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# File: MIT18_05S22_in-class22-script.txt
# Author: Jeremy Orloff
#
# MIT OpenCourseWare: https://ocw.mit.edu
# 18.05 Introduction to Probability and Statistics
# Spring 2022
# For information about citing these materials or our Terms of Use, visit:
# https://ocw.mit.edu/terms.
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Class 22 Confidence intervals for normal data
Jen
Slide 1:
Slide 2: Announcements/Agenda (3 minutes)
Slide 3: Review of critical values (2 minutes)
Quickly --done before, on pset, in reading
Slide 4ab: Concept question critical values (6 minutes)
This is review. Give the answer clearly but don't dwell on it.
1. Within 2 std dev = 95% --> 0.025 in each tail --> z_0.25 \approx 2
2. Within 1 std dev = 68% --> 0.16 in each tail --> -z_0.16 \approx -1
Slide 5: Formulas for confidence intervals (3 minutes)
z, t, chisq for variance
Tell them we will talk about the rationale later
Key point: each gives a range of values for the unknown parameter
Slide 6: Rule of thumb (95\% = 2 \sigma) (3 minutes)
Enough to write down that alpha = 0.5 and
Precise: z_0.25 = 1.96
rule of thumb z_0.25 = 2.
Slide 7: Board question: computing confidence intervals (Work 12 minutes, discuss: 5 minutes)
Reminder: \Chi^2 stat for variance is (n-1)s^2/sigma^2 -->
CI is [(n-1)s^2/c_alpha/2, (n-1)s^2/c_{1-\alpha/2}
DISCUSSION
Avoid the temptation to rederive or explain formulas
Go through each one using the formula.
a-c quickly
(d) X^2 CI for sigma: Reassure them that taking the square root of #3
is okay
(e) rule of thumb: this is a z thing, but
t(99)\approx z so can use t_{.25} \approx 2.(In fact it = 1.98)
Jerry
Slide 8: Conceptual view (4 minutes)
Don't dwell on the connection to NHST. We will look more carefully at this in class 23.
Suggest they look at the applet outside of class. --We won't look at it.
Slide 9: Table discussion: what affects the width of the interval (4 minutes)
Write down formula [xbar - z_alpha/2*sigma/\sqrt{n}, xbar + ...] on board
Slide 10: Intervals and pivoting (5 minutes)
--Key points --don't dwell --this is the next board question
alpha non-rejection region and 1-alpha conf. interval have the same width.
The confidence interval = all mu_0 so that we wouldn't reject H0: mu=mu_0 with the data xbar
Slides 11: BOARD question non-rejection and CI (Work 8 minutes, discussion: 4 minutes, go over solution slide.)
Slide 12: solution
We want to help them get to the picture
Again the key is both intervals have the same radius z_alpha/2 * sigma/sqrt(n)
Slide 13 Polling CIs (3 minutes)
Explain the term proportion: theta is the proportion, i.e. fraction, of trials that result in successes.
1/sqrt(n) is what the media call the margin of error in political polls
Slide 14 BOARD question polling (Work 6 minutes. If 5 minutes left, we can discuss.)