Transcript Confidence Interval - Penn State Statistics Department

Stat 100, This week
• Chapter 20, Try Problems 1-9
• Read Chapters 3 and 4 (Wednesday’s
lecture)
Confidence level
• Probability that procedure provides interval
that captures the population value
• Most commonly used level is 95%
confidence
• Other confidence levels are possible
For Ch. 19 • Margin of error for 95% confidence is
p(1  p)
2
n
For other confidence levels ..
• Change the number “2” in the formula
• Chart on page 345 of book shows other
values
• For example, for 99.7% confidence use “3”
instead of “2”
For 99.7% confidence
p(1  p)
• Margin of error = 3
n
Example
• In a Stat 200 survey of n = 200 students,
65% said they believe there is
extraterrestrial life
• p= .65, n = 200
• For 99.7% CI, margin of error =
• 3  sqrt [.65(1-.65)/200] = 3.034 = .102
• 99.7% CI is 65%  10%, or 55% to 75%
Elements of problem
•
•
•
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Population = all college students
Sample = 200 Stat 200 students
Sample value = 65% believe there is ET
Population value= We’re 99.7% sure that
it’s between 55% and 75%
Chapter 19 Thought Question 1
• Study of n = 199 British married couples
gives 95% CI as .02 to .08 for proportion of
couples in which wife is taller that husband.
• Interpret this interval.
• We can be 95% sure that wife is taller than
husband in somewhere between .02 and .08
of all British married couples (not just the
199 studied)
Chapter 19 Thought Question 2
• Do you think a 99% confidence interval for
Question 1 would be wider or narrower than
the 95% interval?
• Answer = wider. We would be more sure
that the interval would catch true population
value with a wider interval
Chapter 19 Thought Question 3
• Poll result is given that a 95% CI for percent
believing in faith healing in U.S. is 42% to 48%.
• Poll had n =1000
• Suppose the sample size had been n = 5000.
Would the 95% CI have been wider or narrower?
• Answer = narrower. With larger n, the margin of
error is smaller so the interval is narrower.
Chapter 20 Thought Question 1
• Study compares weight loss of men who
only diet compared to those who only
exercise
• 95% confidence intervals for mean weight
loss
> Diet only :
13.4 to 18.0
> Exercise only 6.4 to 11.2
Part a.
• Do you think this means that 95% of men
who diet will lose between 13.4 and 18.0
pounds?
• Answer = NO. A confidence interval does
not estimate individual values.
Part b.
• Can we conclude that there's a difference
between mean weight losses of the two
programs?
• This is a reasonable conclusion. The two
confidence intervals don't overlap.
Thought Question 2
• Suppose the sample sizes had been larger
than they were for question 1.
• How would that change the confidence
intervals?
• Answer = with larger sample size margin of
error is smaller so confidence interval is
narrower
Thought Question 3 of Ch. 20
• We compared confidence intervals for mean
weight loss of the two different treatments.
• What would be a more direct way to
compare the weight losses in question 1?
• Answer = get a single confidence interval
for the difference between the two means.
• This is possible, but we won’t go over the
details
Thought Question 4
• A study compares risk of heart attack for
bald men to risk for men with no hair loss
• A 95% confidence interval for relative risk
is 1.1 to 8.2
• Is it reasonable to conclude that bald men
generally have a greater risk?
Answer
• Relative risk =
risk in group 1/ risk in group 2
• Relative Risk =1 if risks are equal
• Interval 1.1 to 8.2 is completely above 1 so
it seems that the “true” relative risk may be
greater than 1.
• So bald men may have a higher risk – but
note we have very imprecise estimate of
“how much”