A television show conducted an experiment to study what happens when buttered toast is dropped on the floor. When 51 buttered slices of toast were dropped, 30 of them landed with the buttered side up and 21 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 $$ 50 \% $$ 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 th 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. 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 $$ 50 \% $$ of the time. $$ H_{0}: p=0.5 $$ $$ H_{1}: p
eq 0.5 $$ (Type integers or decimals. Do not round.) Identify the test statistic. $$ z=\square $$ (Round to two decimal places as needed.)

Question

A television show conducted an experiment to study what happens when buttered toast is dropped on the floor. When 51 buttered slices of toast were dropped, 30 of them landed with the buttered side up and 21 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 $$ 50 \% $$ 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 th 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. 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 $$ 50 \% $$ of the time. $$ H_{0}: p=0.5 $$ $$ H_{1}: p 
eq 0.5 $$ (Type integers or decimals. Do not round.) Identify the test statistic. $$ z=\square $$ (Round to two decimal places as needed.)

UpStudy AI · Accepted Answer

**Step 1. Define the Hypotheses**

We let $$ p $$ be the population proportion of all buttered toast that lands with the buttered side down when dropped.

The hypotheses are:  
$$
H_{0}: p = 0.5
$$
$$
H_{1}: p \neq 0.5
$$

---

**Step 2. Calculate the Sample Proportion**

The number of toast slices dropped is $$ n = 51 $$. The number that landed buttered side down is $$ x = 21 $$.

The sample proportion is:
$$
\hat{p} = \frac{x}{n} = \frac{21}{51} \approx 0.4118
$$

---

**Step 3. Compute the Standard Error**

Under the null hypothesis ($$ p_0 = 0.5 $$), the standard error is:
$$
\text{SE} = \sqrt{\frac{p_0(1-p_0)}{n}} = \sqrt{\frac{0.5 \times 0.5}{51}} = \sqrt{\frac{0.25}{51}} \approx \sqrt{0.004902} \approx 0.0700
$$

---

**Step 4. Compute the Test Statistic**

The test statistic $$ z $$ is given by:
$$
z = \frac{\hat{p} - p_0}{\text{SE}} = \frac{0.4118 - 0.5}{0.0700} \approx \frac{-0.0882}{0.0700} \approx -1.26
$$

Thus, the test statistic is:
$$
z = -1.26
$$

---

**Conclusion (Addressing the Intent of the Experiment)**

For the two-sided test at the $$ \alpha = 0.01 $$ significance level, the $$ p $$-value corresponding to $$ z = -1.26 $$ would be compared against $$ 0.01 $$. Since the $$ p $$-value would be greater than $$ 0.01 $$, we fail to reject $$ H_{0} $$.

However, 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, the alternative hypothesis would be:
$$
H_{1}: p > 0.5
$$
In that case, because our sample proportion $$ \hat{p} \approx 0.4118 $$ is less than $$ 0.5 $$, the evidence does not support the claim that toast lands buttered side down more than $$ 50\% $$ of the time.

---

**Final Answer:**

Test statistic:  
$$
z = -1.26
$$
Test statistic: $$ z = -1.26 $$

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