A doctor keeps track of the number of babies she delivers in each season. She expects that the distribution will be uniform (the same number of babies in each season). The data she collects is shown in the table below. Find the test statistic, *χ*20, for the chi-square goodness-of-fit test. Round the final answer to one decimal place.

*χ*20=∑*k*(*O*−*E*)2*E*

Season 

 Spr 

 Summ 

 Fall 

 Wint 

Expected 

28

28

28

28

Observed 

35

21

28

28

Provide your answer below:

To find the test statistic \( \chi^2_0 \) using the chi-square goodness-of-fit test formula:

\[
\chi^2_0 = \sum \frac{(O - E)^2}{E}
\]

where:
- \( O \) is the observed frequency, and
- \( E \) is the expected frequency.

### Given:
- Expected frequencies (\( E \)) for each season: 28
- Observed frequencies (\( O \)):
  - Spring (Spr): 35
  - Summer (Summ): 21
  - Fall: 28
  - Winter (Wint): 28

### Step-by-Step Calculation:

1. **For Spring (Spr):**
   \[
   \frac{(O - E)^2}{E} = \frac{(35 - 28)^2}{28} = \frac{7^2}{28} = \frac{49}{28} \approx 1.75
   \]

2. **For Summer (Summ):**
   \[
   \frac{(21 - 28)^2}{28} = \frac{(-7)^2}{28} = \frac{49}{28} \approx 1.75
   \]

3. **For Fall:**
   \[
   \frac{(28 - 28)^2}{28} = \frac{0^2}{28} = \frac{0}{28} = 0
   \]

4. **For Winter (Wint):**
   \[
   \frac{(28 - 28)^2}{28} = \frac{0^2}{28} = \frac{0}{28} = 0
   \]

### Sum of the above results:
\[
\chi^2_0 = 1.75 + 1.75 + 0 + 0 = 3.5
\]

### Final Answer:
The test statistic \( \chi^2_0 \) is **3.5**.

Would you like more details or have any questions?

Here are 8 related questions to expand your understanding:

1. How do you determine the degrees of freedom for this chi-square test?
2. What significance level is typically used when interpreting the chi-square test statistic?
3. What does a chi-square test statistic of 3.5 indicate about the distribution of births across seasons?
4. How would the result change if the observed and expected values were much closer?
5. What is the p-value associated with a chi-square test statistic of 3.5?
6. How does the chi-square test differ when dealing with more than four categories?
7. What happens if the observed frequencies are very close to the expected frequencies?
8. Can the chi-square test be used for continuous data, or is it only for categorical data?

**Tip:** Always ensure that the expected frequency is large enough in each category (typically at least 5) for the chi-square approximation to be valid.

Learn how to calculate the chi-square goodness-of-fit test statistic for a doctor tracking seasonal baby deliveries. Given expected and observed frequencies, find the test statistic using a step-by-step approach. Ideal for high school students and those studying statistics.

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