Question

a television show conducted an experiment to study what happens when buttered toast is dropped on the floor. when 52 buttered slices of toast were dropped, 28 of them landed with the buttered side up and 24 landed with the buttered side down. use a 0.05 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.

h₁: p = 5
(type integers or decimals. do not round.)

identify the test statistic.

z = - 0.56
(round to two decimal places as needed.)

identify the p - value.

Explanation:

Step1: Calculate the sample proportion

The sample proportion \(\hat{p}\) of buttered - side - down toast is \(\hat{p}=\frac{24}{52}\approx0.4615\). The hypothesized proportion \(p = 0.5\) and \(n=52\).

Step2: Calculate the test statistic

The formula for the \(z\) - test statistic in a proportion test is \(z=\frac{\hat{p}-p}{\sqrt{\frac{p(1 - p)}{n}}}\).
Substitute \(\hat{p}=0.4615\), \(p = 0.5\), and \(n = 52\) into the formula:

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Step3: Calculate the P - value

Since this is a two - tailed test (the claim is \(p
eq0.5\)), the P - value is \(2P(Z\lt - 0.56)\).
Using the standard normal table, \(P(Z\lt - 0.56)=0.2877\). So the P - value \(=2\times0.2877 = 0.5754\)

Step4: Make a decision

Since the P - value \(=0.5754>0.05\) (the significance level \(\alpha = 0.05\)), we fail to reject the null hypothesis \(H_0:p = 0.5\)

Step5: Address the second claim (one - tailed)

The new claim is \(H_1:p>0.5\). The \(z\) - statistic is still \(z=-0.56\) (but for a one - tailed test \(H_1:p>0.5\), we consider \(P(Z> - 0.56)=1 - P(Z\lt - 0.56)\))
\(P(Z> - 0.56)=1 - 0.2877=0.7123>0.05\)

Answer:

• For the two - tailed test (\(H_0:p = 0.5\) vs \(H_1:p

eq0.5\)): Fail to reject \(H_0\). There is not sufficient evidence to reject the claim that the proportion of buttered - side - down toast is \(50\%\)

• For the one - tailed test (\(H_0:p = 0.5\) vs \(H_1:p>0.5\)): Fail to reject \(H_0\). There is not sufficient evidence to support the claim that toast will land with the buttered side down more than \(50\%\) of the time.