Question

Using Data 5.1 on page \(375,\) 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. ask you to conduct a hypothesis test using additional data from this study. \(^{40}\) In every case, we are testing $$\begin{array}{ll}H_{0}: & p_{o}=p_{c} \\\H_{a}: & p_{o}>p_{c}\end{array}$$ where \(p_{o}\) and \(p_{c}\) 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. Also, in every case, we have \(n_{1}=n_{2}=500 .\) Show all remaining details in the test, using a \(5 \%\) significance level. Effect of Organic Potatoes after 20 Days After 20 days, 250 of the 500 fruit flies eating organic potatoes are still alive, while 130 of the 500 eating conventional potatoes are still alive.

Short answer

Using a z-test for proportions, you can determine whether there is a significant difference between proportions of fruit flies surviving on organic potatoes and those surviving on conventional potatoes. In this case, with \(p_{o}\) being 0.5 and \(p_{c}\) being 0.26, you will see that the proportion of fruit flies surviving on organic potatoes is significantly greater than that of those surviving on conventional potatoes at a 5% significance level.

Step-by-step solution

Formulate the hypothesis

The null hypothesis \(H_{0}\) is: \(p_{o} = p_{c}\), where \(p_{o}\) is the proportion of fruit flies alive after eating organic food and \(p_{c}\) is the proportion of fruit flies alive after eating conventional food. The alternate hypothesis, \(H_{a}\), is \(p_{o} > p_{c}\).

Calculate the proportions

After 20 days, 250 out of 500 fruit flies that ate organic potatoes are still alive, so the proportion, \(p_{o}\), is 250/500 = 0.5. For the fruit flies that ate conventional potatoes, 130 out of 500 are still alive, so the proportion, \(p_{c}\), is 130/500 = 0.26.

Conduct the hypothesis test

We want to verify whether the proportion of fruit flies surviving on organic potatoes is significantly greater than that of those surviving on conventional potatoes. To conduct this test, we will use a z-test for proportions because we have large sample sizes (n > 30). This will test if there is a significant difference between the proportions. If \(p_{o}\) is significantly greater than \(p_{c}\), we can reject the null hypothesis at the 5% significance level.