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In statistics, a proportion refers to the fraction of the total that possesses a certain characteristic. It's expressed as a ratio or percentage. In the context of our penguin study, we are interested in the survival rates of the tagged penguins. For the metal tagged penguins, the proportion of survivors is calculated as \[ \hat{p_{M}} = \frac{10}{50} = 0.2 \] This means 20% of the metal tagged penguins survived. Similarly, for the electronic tagged penguins, the proportion is \[ \hat{p_{E}} = \frac{18}{50} = 0.36 \] indicating 36% survived.

• Proportion helps us quickly understand the share of a group exhibiting a particular trait.
• It provides a way to compare different groups, such as metal versus electronic tagged penguins.

Using proportions allows us to directly compare different groups and deduce which one performed better in terms of survival, as done in this study.

Standard error is a measure that quantifies the amount of variability or dispersion of a sample statistic from the population parameter. It estimates how much the sample mean (or proportion) would vary if you drew multiple samples from the same population. In our penguin study, the standard error helps us understand the variability in the difference between the two proportions.

• A smaller standard error indicates less variability among sample statistics, meaning our sample estimate is more precise.
• A larger standard error generally indicates more variability, leading to less confidence in the sample statistic as an estimate of the population parameter.

To calculate the standard error of the difference between two proportions, we use the formula: \[ SE = \sqrt{\frac{\hat{p_{M}} \cdot (1-\hat{p_{M}})}{n_{M}} + \frac{\hat{p_{E}} \cdot (1-\hat{p_{E}})}{n_{E}}} = 0.09173 \] This provides a measure of how much the sample proportions (0.2 and 0.36) are expected to vary due to sampling error.

The critical value is a key component in inferential statistics, used to denote the threshold at which the null hypothesis is rejected in a significance test. In constructing confidence intervals, it helps us determine the margin we allow around our point estimate.

• The critical value is derived from a probability distribution, commonly the standard normal distribution in cases involving proportions.
• For a given confidence level, the critical value indicates the z-score at which the remaining probability is equally split between the tails of the distribution.

In our penguin study, to construct a 90% confidence interval, the critical value is \( z_{\alpha / 2} = 1.645 \). This z-score corresponds to the tails containing 5% of the data each (10% of total), leaving 90% in the center. By using this value, we establish how far values can deviate from the estimated difference in proportions, \(\hat{p_{M}} - \hat{p_{E}}\), while still being within the confidence interval.