Let's solve the given problem step by step.

Problem:

We are testing whether the coin is unbiased (fair) at the 5% level of significance, with the provided $Z_{0.05} = 1.96$.

Step 1: Hypotheses

1. Null Hypothesis ($H_0$): The coin is unbiased, meaning $p = 0.5$.
2. Alternative Hypothesis ($H_a$): The coin is biased, meaning $p \neq 0.5$.

Step 2: Test Statistic

The test statistic for proportions is given by:

$Z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0 (1 - p_0)}{n}}}$

Where:

• $\hat{p}$ = observed proportion of heads = $\frac{\text{Number of Heads}}{\text{Total Tosses}} = \frac{216}{400} = 0.54$
• $p_0$ = hypothesized proportion under $H_0$ = 0.5
• $n$ = total number of tosses = 400

Substituting the values:

$Z = \frac{0.54 - 0.5}{\sqrt{\frac{0.5 \times (1 - 0.5)}{400}}}$

Step 3: Simplify Calculation

1. $\sqrt{\frac{0.5 \times 0.5}{400}} = \sqrt{\frac{0.25}{400}} = \sqrt{0.000625} = 0.025$
2. $Z = \frac{0.04}{0.025} = 1.6$

Step 4: Compare $Z$-value with $Z_{0.05}$

• Critical value: $Z_{0.05} = 1.96$ (from the standard normal distribution for a 5% significance level).
• Observed $Z = 1.6$.

Since $|Z| = 1.6$ is less than $1.96$, we fail to reject the null hypothesis.

Step 5: Conclusion

At the 5% level of significance, there is insufficient evidence to conclude that the coin is biased. Therefore, the answer is:

(i) Yes, the coin is unbiased.

Would you like a detailed breakdown of this reasoning or further clarification? 😊

Related Questions:

1. What would happen if the observed $Z$-value was greater than $Z_{0.05}$?
2. How is the formula for $Z$-value derived for proportions?
3. What are Type I and Type II errors in hypothesis testing?
4. How would the result change if we increased the sample size?
5. What is the significance level, and why is it usually set at 5%?

Tip:

Always check the assumptions of your hypothesis test (e.g., large sample size or normality of the sampling distribution for proportions) to ensure the validity of your conclusions.