Transcript Confidence Interval

Stat 100 Work
• Chapter 20, Try Problems 1-9
• Chapter 19, Try Problems 1-7
• Read Chapter 4
Confidence Interval
• An interval of values that is likely to
contain the population value
• The purpose is to use a sample to estimate
a population characteristic.
• Interval is calculated as
sample value ± margin of error
Prob. 12 of CH. 19. Part (a)
• Time Magazine survey: 59% of n=507
American Catholics favor allowing women
to be priests
• Reported margin of error = 4.4%
• Find a confidence interval for the response
to the question and write a sentence
interpreting the interval.
Answer
• Sample value  margin of error
• 59%  4.4%, which is 54.6% to 63.4%
• Interpretation: We can be 95% confident
that between 54.6% and 63.4% of all
American Catholics favoring allowing
women priests.
Elements of problem
• Population = all American Catholics
• Sample = 507 Catholics in survey
• Value of interest = percent favoring women
priests
• Sample value = 59%
• Estimate for population is 54.6% to 63.4%
Prob. 12, CH. 19, part (b)
• Calculate the confidence interval using the
formula given in the book rather than the
reported margin of error
More Exact Margin of Error
for a proportion
• 95% m.e. =2× p(1  p)
n
For women priests question
•
•
•
•
p=0.59 , n=507
2 Sqrt [0.59 (1-.59)/507]= .044, or 4.4%
Value is same as reported
Interval is 59% ± 4.4%
Example pertaining to Ch. 20
• For n=36 college women, mean pulse =
75.3 and SD=8.
• Based on this, determine a confidence
interval for the population mean
Margin of error for mean
• Margin of error =2×SEM = 2[SD/sqrt(n)]
• SEM=8/sqrt(36)=8/6=1.33
• Margin of error = 2×1.33=2.7
Confidence Interval for
Mean Pulse
• sample mean ±margin of error
• 75.3 +/-2.7 ; 72.6 to 78.0
• 95% certain that mean pulse for all women
is between 72.6 and 78.
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?
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”