Confidence Intervals

1
Confidence Intervals

• Proportions

2
The Tattoo Data

• On first day 19.3 of you said you have a tattoo
• n 228 responded
• What percent of all PSU students have a tattoo ?
• Is 19.3 the answer ?

3
General Objective

• Use sample data to estimate the population
  proportion

4
95 Confidence Interval Estimate

• An interval showing our best guesses at the
  population value
• Example statement "we think population
  proportion is between 0.14 and 0.24"

5
Confidence Level for Interval

• Chance that interval really "captures" the
  population value
• 95 is most commonly used for confidence level
• although, any level could be used.

6
Interval for a proportion

• Rule of sample proportions paves the way
• Sample proportions described by bell curve
  centered at true p
• 95 of the time , the sample p is within 2SDs of
  true p
• Interval is sample p /- 2 x SD

7
Formula For SD

• Sqrt p x ( 1- p) / n
• p is supposed to be true p
• Do we know true p?
• No!
• We'll use sample p in SD formula

8
For the tattoo data

• sample p 0.193 and n228
• estimated SD
  sqr. root of (0.193)(1-0.193)/228
  0.026
• 2SDs 2 x 0.026 0.052
• Interval is 0.193 - 0.052 to 0.1930.052
• 95 confident that PSU proportion is between
  0.141 and 0.245

9
Same Survey Last Spring

• sample p 0.15 and n 210
• Can we conclude that tattoo percentage has
  increased since last spring ?

10
Confidence Interval Based on Last Spring

• SD sqrt0.15(1-0.15)/2100.025
• 2 SD 2 x 0.025 0.05
• 95 interval is 0.15 /- 0.05
• 95 sure that PSU proportion is between 0.10 and
  0.20.

11
Comparison

• Intervals from Spring and Fall overlap
• Can't conclude that the population proportions
  differ.

12
Margin of Error

• 95 margin of error 2 SDs
• Earlier in term, we said -
• margin of error 1 / sqr. root(n)
• when p0.5, 2 SD 1/sqr. root(n)

13
Gallup Poll Question

• Do you approve of radio personality Howard Stern?
• Results were 70 did not with n 900
• margin of error given as 3
• Determine 95 conf. interval for population
  percent
• samp est - and margin of error is 67 to 73

14
Interpretation

• Based on the sample, we are 95 sure that between
  67 and 73 of all Americans disapprove of Howard
  Stern

15
Some Warnings

• Sample should be random sample - not volunteer
  sample
• Don't need confidence interval if whole
  population is sampled (rarely done)