1. 8.3 – Estimating a Population Mean
In the previous examples,
we made an unrealistic assumption that the population standard deviation was known and could be used to calculate confidence intervals.

2. Standard Error:
When the standard deviation of a statistic is estimated from the data
When we know  we can use the Z-table to make a confidence interval. But, when we don’t know it, then we have to use something else!

3. Properties of the t-distribution:
σ is unknown
Degrees of Freedom = n – 1
More variable than the normal distribution (it has fatter tails than the normal curve)
Approaches the normal distribution when the degrees of freedom are large (sample size is large).
Area is found to the right of the t-value

4. Properties of the t-distribution:
If n < 15, if population is approx normal, then so is the sample distribution. If the data are clearly non-Normal or if outliers are present, don’t use!
If n > 15, sample distribution is normal, except if population has outliers or strong skewness
If n  30, sample distribution is normal, even if population has outliers or strong skewness

8. Use invT on calculator:
Go to 2nd – VARS - #4 invT
Type in: invT((1+C)/2, n-1)
Example #1: Suppose you want to construct a 90% confidence interval for the mean of a Normal population based on SRS of size 10. What critical value t* should you use?
Degrees of freedom = n – 1 = 10-1 = 9
Calculate: invT((1+.90)/2, 9) = 1.833
t* = 1.833

9. Degrees of Freedom (n-1)
Example #2
Practice finding t* n
Degrees of Freedom (n-1)
Confidence Interval t*
n = 10
99% CI
n = 20
90% CI
n = 40
95% CI
n = 30 9

10. Example #2 Practice finding t* 9 3.250 19 n Degrees of Freedom
Confidence Interval t*
n = 10
99% CI
n = 20
90% CI
n = 40
95% CI
n = 30 9
3.250 19

11. Example #2 Practice finding t* 9 3.250 19 1.729 39 n
Degrees of Freedom
Confidence Interval t*
n = 10
99% CI
n = 20
90% CI
n = 40
95% CI
n = 30 9
3.250 19
1.729 39

12. Example #2 Practice finding t* 9 3.250 19 1.729 39 2.042 29 n
Degrees of Freedom
Confidence Interval t*
n = 10
99% CI
n = 20
90% CI
n = 40
95% CI
n = 30 9
3.250 19
1.729 39
2.042 29

13. Example #2 Practice finding t* 9 3.250 19 1.729 39 2.042 29 2.756 n
Degrees of Freedom
Confidence Interval t*
n = 10
99% CI
n = 20
90% CI
n = 40
95% CI
n = 30 9
3.250 19
1.729 39
2.042 29
2.756

14. Calculator Tip:
Finding P(t)
2nd – Dist – tcdf( lower bound, upper bound, degrees of freedom)

15. Go to: Stat – Tests – TInterval
One-Sample t-interval:
Calculator Tip:
One sample t-Interval
Go to: Stat – Tests – TInterval
Data: If given actual values
Stats: If given summary of values

16. Conditions for a t-interval:
SRS
(problem should say)
(population approx normal and n<15, or moderate size (15≤ n < 30) with moderate skewness or outliers, or large sample size n ≥ 30)
2. Normality
3. Independence
(Population 10x sample size)

17. Robustness:
The probability calculations remain fairly accurate when a condition for use of the procedure is violated
The t-distribution is robust for large n values, mostly because as n increases, the t-distribution approaches the Z-distribution. And by the CLT, it is approx normal.

18. The true mean waste generated per person per day.
Example #3
As part of your work in an environmental awareness group, you want to estimate the mean waste generated by American adults. In a random sample of 20 American adults, you find that the mean waste generated per person per day is 4.3 pounds with a standard deviation of 1.2 pounds. Calculate a 99% confidence interval for  and explain it’s meaning to someone who doesn’t know statistics. P:
The true mean waste generated per person per day.

19. A:
SRS:
Says randomly selected
Normality:
15<n<30. We must assume the population doesn’t have strong skewness. Proceeding with caution!
Independence:
It is safe to assume that there are more than 200 Americans that create waste. N:
One Sample t-interval

20. I:
df =
20 – 1 = 19

22. I:
df = 20 – 1 = 19

23. C:
I am 99% confident the true mean waste generated per person per day is between and pounds.