Math  /  Data & Statistics

QuestionMORE BENEFITS OF EATING ORGANIC
Using specific data, 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. In this case, we are testing H0:po=pcHa:po>pc\begin{array}{l} H_{0}: p_{o}=p_{c} \\ H_{a}: p_{o}>p_{c} \end{array} where pop_{o} and pcp_{c} 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. Use a 5%5 \% significance level.
Effect of Organic Soybeans After 5 Days After 5 days, the proportion of fruit flies eating organic soybeans still alive is 0.89 , while the proportion still alive eating conventional soybeans is 0.85 . The standard error for the difference in proportions is 0.022 .
What is the value of the test statistic? Round your answer to two decimal places. z=z=
What is the pp-value? Round your answer to three decimal places. pp-value == \square What is the conclusion? \square Is there evidence of a difference? \square

Studdy Solution

STEP 1

What is this asking? We're checking if fruit flies eating organic soybeans live longer than those eating conventional soybeans, using some fancy math called a hypothesis test! Watch out! Don't mix up the proportions for organic and conventional soybeans, and remember we're looking for a *greater than* effect, not just a difference.

STEP 2

1. Calculate the test statistic.
2. Calculate the p-value.
3. Draw a conclusion.

STEP 3

The **test statistic**, often called *z*, tells us how far apart our observations are from what we'd expect if there were *no* real difference between the organic and conventional diets.
It's like measuring how surprising our results are!
We calculate it using this formula: z=(p^op^c)0SE z = \frac{(\hat{p}_o - \hat{p}_c) - 0}{SE} where p^o\hat{p}_o is the **observed proportion** for the organic group, p^c\hat{p}_c is the **observed proportion** for the conventional group, and *SE* is the **standard error** of the difference.
Subtracting zero represents the null hypothesis of no difference.

STEP 4

Let's plug in our **values**: p^o=0.89\hat{p}_o = \textbf{0.89}, p^c=0.85\hat{p}_c = \textbf{0.85}, and SE=0.022SE = \textbf{0.022}. z=(0.890.85)00.022 z = \frac{(0.89 - 0.85) - 0}{0.022}

STEP 5

Time to crunch the numbers! z=0.040.0221.82 z = \frac{0.04}{0.022} \approx \textbf{1.82} So, our **test statistic** is approximately **1.82**.

STEP 6

The **p-value** tells us the probability of seeing a test statistic as extreme as ours (or even *more* extreme) if the null hypothesis were actually true (meaning, no real difference between the diets).
A small p-value makes us doubt the null hypothesis.

STEP 7

Since our alternative hypothesis is po>pcp_o > p_c, we're looking for the probability that *z* is greater than or equal to our calculated value of **1.82**.
We can use a *z*-table or calculator to find the area to the *right* of *z* = 1.82.
This gives us a **p-value** of approximately **0.035**.

STEP 8

Our **p-value** (**0.035**) is less than our **significance level** of **0.05**.
This means our result is statistically significant!

STEP 9

We **reject the null hypothesis**!
There *is* evidence to suggest that fruit flies eating organic soybeans have a higher survival rate after 5 days compared to those eating conventional soybeans.

STEP 10

Test statistic: z=1.82z = \textbf{1.82} p-value: 0.035\textbf{0.035} Conclusion: **Reject the null hypothesis**. Is there evidence of a difference? **Yes**

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord