MORE BENEFITS OF EATING ORGANIC

Using data from a study, we find a significant difference in the proportion of fruit flies surviving after 13 days between those eating organic potatoes and those eating conventional (not organic) potatoes. This exercise asks you to conduct a hypothesis test using additional data from this study. In this case, we are testing

where and represent the proportion of fruit flies alive at the end of the given time frame of those eating organic food and those eating conventional food, respectively. We have . Show all remaining details in the test, using a significance level.

Effect of Organic Raisins After 20 Days
After 20 days, 275 of the 500 fruit flies eating organic raisins are still alive, while 170 of the 500 eating conventional raisins are still alive.
Proportions approximated from information given in the paper.

Part 1

Your answer is partially correct.

Give the test statistic and the -value.

Round your answers to three decimal places.

Test statistic

Ask by Ayala Greene. in the United States

Mar 28,2025

Question

MORE BENEFITS OF EATING ORGANIC

Using data from a study, we find a significant difference in the proportion of fruit flies surviving after 13 days between those eating organic potatoes and those eating conventional (not organic) potatoes. This exercise asks you to conduct a hypothesis test using additional data from this study. In this case, we are testing

where and represent the proportion of fruit flies alive at the end of the given time frame of those eating organic food and those eating conventional food, respectively. We have . Show all remaining details in the test, using a significance level.

Effect of Organic Raisins After 20 Days
After 20 days, 275 of the 500 fruit flies eating organic raisins are still alive, while 170 of the 500 eating conventional raisins are still alive.
Proportions approximated from information given in the paper.

Part 1

Your answer is partially correct.

Give the test statistic and the -value.

Round your answers to three decimal places.

Test statistic

Ask by Ayala Greene. in the United States

Mar 28,2025

UpStudy AI · Accepted Answer

**Step 1. Compute the Sample Proportions**

The sample proportions are:  
$$
\hat{p}_o = \frac{275}{500} = 0.55 \quad \text{and} \quad \hat{p}_c = \frac{170}{500} = 0.34.
$$

**Step 2. Compute the Pooled Proportion**

Under $$H_0: p_o = p_c$$, the pooled proportion is:  
$$
\hat{p} = \frac{275 + 170}{500 + 500} = \frac{445}{1000} = 0.445.
$$

**Step 3. Compute the Standard Error**

The standard error for the difference in proportions is:  
$$
SE = \sqrt{\hat{p}(1-\hat{p})\left(\frac{1}{n_o} + \frac{1}{n_c}\right)}.
$$  
Substituting the values:  
$$
SE = \sqrt{0.445 \times 0.555 \times \left(\frac{1}{500} + \frac{1}{500}\right)} = \sqrt{0.445 \times 0.555 \times \frac{2}{500}}.
$$  
First, compute the product in the square root:  
$$
0.445 \times 0.555 \approx 0.246975.
$$  
Then,  
$$
0.246975 \times \frac{2}{500} = 0.246975 \times 0.004 = 0.0009879.
$$  
Finally,  
$$
SE \approx \sqrt{0.0009879} \approx 0.031.
$$

**Step 4. Compute the Test Statistic**

The test statistic is computed by:  
$$
z = \frac{\hat{p}_o - \hat{p}_c}{SE} = \frac{0.55 - 0.34}{0.031} = \frac{0.21}{0.031} \approx 6.774.
$$  
Rounding to three decimal places, we write:  
$$
z \approx 6.774.
$$

**Step 5. Compute the $$p$$-Value**

Since the alternative hypothesis is $$H_a: p_o > p_c$$, this is a one-tailed test. The $$p$$-value is:  
$$
p\text{-value} = P(Z > 6.774).
$$
A $$z$$-score of $$6.774$$ has a probability that is effectively 0. Thus,  
$$
p\text{-value} \approx 0.000.
$$

**Final Answer**

Test statistic $$ = 6.774 $$ and $$ p $$-value $$ \approx 0.000 $$.
Test statistic $$ = 6.774 $$ and $$ p $$-value $$ \approx 0.000 $$.

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