Welcome Back

1
Welcome Back

• Homework Due next Tuesday in class(on the web)
• Were in chapter 8

2
ReviewLarge Sample Confidence Intervals
n gt30 or so for means, np and n(1-p) both gt 5 for
proportions

• 1-a confidence interval for a mean
• x /- za/2 s/sqrt(n)
• 1-a confidence interval for a proportion
• p /- za/2 p(1-p)/sqrt(n)
• 1-a confidence interval for the difference
  between two means
• x1 x2 /- za/2 sqrt(s21/n1s22/n2)

3
In General
)
(
Estimate (that is normally distributed via the
CentralLimit Theorem)
standard deviation Za/2
of estimate
/-
This gives an interval (Lower Bound , Upper
Bound) Interpretation This is a plausible range
for the true value of the number that were
estimating. a is a tuning parameter for level of
plausibility smaller a more conservative
estimate.
4
Large Sample Confidence Intervals
np and n(1-p) gt 5 for all ps

• 1-a confidence interval for difference between
  two proportions
• p1-p2 /- za/2 sqrt(p1(1-p1)/n1)(p2(1-p2)/n2)
  

5
Designing an Experiment and Choosing a Sample Size

• Example Compare the shrinkage in a tumor due to
  a new cancer treatment relative to standard
  treatment
• 100 patients randomly assigned to new treatment
  or standard treatment
• xinew reduction in tumor size for person i
  under new treatment
• xjstd reduction in tumor size for person j
  under std treatment
• xnew and s2new
• xstd and s2std

Mean and sample variance of the changes in size
for the new and standard treatments
6
Suppose the data are

• xnew 25.3
• snew 2.0
• xstd 24.8
• sstd 2.3

95 Confidence Interval for difference x1 x2
/- za/2 sqrt(s21/n1s22/n2) 0.5 /- 0.84
What can we conclude?
7

• Theres no difference?
• Cant see a difference?
• Theres a difference, but its too small to care
  about?

8
There is a difference between

• Cant see a difference
• Theres no difference

Situation for Cancer example
(In cancer experiment, we can assume we care
about small differences.)
9

• Cant see a difference (that is big enough to
  care about) wasted experiment
• AVOID / PREVENT THE WASTE AND ASSOCIATED TEARS
  USE SAMPLE SIZE PLANNING

10
Sample Size Planning

• Length of a 1-a level confidence interval
  is 2 za/2 std deviation of estimate

2za/2s/sqrt(n) 2za/2p(1-p)/sqrt(n) 2za/2sqrt((s2
1/n1)(s22/n2)) 2za/2sqrt(p1(1-p1)/n1)(p2(1-p2)
/n2)
11

• Suppose we want a 95 confidence interval no
  wider than W units.
• a is fixed. Assume a value for the standard
  deviation (or variance) of the estimator.
• Solve for an n (or n1 and n2) so that the width
  is less than W units.
• When there are two sample sizes (n1 and n2), we
  often assume that n1 n2.

12
Cancer example

• Let W 0.1. Want 95 CI for difference between
  means with width less than W.
• Suppose s2new s2std 6 (conservative guess)
• W gt 2za/2sqrt((s2new/n1)(s2std/n2))
• 0.1 gt 2(1.96)sqrt(6/n 6/n)
• 0.1 gt 3.92sqrt(12/n)
• 0.01 gt (3.922)12/n
• n gt 18439.68 (each group)

Books B (our W)/2