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
-value
(Round to three decimal places as needed )

Ask by Daniel Davies. 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
-value
(Round to three decimal places as needed )

Ask by Daniel Davies. in the United States

Mar 21,2025

UpStudy AI · Accepted Answer

**Test Statistic Calculation:**

The sample proportion is 
$$
\hat{p}=\frac{25}{54}\approx0.463.
$$
The test statistic is calculated by
$$
z=\frac{\hat{p}-p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}
$$
with $$p_0=0.5$$ and $$n=54$$. Substituting,
$$
z=\frac{0.463-0.5}{\sqrt{\frac{0.5(0.5)}{54}}}=\frac{-0.037}{\sqrt{\frac{0.25}{54}}}\approx\frac{-0.037}{0.0681}\approx-0.54.
$$

**P-value Calculation (Two-Tailed Test):**

For a two-tailed test, we calculate the P–value as
$$
\text{P–value} = 2P(Z\leq -0.54).
$$
Looking up a standard normal table or using a calculator, 
$$
P(Z\leq -0.54)\approx0.294.
$$
Thus,
$$
\text{P–value} \approx 2(0.294)=0.588.
$$
Rounded to three decimal places, the P–value is 
$$
\boxed{0.588}.
$$

**Conclusion for the Claim (More Than 50% Case):**

If 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, we would have set up the hypotheses as:
$$
\begin{array}{l}
H_{0}: p=0.5 \
H_{1}: p>0.5.
\end{array}
$$
The test statistic remains $$z=-0.54$$. For a one–sided test the P–value would be
$$
\text{P–value} = P(Z\geq -0.54) \approx 0.705.
$$
Because this P–value is much larger than the significance level $$0.01$$, we fail to reject the null hypothesis.

**Final Statement:**

There is insufficient statistical evidence at the 0.01 significance level to support the claim that toast will land with the buttered side down more than $$50\%$$ of the time.
**Test Statistic and P-value:** - **Test Statistic (z):** -0.54 - **P-value:** 0.588 **Conclusion:** There is not enough evidence to support the claim that buttered toast lands with the buttered side down more than 50% of the time.

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