is conducted in which the die is rolled 240 times. the outcomes are in the table below. at a significance level of \\( \alpha=0.05 \\), is there enough evidence to support the hypothesis that the die is unbiased? (7 items \\( \times 3 \\) points)

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$$\begin{tabular}{|c|c|c|c|c|c|c|} \\hline outcome & 1 & 2 & 3 & 4 & 5 & 6 \\\\ \\hline frequency & 34 & 44 & 30 & 46 & 51 & 35 \\\\ \\hline \\end{tabular}$$

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$$\begin{tabular}{|l|} \\hline a. the null hypothesis \\( h_0 \\) : \\\\ b. the alternative hypothesis \\( h_a \\) : \\\\ c. the level of significance \\( \\alpha= \\) \\\\ d. the degrees of freedom: \\\\ e. the test statistic: \\\\ f. the critical value: \\\\ g. decision: \\\\ \\hline \\end{tabular}$$

Explanation:

Step1: State hypotheses

• Null hypothesis \(H_0\): The die is unbiased (\(p_1 = p_2=\cdots=p_6=\frac{1}{6}\))
• Alternative hypothesis \(H_a\): The die is biased (\(\exists i

eq j\) such that \(p_i
eq p_j\))

Step2: Determine significance level

The level of significance \(\alpha = 0.05\)

Step3: Calculate degrees of freedom

For a chi - square goodness - of - fit test with \(k = 6\) categories, the degrees of freedom \(df=k - 1=6 - 1 = 5\)

Step4: Calculate expected frequencies

The expected frequency \(E_i=n\times p_i\), where \(n = 240\) and \(p_i=\frac{1}{6}\) for \(i = 1,\cdots,6\). So \(E_i=240\times\frac{1}{6}=40\) for all \(i\)

Step5: Calculate test statistic

The chi - square test statistic is \(\chi^2=\sum_{i = 1}^{k}\frac{(O_i - E_i)^2}{E_i}\)

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Step6: Find critical value

Using a chi - square distribution table with \(df = 5\) and \(\alpha=0.05\), the critical value \(\chi_{0.05,5}^2 = 11.0705\)

Step7: Make a decision

Since \(\chi^2=8.35<\chi_{0.05,5}^2 = 11.0705\), we fail to reject the null hypothesis.

Answer:

a. \(H_0\): The die is unbiased (\(p_1 = p_2=\cdots=p_6=\frac{1}{6}\))
b. \(H_a\): The die is biased (\(\exists i
eq j\) such that \(p_i
eq p_j\))
c. \(\alpha = 0.05\)
d. \(df = 5\)
e. \(\chi^2=8.35\)
f. \(\chi_{0.05,5}^2 = 11.0705\)
g. Fail to reject \(H_0\)