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In studies comparing two groups, like in the penguin survival scenario, the difference in proportions helps us understand how one group fares relative to the other.
We calculate proportions using the ratio of successes (or events) in each group to the total in that group.
For the metal tagged penguins, the survival proportion is calculated as 10 survivors out of 50 total penguins, or \( p_M = \frac{10}{50} = 0.20 \).
Similarly, for electronic tagged penguins, the proportion is 18 out of 50, or \( p_E = \frac{18}{50} = 0.36 \).
Therefore, the difference in proportions \( p_M - p_E = 0.20 - 0.36 = -0.16 \).

This negative difference indicates metal tagged penguins have a lower survival rate compared to electronic tagged ones.
Understanding the difference in proportions is crucial because it shows the practical significance of the tags on survival rates.

The Standard Error (SE) is a statistical measure that provides an estimate of the accuracy of a sample proportion.
In this context, we use SE to measure the variability of the difference in proportions between the two groups of penguins.
The formula for SE for two proportions is:
\[ SE = \sqrt{\left[\frac{p_{M}(1-p_{M})}{n_{M}}\right] + \left[\frac{p_{E}(1-p_{E})}{n_{E}}\right]} \]
where \( n_M \) and \( n_E \) are the sample sizes for the metal and electronic tagged penguins, respectively.
By plugging in the values, \( p_M = 0.20 \), \( p_E = 0.36 \), \( n_M = 50 \), and \( n_E = 50 \), the SE is calculated as
\[ SE = \sqrt{\left[\frac{0.20 \times 0.80}{50}\right] + \left[\frac{0.36 \times 0.64}{50}\right]} = 0.071 \].

The standard error tells us how much the difference in proportions could vary if we repeated this experiment multiple times.
A smaller SE indicates more precise estimates.

The z-value is a critical value that represents the number of standard deviations a data point is from the mean of a distribution.
It is used to calculate confidence intervals by determining how wide the interval should be for a specified level of confidence.
In a normal distribution, certain z-values correspond to specific confidence levels.
For a 90% confidence interval, the z-value is typically 1.645.

This z-value means that 90% of data points fall within 1.645 standard deviations from the mean.
In our example, by using the z-value of 1.645 along with our Standard Error (SE), we construct the confidence interval for the difference in proportions.
The calculation is done using the formula:
\[ (p_M - p_E) \pm z \times SE \]
Plug in \( p_M - p_E = -0.16 \), \( z = 1.645 \), \( SE = 0.071 \), which gives us a confidence interval of \(-0.277, -0.043\).

This interval estimation tells us reliably where the true difference in proportions may lie with 90% confidence.