1. Stat 100 March 20
Chapter 19, Problems 1-7
Reread Chapter 4

2. 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

3. 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.

4. 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%

5. Calculating a Confidence Interval
Sample estimate ± margin of error
Last time we looked at percents
Today, we look at estimating averages (means)

6. 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)]

7. 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.

9. 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

10. 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

11. 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.

12. 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.

13. 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.

14. 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

15. 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

16. 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

17. Confidence interval for men
Margin of error =2×[SD/sqrt(n)] = 2×[1.85/sqrt(321)]= 0.20
Interval is ± 0.10, or from 1.45 to 1.85 classes skipped per week.
This estimates mean for all PSU men

18. 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.