To conduct a Chi-square ($\chi^2$) goodness-of-fit test, we will follow these steps:

1. State the hypotheses:

   • Null hypothesis ( $H_0$ ): The distribution of appointments follows an even distribution.
   • Alternative hypothesis ( $H_a$ ): The distribution of appointments does not follow an even distribution.

2. Step-by-step process:

a. Calculate the test statistic $\chi^2$

To calculate the $\chi^2$ test statistic, you need the observed frequencies ( $O_i$ ) and the expected frequencies ( $E_i$ ).

The test statistic is given by:

$\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}$

Where:

• $O_i$ = observed frequency for category $i$
• $E_i$ = expected frequency for category $i$

Let's assume you provide data, or I can generate an example to calculate $\chi^2$. If you have specific observed and expected values, please share them.

b. Determine the critical value

To find the critical value, we need:

• Degrees of freedom (df): $df = \text{number of categories} - 1$
• Significance level: 2.5% ($\alpha = 0.025$)

Once you provide the number of categories, I can look up the critical value from the Chi-square distribution table.

c. Conclude

• Reject $H_0$ if the calculated $\chi^2$ statistic is greater than the critical value.
• Fail to reject $H_0$ if the calculated $\chi^2$ statistic is less than or equal to the critical value.

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Please provide the observed frequencies and the expected values or number of categories so that I can help you calculate the test statistic and finalize the conclusion!

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Would you like me to generate an example? Here are some follow-up questions:

1. How many categories of appointments do you have in your data?
2. What are the observed frequencies of each category?
3. What is the expected distribution (even or specific)?
4. Are there any special conditions or assumptions for the problem?
5. Would you like a full explanation on critical values?

Tip: The degrees of freedom are crucial in determining the critical value in a Chi-square test, always subtract 1 from the number of categories!