Chapter 9.2, Problem 48E

To determine

To identify: The errors in the provided solution and correct them.

Explanation of Solution

Given:

  $\begin{array}{l}\text{Number of times coin has been tossed}\left(n\right)=4040\\ \text{Number of times head obtained}\left(x\right)=2048\end{array}$

Formula used:

The formula to compute the test statistic is:

  $z=\frac{\widehat{p}-p_0}{\sqrt{\frac{p_0\left(1-p_0\right)}{n}}}$

Calculation:

The sample proportion is:

  $\begin{array}{c}\widehat{p}=\frac{x}{n}\\ =\frac{2048}{4040}\\ =0.507\end{array}$

Here, it is required to check whether the coin is fair or not. Mistake has been committed in stating the hypotheses. Thus, the appropriate null and alternative hypotheses are:

  $\begin{array}{l}{H}_0:p=0.5\\ H_0:p\ne 0.5\end{array}$

The correct test statistic can be computed as:

  $\begin{array}{c}z=\frac{\widehat{p}-p_0}{\sqrt{\frac{p_0\left(1-p_0\right)}{n}}}\\ =\frac{0.507-0.50}{\sqrt{\frac{0.50\left(0.50\right)}{4040}}}\\ =0.890\end{array}$

The p-value can be computed as:

  $\begin{array}{c}p-value=2\times P\left(Z>\left|z\right|\right)\\ =2\times P\left(Z>\left|0.890\right|\right)\\ =0.373\end{array}$

Here, p-value is 0.373. Here, p-value >level of significance.

There is a mistake in the decision rule. Thus, the decision is not to reject the null hypothesis.

Conclusion:

Thus, at 5% significance level there is sufficient evidence to conclude that the coin is fair.