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When comparing two groups, such as Internet users and non-users, we often want to understand how they differ in a particular characteristic—in this case, trust. To find this difference, we calculate the difference between the proportions of each group that displays this characteristic.

In the example of Internet users versus non-users:

• First, determine the proportion of Internet users who agree with the statement "most people can be trusted." This is done by dividing the number of trusted Internet users by the total number of Internet users, giving us a proportion of 0.460.
  
• Similarly, calculate the proportion for non-Internet users using the same method. This proportion turns out to be 0.269.
  

The difference in these sample proportions \( \hat{p}_1 - \hat{p}_2 \) is then found by subtracting the non-user proportion from the user proportion. This lays the foundation for further statistical analysis, like confidence intervals, which help us understand the variability of these differences.

Understanding the variability in the difference of proportions requires calculating the standard error. The standard error gives us an idea of how much we expect our sample results to vary from the true population values.

The formula for standard error when comparing two proportions is:
\[SE = \sqrt{ \frac{\hat{p_1}(1-\hat{p_1})}{n_1} + \frac{\hat{p_2}(1-\hat{p_2})}{n_2} }\]
Here, \( \hat{p}_1 \) and \( \hat{p}_2 \) are the proportions of the two groups, and \( n_1 \) and \( n_2 \) are the sample sizes.

In our example, after substituting the values, we find that the standard error is 0.0198.

• This tells us the extent to which the difference in sample proportions may fluctuate, helping in constructing a range (the confidence interval) within which the true population difference likely falls.
  

Understanding and calculating the standard error is crucial because it helps assess the reliability and precision of our difference estimates.

A z-score is instrumental in determining how far away our sample result is from the mean, and it helps us quantify the confidence level in our interval estimates.

For a confidence interval, the z-score links directly to the desired level of confidence—how sure we want to be that our interval estimate contains the true population parameter. The choice of a 90% confidence level gives us a z-score of approximately 1.645.

This is interpreted as follows:

• By multiplying this z-score by the standard error calculated earlier (0.0198), we obtain the margin of error—0.0326 in this example.
  
• The margin of error is added to and subtracted from the difference in sample proportions, providing the bounds of our confidence interval.
  

Thus, by interpreting the z-score, we confidently say that, with a 90% chance, the true proportion difference lies within the calculated confidence interval of 0.158 to 0.224. This interval gives a reliable estimate of how greatly trust differs between our two groups.