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)
Identify the test statistic.

(Round to two decimal places as needed.)
Identify the P-value

(Round to three decimal places as needed )
State the conclusion about the null hypothesis, as well as the final conclusion that addresses the original claim.
the null hypothesis. There sufficient evidence to the claim that buttered toast will land with the buttered side down of the time.

Ask by Weber Knight. 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)
Identify the test statistic.

(Round to two decimal places as needed.)
Identify the P-value

(Round to three decimal places as needed )
State the conclusion about the null hypothesis, as well as the final conclusion that addresses the original claim.
the null hypothesis. There sufficient evidence to the claim that buttered toast will land with the buttered side down of the time.

Ask by Weber Knight. in the United States

Mar 21,2025

UpStudy AI · Accepted Answer

**Step 1. Define the hypotheses.**

Let $$ p $$ denote the population proportion of buttered toast that will land with the buttered side down. To test the claim that toast lands with the buttered side down $$50\%$$ of the time, we set up the hypotheses as

$$
\begin{array}{l}
H_{0}: p = 0.5 \
H_{1}: p \neq 0.5
\end{array}
$$

**Step 2. Compute the test statistic.**

The sample size is $$ n = 54 $$ and the number of toast landing buttered side down is $$ X = 25 $$. Therefore, the sample proportion is

$$
\widehat{p} = \frac{25}{54} \approx 0.463.
$$

The standard error under $$ H_0 $$ is computed using

$$
\text{SE} = \sqrt{\frac{p_0 (1-p_0)}{n}} = \sqrt{\frac{0.5 \times 0.5}{54}} = \sqrt{\frac{0.25}{54}} \approx 0.0681.
$$

The test statistic is then

$$
z = \frac{\widehat{p} - p_0}{\text{SE}} = \frac{0.463 - 0.5}{0.0681} \approx -0.54.
$$

**Step 3. Find the P-value.**

Since the alternative hypothesis is two-sided, the P-value is given by

$$
\text{P-value} = 2 \times P(Z \leq -0.54).
$$

Looking up the standard normal distribution, we find

$$
P(Z \leq -0.54) \approx 0.294.
$$

Thus,

$$
\text{P-value} \approx 2 \times 0.294 = 0.588.
$$

**Step 4. Conclusion regarding the null hypothesis.**

With a significance level of $$\alpha = 0.01$$, we compare the P-value to $$\alpha$$:

$$
0.588 > 0.01.
$$

Since the P-value is greater than the significance level, we do not have sufficient evidence to reject the null hypothesis.  
 
Fill in the blanks:  
$$
\textbf{Fail to reject} \quad \text{the null hypothesis. There is insufficient evidence to support the claim that buttered toast will land with the buttered side down }50\%\text{ of the time.}
$$

**Step 5. Addressing the experiment’s main intent (more than $$50\%$$ landing buttered side down).**

If the intent of the experiment were to assess the claim that toast lands with the buttered side down more than $$50\%$$ of the time, the alternative hypothesis would be:

$$
H_{1}: p > 0.5.
$$

However, since our observed sample proportion $$\widehat{p} \approx 0.463$$ is below $$0.5$$ and our test statistic is $$ z \approx -0.54 $$, there is no evidence in favor of $$ p > 0.5 $$. In fact, this outcome suggests that the buttered side down is occurring less than $$50\%$$ of the time.

Thus, addressing the experiment's intent: There is insufficient evidence to conclude that buttered toast will land with the buttered side down more than $$50\%$$ of the time.
**Hypotheses:** - $$ H_{0}: p = 0.5 $$ - $$ H_{1}: p \neq 0.5 $$ **Test Statistic:** $$ z = -0.54 $$ **P-value:** 0.588 **Conclusion:** Fail to reject the null hypothesis. There is insufficient evidence to support the claim that buttered toast will land with the buttered side down 50% of the time. **Regarding the Experiment’s Intent:** There is insufficient evidence to conclude that buttered toast will land with the buttered side down more than 50% of the time.

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