A television show conducted an experiment to study what happens when buttered toast is dropped on the floor. When 54 buttered slices of toast were dropped, 29 of them landed with the buttered side up and 25
landed with the buttered side down. Use a 0.01 significance level to test the claim that toast will land with the buttered side down 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 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 of the time.

(Type integers or decimals. Do not round.)

Ask by Knight King. in the United States

Mar 21,2025

Question

A television show conducted an experiment to study what happens when buttered toast is dropped on the floor. When 54 buttered slices of toast were dropped, 29 of them landed with the buttered side up and 25
landed with the buttered side down. Use a 0.01 significance level to test the claim that toast will land with the buttered side down 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 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 of the time.

(Type integers or decimals. Do not round.)

Ask by Knight King. in the United States

Mar 21,2025

UpStudy AI · Accepted Answer

To test the claim that buttered toast will land with the buttered side down $$ 50\% $$ of the time, we need to set up our null and alternative hypotheses. 1. **Null Hypothesis ($$ H_0 $$)**: The proportion of toast landing buttered side down is $$ 0.5 $$. $$ H_0: p = 0.5 $$ 2. **Alternative Hypothesis ($$ H_a $$)**: The proportion of toast landing buttered side down is greater than $$ 0.5 $$. $$ H_a: p > 0.5 $$ Next, we will calculate the test statistic and the P-value using the normal approximation to the binomial distribution. ### Step 1: Calculate the sample proportion Let $$ X $$ be the number of toast slices that landed buttered side down. From the problem, we have: - $$ n = 54 $$ (total slices) - $$ X = 25 $$ (slices landed buttered side down) The sample proportion $$ \hat{p} $$ is calculated as: $$ \hat{p} = \frac{X}{n} = \frac{25}{54} $$ ### Step 2: Calculate the standard error The standard error (SE) for the proportion is given by: $$ SE = \sqrt{\frac{p_0(1 - p_0)}{n}} $$ where $$ p_0 = 0.5 $$. ### Step 3: Calculate the test statistic The test statistic $$ z $$ is calculated using: $$ z = \frac{\hat{p} - p_0}{SE} $$ ### Step 4: Calculate the P-value The P-value can be found using the standard normal distribution. Now, let's perform the calculations step by step. First, we will calculate $$ \hat{p} $$, $$ SE $$, and then $$ z $$. Calculate the value by following steps: - step0: Calculate: $$\frac{25}{54}$$ Calculate or simplify the expression $$ \sqrt((0.5*(1-0.5))/54) $$. Calculate the value by following steps: - step0: Calculate: $$\sqrt{\frac{\left(0.5\left(1-0.5\right)\right)}{54}}$$ - step1: Remove the parentheses: $$\sqrt{\frac{0.5\left(1-0.5\right)}{54}}$$ - step2: Subtract the numbers: $$\sqrt{\frac{0.5\times 0.5}{54}}$$ - step3: Multiply the numbers: $$\sqrt{\frac{0.25}{54}}$$ - step4: Divide the terms: $$\sqrt{\frac{1}{216}}$$ - step5: Use the properties of radicals: $$\frac{\sqrt{1}}{\sqrt{216}}$$ - step6: Simplify the expression: $$\frac{1}{6\sqrt{6}}$$ - step7: Multiply by the Conjugate: $$\frac{\sqrt{6}}{6\sqrt{6}\times \sqrt{6}}$$ - step8: Multiply the numbers: $$\frac{\sqrt{6}}{36}$$ ### Step 1: Calculate the sample proportion The sample proportion $$ \hat{p} $$ is: $$ \hat{p} = \frac{25}{54} \approx 0.46296 $$ ### Step 2: Calculate the standard error The standard error (SE) is: $$ SE = \sqrt{\frac{0.5(1 - 0.5)}{54}} \approx 0.068041 $$ ### Step 3: Calculate the test statistic Now we can calculate the test statistic $$ z $$: $$ z = \frac{\hat{p} - p_0}{SE} = \frac{0.46296 - 0.5}{0.068041} $$ Let's calculate $$ z $$. Calculate the value by following steps: - step0: Calculate: $$\frac{\left(0.46296-0.5\right)}{0.068041}$$ - step1: Remove the parentheses: $$\frac{0.46296-0.5}{0.068041}$$ - step2: Subtract the numbers: $$\frac{-0.03704}{0.068041}$$ - step3: Convert the expressions: $$\frac{-\frac{463}{12500}}{\frac{68041}{1000000}}$$ - step4: Multiply by the reciprocal: $$-\frac{463}{12500}\times \frac{1000000}{68041}$$ - step5: Reduce the numbers: $$-463\times \frac{80}{68041}$$ - step6: Multiply: $$-\frac{463\times 80}{68041}$$ - step7: Multiply: $$-\frac{37040}{68041}$$ ### Step 3: Calculate the test statistic The test statistic $$ z $$ is approximately: $$ z \approx -0.544378 $$ ### Step 4: Calculate the P-value Since we are conducting a one-tailed test (testing if the proportion is greater than $$ 0.5 $$), we will find the P-value corresponding to the calculated $$ z $$ value. Now, let's calculate the P-value using the standard normal distribution. The P-value is given by: $$ P(Z > z) $$ Let's calculate the P-value for $$ z \approx -0.544378 $$. It seems I cannot directly calculate the P-value from the z-score. However, I can explain how to find it. To find the P-value for $$ z \approx -0.544378 $$: 1. Use a standard normal distribution table or a calculator to find $$ P(Z < -0.544378) $$. 2. Since we are interested in $$ P(Z > -0.544378) $$, we can use the complement: $$ P(Z > -0.544378) = 1 - P(Z < -0.544378) $$ Given that $$ z $$ is negative, we can expect the P-value to be greater than $$ 0.5 $$. ### Conclusion 1. If the P-value is greater than $$ 0.01 $$, we fail to reject the null hypothesis $$ H_0 $$. 2. If the P-value is less than or equal to $$ 0.01 $$, we reject the null hypothesis. Since we are testing the claim that toast will land with the buttered side down more than $$ 50\% $$ of the time, if we fail to reject $$ H_0 $$, it suggests that there is not enough evidence to support the claim. Now, let's summarize the findings: - **Null Hypothesis**: $$ H_0: p = 0.5 $$ - **Alternative Hypothesis**: $$ H_a: p > 0.5 $$ - **Test Statistic**: $$ z \approx -0.544378 $$ - **P-value**: (to be determined from a standard normal distribution table) Based on the calculated $$ z $$ value, we can conclude that there is insufficient evidence to support the claim that buttered toast lands buttered side down more than $$ 50\% $$ of the time at the $$ 0.01 $$ significance level. **Null Hypothesis ($$ H_0 $$)**: The proportion of buttered toast landing buttered side down is $$ 0.5 $$. **Alternative Hypothesis ($$ H_a $$)**: The proportion of buttered toast landing buttered side down is greater than $$ 0.5 $$.

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