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The pesticide diazinon is commonly used to treat infestations of the German cockroach, *Blattella germanica*. A study investigated the persistence of this pesticide on various types of surfaces. Researchers applied a 0.5% emulsion of diazinon to glass and plasterboard. After 14 days, they randomly assigned 90 cockroaches to two groups of 45, placed one group on each surface, and recorded the number that died within 48 hours.

On glass, 21 cockroaches died, while on plasterboard, 30 died. We wish to determine if this is convincing evidence that the proportion of all cockroaches that will die when exposed to diazinon on plasterboard (\( p_p \)) is greater than the proportion of all cockroaches that will die when exposed to diazinon on glass (\( p_g \)).

Answer :

The researchers conducted a study to determine if there is convincing evidence that the proportion of all cockroaches that will die when exposed to diazinon on plasterboard (pp) is greater than the proportion that will die when exposed to diazinon on glass (pg). They applied a 0.5% emulsion of diazinon to glass and plasterboard surfaces and recorded the number of cockroaches that died within 48 hours. On glass, 21 out of 45 cockroaches died, while on plasterboard, 30 out of 45 cockroaches died.

To analyze the data, we can use a hypothesis test. Our null hypothesis, H0, is that there is no difference in the proportion of cockroaches that will die on plasterboard and glass surfaces (pp = pg). The alternative hypothesis, Ha, is that the proportion on plasterboard is greater than on glass (pp > pg).

We can calculate the test statistic using the formula:
z = (phatp - phatg) / sqrt(phat(1-phat)(1/n1 + 1/n2))
where phatp and phatg are the sample proportions, and n1 and n2 are the sample sizes.

Based on the given data, the test statistic is approximately 1.39.

To determine if this is convincing evidence, we compare the test statistic to the critical value at a chosen significance level (e.g., α = 0.05). If the test statistic is greater than the critical value, we reject the null hypothesis and conclude that there is convincing evidence that the proportion on plasterboard is greater than on glass.

Since the test statistic is not provided, we cannot determine if there is convincing evidence based on this information alone. Additionally, it would be helpful to know the critical value at α = 0.05 and the sample sizes (n1 and n2) to perform the hypothesis test accurately.

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