Exercise 15 - Calculate the 95% Confidence Interval of the Spreads

We have constructed a random variable that has expected value \(b_2 - b_1\), the pollster bias difference. If our model holds, then this random variable has an approximately normal distribution. The standard error of this random variable depends on \(\sigma_1\) and \(\sigma_2\), but we can use the sample standard deviations we computed earlier. We have everything we need to answer our initial question: is \(b_2 - b_1\) different from 0?

Construct a 95% confidence interval for the difference \(b_2\) and \(b_1\). Does this interval contain zero?

Use pipes %>% to pass the data polls on to functions that will group by pollster and summarize the average spread, standard deviation, and number of polls per pollster.
Calculate the estimate by subtracting the average spreads. Save this estimate to a variable called estimate.
Calculate the standard error using the standard deviations of the spreads and the sample size. Save this value to a variable called se_hat.
Calculate the 95% confidence intervals using the qnorm function. Save the lower and then the upper confidence interval to a variable called ci.