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Conduct the hypothesis test and provide the test statistic, the critical value, and state the conclusion.

A person purchased a slot machine and tested it by playing it 1,279 times. There are 10 different categories of outcomes, including no win, win jackpot, win with three bells, and so on. When testing the claim that the observed outcomes agree with the expected frequencies, the author obtained a test statistic of [tex]\chi^2=17.973[/tex]. Use a 0.05 significance level to test the claim that the actual outcomes agree with the expected frequencies. Does the slot machine appear to be functioning as expected?

The test statistic is 17.973.

The critical value is [tex]\square[/tex].
(Round to three decimal places as needed.)

State the conclusion.
[tex]\square[/tex] [tex]H_0[/tex]. There [tex]\square[/tex] sufficient evidence to warrant rejection of the claim that the observed outcomes agree with the expected frequencies. The slot machine [tex]\square[/tex] to be functioning as expected.

Sagot :

GinnyAnswer
To conduct the hypothesis test, let's go through the steps in detail.

### Step 1: State the Hypotheses

- Null Hypothesis (H₀): The observed outcomes agree with the expected frequencies. (The slot machine is functioning as expected.)
- Alternative Hypothesis (H₁): The observed outcomes do not agree with the expected frequencies. (The slot machine is not functioning as expected.)

### Step 2: Calculate the Test Statistic

Given:
- The test statistic ([tex]\(\chi^2\)[/tex]) is 17.973.

### Step 3: Determine the Critical Value

We need to determine the critical value from the chi-square distribution table using the following parameters:
- Significance level ([tex]\(\alpha\)[/tex]) = 0.05.
- Degrees of freedom (df) = number of categories - 1 = 10 - 1 = 9.

From the chi-square distribution table, the critical value for a significance level of 0.05 and 9 degrees of freedom is:
- Critical value = 16.919 (rounded to three decimal places).

### Step 4: Compare the Test Statistic to the Critical Value

- Test statistic ([tex]\(\chi^2\)[/tex]) = 17.973.
- Critical value = 16.919.

Since the test statistic (17.973) is greater than the critical value (16.919), we reject the null hypothesis.

### Step 5: State the Conclusion

Because the test statistic exceeds the critical value, we have sufficient evidence to reject the null hypothesis.

- Conclusion:
- Reject H₀. There is sufficient evidence to warrant rejection of the claim that the observed outcomes agree with the expected frequencies. The slot machine does not appear to be functioning as expected.

### Summary of Results

- The test statistic is 17.973.
- The critical value is 16.919.
- Conclusion: Reject H₀. There is sufficient evidence to warrant rejection of the claim that the observed outcomes agree with the expected frequencies. The slot machine does not appear to be functioning as expected.
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