Transcript Stat 100, This week

Stat 100 March 20
• Chapter 19, Problems 1-7
• Reread Chapter 4
Basic statistical problem
• Use a sample to estimate something about a
population
• Example: Use sample of n = 50 students to
estimate mean number of classes missed per
week by all PSU students
Margin of Error
• A sample is not likely to match the
population exactly
• Margin of error = likely upper bound on
sampling error
• Difference between sample result and
population value is likely to be smaller than
the margin of error.
Confidence Interval
• Confidence interval is an interval that is
likely to catch a population value.
• Confidence level = probability procedure
provides interval that captures population
value.
• Most common confidence level is 95%
Calculating a Confidence Interval
• Sample estimate ± margin of error
• Last time we looked at percents
• Today, we look at estimating averages
(means)
Margin of Error for Sample Mean
• SEM = “standard error of the mean”
• SEM= SD of data / sqrt(n)
• Margin of error for a mean = 2 × SEM
=2 × [SD of data / sqrt(n)]
Example of Estimating a
Population Mean
• Spring ‘98 Stat 100 survey included
question about hours of sleep the previous
night.
• For n = 190 students, mean was 7.1 hours
and standard deviation was 1.95 hours.
Nightly Hours of Sleep
n=190 Spring '98 students
25
Percent
20
Mean = 7.11 ,
SD = 1.95
15
10
5
0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Hours of Sleep
Basic elements of the problem
• Population = all 40,000 PSU students
• Sample = 190 students in Stat 100
• Value of interest = mean hours of sleep the
previous night
• Sample mean = 7.1 hours
• Objective: Estimate mean hours of sleep for
population
Margin of error for the
sample mean
• Margin of error= 2 × SEM
= 2 × [SD/sqrt(n)]
• SD of data = 1.95, and n = 190
• Margin of Error = 2 × [1.95/sqrt(190)]
= 0.28 hours (About 0.3 hours)
• Interpretation: It is likely that the sample
mean is within 0.3 hours of the population
mean
95% Confidence Interval for
Population Mean
• Sample mean ± margin of error
• 7.1 ± 0.3 hours, or from 6.8 to 7.4 hours.
• 95% confident that in the population, mean
hours of sleep is between 6.8 and 7.4.
Using interval to test hypotheses
• Somebody claims that average amount of sleep for
students is 6 hours per night.
• What does our interval indicate about this claim?
• Interval estimate of population mean was 6.8 to
7.4 hours.
• Seems safe to reject claim that mean is 6 because
evidence (the interval) is that mean is higher.
Another hypothesis • Claim is made that mean hours of sleep is 7
hours for college students.
• Claim acceptable: It’s consistent with our
estimate that the mean is between 6.8 and
7.4 hrs.
How many classes do
you skip per week?
• For n = 554 women in Stat 200
Mean = 1.09 and SD = 1.32
• For n = 321 men in Stat 200
mean=1.65, SD=1.85
Some problems
• Use the data to estimate mean classes
skipped per week for all PSU women.
• Use the data to estimate mean classes
skipped per week for all PSU men
• Determine if there’s a difference between
men and women when it comes to skipping
Confidence interval for women
• Margin of error =2×[SD/sqrt(n)] =
2×[1.32/sqrt(554)] =0.12
• Interval is 1.09 ± 0.12 , or from 0.97 to 1.21
classes skipped per week.
• This estimates mean for all 20,000 women
at PSU
Confidence interval for men
• Margin of error =2×[SD/sqrt(n)] =
2×[1.85/sqrt(321)]= 0.20
• Interval is 1.65 ± 0.10, or from 1.45 to 1.85
classes skipped per week.
• This estimates mean for all PSU men
Is there a difference?
• Estimated mean for women:
Between 0.97 and 1.21
• Estimated mean for men:
Between 1.45 and 1.85
• Range of estimates clearly higher for men.
Safe to conclude men skip more classes.