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. identify the test statistic. z = - 0.56 (round to two decimal places as needed.) identify the p - value. p - value = (round to three decimal places as needed.)

Explanation:

Step1: Determine the type of test

Since the claim is about a proportion (\(p = 0.5\)) and we are using the normal approximation to the binomial, and the test statistic \(z=- 0.56\), we need to find the P - value for a two - tailed test first (because the original claim is \(p = 0.5\), no direction is specified in the first part of the problem). The formula for the P - value of a two - tailed \(z\) test is \(P - value=2\times(1 - \Phi(|z|))\), where \(\Phi(z)\) is the cumulative distribution function of the standard normal distribution.

Step2: Calculate the P - value for the two - tailed test

We know that \(z=-0.56\), so \(|z| = 0.56\). Using a standard normal table or a calculator with a normal distribution function (e.g., in Excel: \(=2*(1 - NORM.S.DIST(0.56,TRUE))\)), we have:
\(P - value=2\times(1 - 0.7123)=2\times0.2877 = 0.5754\approx0.575\)

If we consider the second part (claim \(p>0.5\), one - tailed test), the P - value is \(1-\Phi(|z|)\) (because \(z=-0.56\) and for \(H_1:p > 0.5\), we look at the right - tail). Using the standard normal table or calculator (\(=1 - NORM.S.DIST(0.56,TRUE)\)), \(P - value=0.2877\approx0.288\)

Answer:

For the two - tailed test (claim \(p = 0.5\)): \(P - value = 0.575\)
For the one - tailed test (claim \(p>0.5\)): \(P - value=0.288\)