levision show conducted an experiment to study what happens when buttered toast is dropped on the floor. When 48 buttered slices of toast were dropped, 30 of them landed with the buttered side up and 18 led 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 roximation 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 addresses the intent of the experiment.
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 tered side down of the time.

pe integers or decimals Do not round.)
entify the test statistic

ound to two decimal places as needed.)

Ask by Watson Wheeler. in the United States

Mar 21,2025

Question

levision show conducted an experiment to study what happens when buttered toast is dropped on the floor. When 48 buttered slices of toast were dropped, 30 of them landed with the buttered side up and 18 led 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 roximation 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 addresses the intent of the experiment.
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 tered side down of the time.

pe integers or decimals Do not round.)
entify the test statistic

ound to two decimal places as needed.)

Ask by Watson Wheeler. in the United States

Mar 21,2025

UpStudy AI · Accepted Answer

**Step 1. Formulate the hypotheses** Let $$ p $$ be the population proportion of all buttered toast that will land with the buttered side down when dropped. For testing the claim that toast lands with the buttered side down $$ 50\% $$ of the time: $$ \text{Null hypothesis: } H_{0}: p = 0.5 \ \text{Alternative hypothesis: } H_{a}: p \neq 0.5 $$ When the intent is to assess the claim that toast will land with the buttered side down more than $$ 50\% $$ of the time, the alternative hypothesis becomes: $$ H_{a}: p > 0.5 $$ --- **Step 2. Calculate the sample proportion** Total slices dropped: $$ n = 48 $$ Number of slices landing buttered side down: $$ x = 18 $$ The sample proportion is: $$ \hat{p} = \frac{x}{n} = \frac{18}{48} = 0.375 $$ --- **Step 3. Identify the test statistic** We use the normal approximation to the binomial distribution. The test statistic is given by: $$ z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0 (1-p_0)}{n}}} $$ Here, $$ p_0 = 0.5 $$. First, compute the standard error: $$ \text{SE} = \sqrt{\frac{0.5 \times 0.5}{48}} = \sqrt{\frac{0.25}{48}} = \sqrt{0.0052083} \approx 0.07217 $$ Now, compute the $$ z $$-statistic: $$ z = \frac{0.375 - 0.5}{0.07217} = \frac{-0.125}{0.07217} \approx -1.73 $$ --- **Step 4. Determine the P-value and decision** *For the two-tailed test (claiming $$ p = 0.5 $$):* The p-value is computed as: $$ \text{p-value} = 2 \times P(Z < -1.73) $$ Using standard normal tables or a calculator, $$ P(Z < -1.73) \approx 0.0418, $$ so $$ \text{p-value} \approx 2 \times 0.0418 = 0.0836. $$ Since $$ 0.0836 > 0.01 $$ (the significance level), we do not reject the null hypothesis. There is insufficient evidence to conclude that the proportion is different from $$ 50\% $$. *For the one-tailed test (claiming $$ p > 0.5 $$):* In this case, the p-value is: $$ \text{p-value} = P(Z \geq -1.73) $$ Since $$ P(Z \geq -1.73) = 1 - P(Z < -1.73) \approx 1 - 0.0418 = 0.9582, $$ again $$ 0.9582 > 0.01 $$ so we do not reject the null hypothesis. The data do not support the claim that $$ p > 0.5 $$. --- **Step 5. Conclusion addressing the experiment’s intent** - When testing whether the proportion of buttered toast landing buttered side down is $$ 50\% $$, there is not enough evidence at the $$ \alpha = 0.01 $$ level to reject $$ H_0: p = 0.5 $$. - Supposing the experiment was intended to assess the claim that toast lands buttered side down more than $$ 50\% $$ of the time, the one-tailed test also fails to provide sufficient evidence to support the claim (with a p-value of approximately $$ 0.9582 $$). Thus, we conclude that there is no significant evidence at the $$ \alpha = 0.01 $$ level to claim that buttered toast lands with the buttered side down more than $$ 50\% $$ of the time. **Conclusion:** - **Two-Tailed Test:** There is not enough evidence to conclude that the proportion of buttered toast landing buttered side down is different from $$ 50\% $$ at the $$ 0.01 $$ significance level. - **One-Tailed Test:** There is not enough evidence to support the claim that buttered toast lands buttered side down more than $$ 50\% $$ of the time at the $$ 0.01 $$ significance level. Therefore, based on the experiment, we do not have sufficient evidence to support the claims regarding the proportion of buttered toast landing with the buttered side down.

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