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Manufacturers are testing a die to make sure that it is fair (has a uniform distribution). They roll the die 78 times and record the outcomes in the table below.

They conduct a chi-square Goodness-of-Fit hypothesis test at the [tex]$1 \%$[/tex] significance level.
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline Outcome & 1 & 2 & 3 & 4 & 5 & 6 \\
\hline Expected & 13 & 13 & 13 & 13 & 13 & 13 \\
\hline Observed & 7 & 10 & 14 & 16 & 9 & 22 \\
\hline
\end{tabular}

(a) Select the correct null and alternative hypotheses for this test.

Select all that apply:
- [tex]$H_0$[/tex]: The die has the uniform distribution.
- [tex]$H_a$[/tex]: The die does not have the uniform distribution.

Sagot :

GinnyAnswer
To determine the null and alternative hypotheses for this chi-square Goodness-of-Fit test, let’s first understand what the test aims to accomplish.

The chi-square Goodness-of-Fit test is used to determine if a sample data fits a population with a specific distribution. In this context, we are testing whether a die has a uniform distribution, which means each outcome (1 through 6) should be equally likely.

Step-by-Step Solution:

1. Understand the Null Hypothesis ([tex]\(H_0\)[/tex]):
- The null hypothesis ([tex]\(H_0\)[/tex]) represents the idea that there is no difference between the observed and expected frequencies. In other words, any deviations from the expected frequencies are due to random chance.
- In this case, [tex]\(H_0\)[/tex] would state that the die is fair and follows a uniform distribution. This means that the probability of rolling any number from 1 to 6 is the same (i.e., [tex]\(\frac{1}{6}\)[/tex] for a fair die).

2. Formulate the Null Hypothesis ([tex]\(H_0\)[/tex]):
- [tex]\(H_0\)[/tex]: The die has the uniform distribution.

3. Understand the Alternative Hypothesis ([tex]\(H_a\)[/tex]):
- The alternative hypothesis ([tex]\(H_a\)[/tex]) represents the idea that there is a significant difference between the observed and expected frequencies. This signifies that the observed deviations are not due to random chance, suggesting the die does not follow a uniform distribution.
- In this scenario, [tex]\(H_a\)[/tex] would state that the die is not fair and does not follow a uniform distribution. This means the probabilities of rolling the numbers from 1 to 6 are not equal.

4. Formulate the Alternative Hypothesis ([tex]\(H_a\)[/tex]):
- [tex]\(H_a\)[/tex]: The die does not have the uniform distribution.

Therefore, the correct hypotheses for this test are:
- [tex]\(H_0\)[/tex]: The die has the uniform distribution.
- [tex]\(H_a\)[/tex]: The die does not have the uniform distribution.

Select the correct null and alternative hypotheses from the given options:
- [tex]\(H_0\)[/tex]: The die has the uniform distribution. (Correct)
- [tex]\(H_a\)[/tex]: The die has the uniform distribution. (Incorrect)
- [tex]\(H_0\)[/tex]: The die does not have the uniform distribution. (Incorrect)
- [tex]\(H_a\)[/tex]: The die does not have the uniform distribution. (Correct)
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