Worksheet 23: Estimation for quantitative variables

Worksheet 23: Estimation for quantitative variables#

Your name:

Your student ID number:

Recap#

  1. Suppose you surveyed \(n=100\) Stanford students and \(55\) of them say they support the proctoring pilot.

    a. What is the estimate \(\hat{\pi}_n\)?

    b. What is the standard deviation of \(\hat{\pi}_n\)?

    c. What is a 68% confidence interval for \(\pi\)?

Quantitative variables#

  1. How does the sample size \(n\) effect the distribution of the sample mean \(\hat{\mu}_n\)?

  2. What do you notice about the distribution of \(\hat{\mu}_n\)?

  3. Suppose that \(n=100\) and \(\hat{\sigma}_x=200\). What is the standard deviation of \(\hat{\mu}_n\)?

  4. Suppose that \(\hat{\mu}_n=310\) and the standard deviation of \(\hat{\mu}_n\) is \(20\). What is a 95% confidence interval for \(\mu\)?

Mini crosswords#

  1. The sample mean of the difference in crossword times is \(-7.3\) seconds. Is this evidence that Clikey are better than Andel?

  2. The sample size is \(n=33\) and the standard deviation of \(x_1,\ldots,x_n\) is \(\hat{\sigma}_x= 75\) seconds. What is a confidence interval for \(\mu\)?

  3. Suppose that \(\mu = -5\). How large does \(n\) need to be so that a 95% confidence interval centered at \(-5\) will only include negative numbers? (Assume that \(\sigma_x = 75\))

Error bars#

  1. Confidence intervals are often represented visually as error bars to represent uncertainty. How would design a visualization that emphasizes uncertainty?

Confidence intervals theory#

  1. Write down the formula for a \(1-\alpha\) confidence interval for \(\mu\):