Questions: A television show conducted an experiment to study what happens when buttered toast is dropped on the floor. When 43 buttered slices of toast were dropped, 24 of them landed with the buttered side up and 19 landed with the buttered side down. Use a 0.10 significance level to test the claim that toast will land with the buttered side down 50% of the time. 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 land with the buttered side down more than 50% of the time, write a conclusion that addresses the intent of the experiment. Let p denote the population proportion of all buttered toast that will land with the buttered side down when dropped. Identify the null and alternative hypotheses to test the claim that buttered toast will land with the buttered side down 50% of the time. H0: p = 0.50 H1: p ≠ 0.50

Solution Steps

We define the null and alternative hypotheses as follows: \[ H_0: p = 0.5 \quad \text{(the proportion of toast landing buttered side down is 50\%)} \] \[ H_1: p \neq 0.5 \quad \text{(the proportion of toast landing buttered side down is not 50\%)} \]

The test statistic \(Z\) is calculated using the formula: \[ Z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}} \] Substituting the values: \[ \hat{p} = \frac{19}{43} \approx 0.4419, \quad p_0 = 0.5, \quad n = 43 \] \[ Z = \frac{0.4419 - 0.5}{\sqrt{\frac{0.5(1 - 0.5)}{43}}} \approx -0.7625 \]

The P-value associated with the test statistic \(Z = -0.7625\) is: \[ \text{P-value} = 0.4458 \]

For a significance level of \(\alpha = 0.10\) in a two-tailed test, the critical regions are: \[ Z < -1.6449 \quad \text{or} \quad Z > 1.6449 \]

Since the test statistic \(Z = -0.7625\) does not fall into the critical region, we fail to reject the null hypothesis: \[ \text{Fail to reject } H_0. \text{ There is not sufficient evidence to support the claim that toast will land with the buttered side down more than 50\% of the time.} \]

The intent of the experiment was to assess the claim that toast will land with the buttered side down more than 50% of the time. For this, we perform a one-tailed test (right-tailed).

Using the same test statistic formula for the one-tailed test: \[ Z = -0.7625 \]

The P-value for the one-tailed test is: \[ \text{P-value (Intent)} = 0.7771 \]

For a significance level of \(\alpha = 0.10\) in a right-tailed test, the critical region is: \[ Z > 1.2816 \]

Since the test statistic \(Z = -0.7625\) does not exceed the critical value, we again fail to reject the null hypothesis: \[ \text{Fail to reject } H_0. \text{ There is not sufficient evidence to support the claim that toast will land with the buttered side down more than 50\% of the time.} \]

Final Answer