Question

Metal Tags on Penguins and Breeding Success Data 1.3 on page 10 discusses a study designed to test whether applying metal tags is detrimental to penguins. Exercise 6.148 investigates the survival rate of the penguins. The scientists also studied the breeding success of the metal- and electronic-tagged penguins. Metal-tagged penguins successfully produced offspring in \(32 \%\) of the 122 total breeding seasons, while the electronic-tagged penguins succeeded in \(44 \%\) of the 160 total breeding seasons. Construct a \(95 \%\) confidence interval for the difference in proportion successfully producing offspring \(\left(p_{M}-p_{E}\right)\). Interpret the result.

Short answer

The \(95\%\) confidence interval for the difference in proportion successfully producing offspring between metal-tagged and electronic-tagged penguins is \((-0.2303, -0.0097)\). This indicates that there is a meaningful difference in breeding success rates between the two groups, with electronic-tagged penguins having a higher success rate. So, the study provides evidence that metal tagging may be detrimental to penguin breeding success.

Step-by-step solution

Identify the Sample Proportions and Sizes

First, identify the sample proportions and sizes. For metal-tagged penguins, there were 122 breeding seasons and a success rate of \(32\% = 0.32\), giving a sample proportion \(p_M\) of \(0.32\). This gives a number of successes \(x_M = 0.32 * 122 = 39.04\). Similarly, for electronic-tagged penguins, there were 160 breeding seasons and a success rate of \(44\% = 0.44\), giving a sample proportion \(p_E\) of \(0.44\) and a number of successes \(x_E = 0.44 * 160 = 70.4\).

Calculate the Difference in Sample Proportions

Subtract the two sample proportions to find the observed difference: \(p_M - p_E = 0.32 - 0.44 = -0.12\). This means that, on average, metal-tagged penguins had \(12\%\) fewer successful breeding seasons than electronic-tagged penguins.

Calculate the Standard Error

Next, calculate the standard error (SE) of the difference in proportions. The formula for the SE of a difference in proportions is \(\sqrt{\frac{{p_M*(1-p_M)}}{{n_M}} + \frac{{p_E*(1-p_E)}}{{n_E}}}\). Substituting our values in, we get a SE of \(\sqrt{\frac{{0.32*(1-0.32)}}{{122}} + \frac{{0.44*(1-0.44)}}{{160}}} = 0.0564\).

Calculate the Confidence Interval

A \(95\%\) confidence interval for the difference in proportions is given by \((p_M - p_E) \pm Z*SE\), where \(Z\) is the Z-value from the standard normal distribution for the desired level of confidence. For a \(95\%\) confidence level, the Z-value is 1.96. So, the confidence interval is \(-0.12 \pm 1.96*0.0564 = (-0.2303, -0.0097)\). This means we are \(95\%\) confident that the true difference in proportions \(p_M - p_E\) lies somewhere between \(-.2303\) and \(-0.0097\).

Interpret the Result

Since the confidence interval does not contain 0, this indicates that there is a significant difference in breeding success rates between metal-tagged and electronic-tagged penguins, with electronic-tagged penguins having a higher success rate. So, the study provides evidence that metal tagging may be detrimental to penguins' breeding success.

Key concepts

Sample Proportions

Understanding sample proportions is fundamental in statistics, especially when we deal with differences in success rates from two groups or categories. A sample proportion represents the fraction of observations in a sample that has a particular characteristic. For instance, in our penguin study, we analyzed the breeding success of metal-tagged versus electronic-tagged penguins. The sample proportion for the metal-tagged penguins with successful breeding (\( p_M \)) was calculated by dividing the number of successful breedings by the total number of breeding seasons observed, yielding \( p_M = 0.32 \) Thus, it indicates that 32% of metal-tagged penguins had a successful breeding season.Similarly, the sample proportion for electronic-tagged penguins was calculated, resulting in a higher proportion of \( p_E = 0.44 \) which reflects 44% successful breeding seasons for this group. These proportions are essential pieces of information because they allow us to compare the success rates between the two groups and assess if the tagging method has any impact on the penguins' breeding success.

Moreover, it is vital to note that sample proportions can fluctuate with different samples. Hence, they are often used to make inferences about the population proportions, which leads us to construct confidence intervals to account for this variability.

Standard Error of Proportions

When comparing sample proportions, like in the penguin study, we need to acknowledge that there's a certain amount of error due to sampling variability. This is where the standard error (SE) comes into play—it measures how much the sample proportions can be expected to fluctuate around the true population proportion.In our exercise, we calculated the standard error for the difference in sample proportions, which provides us with an understanding of the distribution of that difference. The calculation follows the formula \( SE = \sqrt{\frac{{p_M*(1-p_M)}}{{n_M}} + \frac{{p_E*(1-p_E)}}{{n_E}}} \) where \( p_M \) and \( p_E \) are the sample proportions, while \( n_M \) and \( n_E \) are the sample sizes. The outcome, SE = 0.0564, indicates how precise or stable the difference in sample proportions is. A smaller standard error would mean that our sample proportions are likely closer to the true population proportions.

It is this standard error that allows us to calculate confidence intervals to estimate the true difference between the population proportions, giving us a range where we believe the true difference lies with a certain level of confidence.

Statistical Significance

The concept of statistical significance plays a pivotal role in hypothesis testing and helps us determine whether the observed differences or effects we see in our sample are likely due to chance or not. In the context of the penguin study, we were interested in knowing whether there's a significant difference in breeding success between the two types of tags used. To ascertain this, we looked at the confidence interval we calculated for the difference in proportions, which was \( (-0.2303, -0.0097) \).The key takeaway here is that the interval does not include zero, which implies that the difference in breeding success rates is significant and not likely due to random chance. Statistically speaking, when zero is not in the confidence interval, it suggests that the observed effect (in this case, the lower breeding success in metal-tagged penguins) is indeed present in the population.

Therefore, concluding that metal tagging can negatively affect breeding success is supported by the data, given the statistical significance of the results. The absence of zero in the confidence interval essentially strengthens our confidence that the difference we see is real and potentially biologically or practically important.