9. Statistical Inference: Confidence Intervals and T-Tests. Suppose we wish to use a sample to estimate the mean of a population The sample mean will not necessarily be exactly the same as the population mean. Imagine that we take a sample of 3 from a population of 10,000 cases.
S1: 1,2,9 mean=4
S2: 5,4,9 mean=6
S3: 3,7,5 mean=5
S4: 1,1,2 mean=1.3
S5: 7,9,5 mean=7
And so forth μ=5.5
Column one shows the population distribution values of 1,2,3,4,5,6,7,8,9,10
Column two is the distribution of 3-draw means from column one; column three is the distribution of 30-draw means from column one.Distribution of Sample Mean by Same Size
Central Limit Theorem values of 1,2,3,4,5,6,7,8,9,10
As Sample Size Gets Large Enough
Almost Normal regardless of shape of population
When the Population is Normal case is normally distributed
n = 4X = 5
n =16X = 2.5
When The Population is Not Normal case is normally distributed
n =30X = 1.8
n = 4X = 5
1. Calculate the standard deviation for
Notation hint: population notation is mostly greek; sample latin.
William Gosset, a.k.a. “Student”
Small Sample? Hedge your bet!
Calculate standard dev. of mean:
Sample of 15 students slept an average of 6.4 hours last night with standard deviation of 1 hour.
Need t with n-1 = 15-1 = 14 d.f.
For 95% confidence, t14 = 2.145
For large samples:
Z and t values become almost identical, so CIs are almost identical.
Anytime our hypothesis specifies direction,
eg, Meanw-Meanm>0 rather than simply
Meanw-Meanm≠0 we can and should use a one tail test.
For our one tail test example (Meanw-Meanm>0), we could reject the null if our sample was > than 1.645 standard deviations from the mean. In the two tail situation (Meanw-Meanm≠0) we cannot reject the null unless our sample is > than 1.96 standard deviations from the mean.
When the one tail test is appropriate, using it (which we always should) makes it more likely we will reject the null and accept our hypothesis