Question

Metal Tags on Penguins and Survival Data 1.3 on page 10 discusses a study designed to test whether applying metal tags is detrimental to penguins. One variable examined is the survival rate 10 years after tagging. The scientists observed that 10 of the 50 metal tagged penguins survived, compared to 18 of the 50 electronic tagged penguins. Construct a \(90 \%\) confidence interval for the difference in proportion surviving between the metal and electronic tagged penguins \(\left(p_{M}-p_{E}\right)\). Interpret the result.

Short answer

The \(90 \%\) confidence interval for the difference in survival proportions between metal-tagged and electronic-tagged penguins is \([-0.30945, -0.05054]\).

Step-by-step solution

Calculate the Proportions

First, calculate the proportions of surviving penguins for both groups. This can be done by dividing the number of survivors in each group by the total number of individuals in that group: \n\[\hat{p_{M}} = \frac{10}{50} = 0.2\] and\n\[\hat{p_{E}} = \frac{18}{50} = 0.36\]

Compute the Standard Error

Next, calculate the standard error using the following formula: \n\[\sqrt{\frac{\hat{p_{M}} \cdot (1-\hat{p_{M}})}{n_{M}} + \frac{\hat{p_{E}} \cdot (1-\hat{p_{E}})}{n_{E}}} = \sqrt{\frac{0.2 \cdot 0.8}{50} + \frac{0.36 \cdot 0.64}{50}} = 0.09173...\]

Determine the Critical Value

Then find the critical value from the standard normal distribution that corresponds to the midpoint of the desired \(90 \%\) confidence interval. The critical value \( z_{\alpha / 2} = 1.645\) for a \(90 \%\) confidence interval.

Construct the Confidence Interval

Finally, construct the difference's confidence interval by adding and subtracting the critical value times the standard error from the difference in survival rates: \n\[\hat{p_{M}} - \hat{p_{E}} \pm z_{\alpha /2} * SE = 0.2 - 0.36 \pm 1.645 * 0.09173 = [-0.30945, -0.05054]\]

Interpret the Results

Interpretation of the confidence interval says that with \(90 \%\) confidence, the true difference in survival proportion between metal-tagged penguins and electronic-tagged penguins lies between \(-0.30945\) and \(-0.05054\). This interval does not include zero, suggesting that there may be a real difference in survival rates between the two tagging groups, with the electronic-tagged penguins surviving at a higher rate.