Step 1 :Given that the sample proportion \(\hat{p} = \frac{31}{49}\), the hypothesized population proportion \(p_0 = 0.5\), and the sample size \(n = 49\).
Step 2 :The test statistic for a proportion is calculated using the formula: \[z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}\]
Step 3 :Substituting the given values into the formula, we get \(z = 1.8571428571428577\).
Step 4 :Final Answer: The test statistic is \(\boxed{1.86}\).
Step 5 :The P-value is the probability that a random variable is more extreme than the observed data, assuming the null hypothesis is true. In this case, we are dealing with a two-tailed test, so we need to find the probability that a standard normal random variable is greater than the absolute value of the test statistic.
Step 6 :Calculating the P-value, we get \(p_{value} = 0.0632908322333452\).
Step 7 :Final Answer: The P-value is \(\boxed{0.063}\).
Step 8 :The null hypothesis in this case is that the toast will land with the buttered side down 50% of the time. We reject the null hypothesis if the P-value is less than the significance level. In this case, the significance level is 0.01.
Step 9 :Since the P-value is greater than the significance level, we do not reject the null hypothesis.
Step 10 :Final Answer: There is not sufficient evidence to reject the claim that buttered toast will land with the buttered side down $50 \%$ of the time.
Step 11 :Since we did not reject the null hypothesis, there is not sufficient evidence to support the claim that the toast will land with the buttered side down more than 50% of the time.
Step 12 :Final Answer: Based on these results, there is not sufficient evidence to support the claim that buttered toast will land with the buttered side down more than $50 \%$ of the time.