To solve the problem presented, here’s the step-by-step breakdown:

(a) State the Null and Alternate Hypotheses

The null and alternative hypotheses in this case are as follows:

• Null Hypothesis ($H_0$): The number of correct answers follows a binomial distribution with 4 trials and success probability 0.5. This assumes students are guessing.
• Alternative Hypothesis ($H_1$): The number of correct answers does not follow a binomial distribution with 4 trials and success probability 0.5. This implies students are not guessing randomly.

Steps to Perform the Chi-Square Test

1. Calculate the Expected Frequencies:

   • Under the null hypothesis, calculate the probabilities of getting 0, 1, 2, 3, and 4 correct answers using the binomial formula: $P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$ where $n = 4$, $p = 0.5$, and $k = 0, 1, 2, 3, 4$.

   • Multiply each probability by the total number of students (84) to find the expected frequencies.

2. Perform the Chi-Square Test:

   • Use the formula: $\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}$ where $O_i$ is the observed frequency and $E_i$ is the expected frequency.

   • Calculate $\chi^2$ for all categories (0, 1, 2, 3, and 4 correct answers).

3. Find the P-Value:

   • Use the calculated $\chi^2$ value with the degrees of freedom (df). For this case: $\text{df} = \text{Number of categories} - 1 - \text{Number of parameters estimated}$ Here, df = $5 - 1 - 0 = 4$.

4. Compare to Significance Level ($\alpha = 0.05$):

   • Reject $H_0$ if the P-value is less than 0.05.

Would you like detailed calculations for the expected frequencies, $\chi^2$, and P-value? Let me know!

5 Related Questions:

1. How are the probabilities for each category calculated in a binomial distribution?
2. Why do we subtract 1 and account for estimated parameters in degrees of freedom?
3. What happens if an expected frequency is too small in a chi-square test?
4. Can we use alternative methods like simulation to validate the chi-square test result?
5. How is the P-value interpreted in a chi-square test?

Tip:

Always ensure expected frequencies are greater than 5 for the chi-square test to be valid; if not, consider combining categories.