To solve this, we need to compute the test statistic and compare it with the critical value.

Step 1: Expected Frequencies

The total number of observations is:

$n = 17 + 21 + 25 + 17 = 80$

Using the null hypothesis proportions, the expected frequencies for each category are:

$E_A = 0.1 \times 80 = 8.0$ $E_B = 0.4 \times 80 = 32.0$ $E_C = 0.2 \times 80 = 16.0$ $E_D = 0.3 \times 80 = 24.0$

Step 2: Chi-Square Test Statistic Formula

The chi-square test statistic is given by:

$\chi^2 = \sum \frac{(O - E)^2}{E}$

For each category:

1. $A: \frac{(17 - 8)^2}{8} = \frac{81}{8} = 10.125$
2. $B: \frac{(21 - 32)^2}{32} = \frac{121}{32} = 3.781$
3. $C: \frac{(25 - 16)^2}{16} = \frac{81}{16} = 5.062$
4. $D: \frac{(17 - 24)^2}{24} = \frac{49}{24} = 2.042$

Adding these values gives the chi-square test statistic:

$\chi^2 = 10.125 + 3.781 + 5.062 + 2.042 = 21.010$

Step 3: Critical Value

Degrees of freedom (df) = $k - 1 = 4 - 1 = 3$.

Using a chi-square table at $\alpha = 0.005$ and $df = 3$:

$\chi^2_{\text{critical}} = 12.838$

Final Answers:

• Test Statistic: $\chi^2 = 21.010$
• Critical Value: $\chi^2 = 12.838$

Conclusion:

Since $\chi^2 = 21.010 > 12.838$, we reject the null hypothesis at the 0.005 significance level.

Would you like further explanation or assistance with related concepts?
Here are 5 follow-up questions to explore further:

1. What is the meaning of a goodness-of-fit test in hypothesis testing?
2. How do you determine degrees of freedom for a chi-square test?
3. What happens if the test statistic is less than the critical value?
4. How does the significance level (alpha) affect the critical value?
5. How do we calculate expected frequencies in cases with unequal proportions?

Tip: Always ensure that the expected frequencies are at least 5 for the chi-square test to be valid