Question

a television show conducted an experiment to study what happens when buttered toast is dropped on the floor. when 47 buttered slices of toast were dropped, 25 of them landed with the buttered side up and 22 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.
state the conclusion about the null hypothesis, as well as the final conclusion that addresses the original claim.
fail to reject the null hypothesis. there is not sufficient evidence to warrant rejection of the claim that buttered toast will land with the buttered side down 50% of the time.
write a conclusion that addresses the test that the toast will land with the buttered side down more than 50% of the time.
the intent of the experiment was to test ( h_0:p) and ( h_1:p). based on these results, there sufficient evidence to support of the claim that buttered toast will land with the buttered side down more than 50% of the time.
(type integers or decimals. do not round.)

Explanation:

Step1: State the hypotheses

The null hypothesis \(H_0:p = 0.5\) (claim that buttered toast will land with the buttered side down \(50\%\) of the time). The alternative hypothesis \(H_1:p>0.5\) (claim that buttered toast will land with the buttered side down more than \(50\%\) of the time).

Step2: Calculate the sample proportion \(\hat{p}\)

\(\hat{p}=\frac{x}{n}\), where \(x = 22\) (number of buttered - side - down landings) and \(n=47\). So \(\hat{p}=\frac{22}{47}\approx0.468\).

Step3: Calculate the test statistic \(z\)

The formula for the test statistic in a one - sample proportion test is \(z=\frac{\hat{p}-p}{\sqrt{\frac{p(1 - p)}{n}}}\). Substituting \(p = 0.5\), \(\hat{p}=0.468\), and \(n = 47\):

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Step4: Calculate the P - value

For a right - tailed test (\(H_1:p>0.5\)), the P - value is \(P(Z>z)\). Since \(z=-0.44\), \(P(Z > - 0.44)=1 - P(Z\leq - 0.44)\). From the standard normal table, \(P(Z\leq - 0.44)=0.3300\), so \(P - value=1 - 0.3300 = 0.6700\).

Step5: Make a decision

Since the P - value (\(0.6700\))>significance level (\(\alpha = 0.05\)), we fail to reject \(H_0\).

Answer:

The intent of the experiment was to test \(H_0:p = 0.5\) and \(H_1:p>0.5\). 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.