A television show conducted an experiment to study what happens when buttered toast is dropped on the floor. Whien 48 buttered stices of toast were dropped, 30 of them landed with the buttered side up and 48 landed with the buttered side down. Use a 0.01 significance level to test the claim that toast will land with the buttered side down \( 50 \% \) of the lime Use the \( P \)-value method Use the normal distribution as an approximation to the binomial distribution. After that, supposing the intent of the experiment was to assess the claim that toast will fand with the buttered side down more than \( 50 \% \) of fie lime, write a condusion that addresses the intent of the experiment. Let p denote the pepulation proportion of ail buttered toast that will land with the buttered side down when dropped identify ihe nuil and atternative hypotheses to test the claim that buttered toast will land with the buttered side down 50\% of the lime. \[ \begin{array}{l} H_{0} \cdot \rho=05 \\ H_{1} \rho \neq 05 \end{array} \] (Type integers or decimals. Do not round) Identify the test statistic. \[ z=\square \] (Round to two decimal places as needed.)

**Step 1. State the Hypotheses** Let \( p \) denote the population proportion of all buttered toast that will land with the buttered side down when dropped. To test the claim that toast will land with the buttered side down \( 50\% \) of the time, we set up the hypotheses as \[ \begin{aligned} H_0 &: p = 0.5, \\ H_1 &: p \neq 0.5. \end{aligned} \] **Step 2. Gather the Sample Information** According to the experiment, a total of 78 toast slices were observed (since 30 landed buttered side up and 48 landed buttered side down). Here, the number of successes (landing buttered side down) is \[ x = 48, \quad n = 78. \] Thus, the sample proportion is \[ \hat{p} = \frac{48}{78} \approx 0.6154. \] **Step 3. Identify the Test Statistic** Using the normal approximation to the binomial distribution, the test statistic \( z \) is given by \[ z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}} \] where \( p_0 = 0.5 \). Substituting the values: \[ z = \frac{0.6154 - 0.5}{\sqrt{\frac{0.5 \times 0.5}{78}}}. \] First, compute the standard error: \[ \text{Standard error} = \sqrt{\frac{0.25}{78}} \approx \sqrt{0.003205} \approx 0.0566. \] Then, the \( z \)-statistic is \[ z = \frac{0.1154}{0.0566} \approx 2.04. \] **Step 4. Conclusion for the Two-Sided Test** Using a significance level of \(\alpha = 0.01\), we compare the \(P\)-value corresponding to \( z = 2.04 \) with \(\alpha\). For a two-sided test, the \(P\)-value is \[ P\text{-value} = 2 \times \Pr(Z > 2.04). \] Since typically \(\Pr(Z > 2.04) \approx 0.0207\), we have \[ P\text{-value} \approx 2 \times 0.0207 = 0.0414. \] Because \( 0.0414 > 0.01 \), we fail to reject \( H_0 \) at the 0.01 significance level. Thus, there is insufficient evidence to conclude that the proportion of toast landing with the buttered side down differs from \(50\%\). **Step 5. Addressing the Alternate Experimental Intent** Suppose the intent of the experiment was instead to assess the claim that toast will land with the buttered side down more than \(50\%\) of the time. In that case, the hypotheses would be \[ \begin{aligned} H_0 &: p = 0.5, \\ H_1 &: p > 0.5. \end{aligned} \] The test statistic remains \[ z \approx 2.04. \] For the one-sided test, the \(P\)-value is \[ P\text{-value} = \Pr(Z > 2.04) \approx 0.0207. \] Since \( 0.0207 > 0.01 \), even in this one-tailed test we fail to reject \(H_0\). **Step 6. Final Conclusion** Whether testing the claim two-sided (\(p \neq 0.5\)) or one-sided (\(p > 0.5\)) at the \(\alpha = 0.01\) significance level, the \(P\)-value exceeds \(0.01\). Therefore, we do not have sufficient evidence to support the claim that toast will land with the buttered side down \( 50\% \) of the time or more specifically, more than \(50\%\) of the time. **Test Statistic** \[ z \approx 2.04. \]