Lecture 7 - Confidence Intervals and Hypothesis Testing
Lecture 7 - Confidence Intervals and Hypothesis Testing
Statistics 102
Colin Rundel
February 6, 2013
CLT - Conditions
From last time
Certain conditions must be met for the CLT to apply:
1 Independence: Sampled observations must be independent.
This is difficult to verify, but is more likely if random sampling/assignment is used, and n < 10% of the population.
2 Sample size/skew: the population distribution must be nearly normal or n > 30 and the population distribution is not extremely skewed.
This is also difficult to verify for the population, but we can check it using the sample data, and assume that the sample mirrors the population.
Statistics 102 (Colin Rundel)
Lec 7
February 6, 2013 2 / 26
Example - Review
From last time
To the right is a plot of a population distribution. Match each of the following descriptions to one of the three plots below.
(1) a single random sample of 100 observations from this population (2) a distribution of 100 sample means from random samples with size 7
Population ? = 10 = 7
(3) a distribution of 100 sample means from random
0
10 20 30 40 50
samples with size 49
30 25 20 15 10
5 0
4 6 8 10 12 14 16 18
Plot A
30 25 20 15 10
5 0
0
5 10 15 20 25 30 35
Plot B
20 15 10
5 0
8
9 10 11 12
Plot C
Statistics 102 (Colin Rundel)
Lec 7
February 6, 2013 3 / 26
Confidence intervals Why do we report confidence intervals?
Confidence intervals
A plausible range of values for the population parameter is called a confidence interval. Using only a point estimate to estimate a parameter is like fishing in a murky lake with a spear, and using a confidence interval is like fishing with a net.
We can throw a spear where we saw a fish but we are more likely to miss. If we toss a net in that area, we have a better chance of catching the fish.
If we report a point estimate, we probably will not hit the exact population parameter. If we report a range of plausible values ? a confidence interval ? we have a good shot at capturing the parameter.
Statistics 102 (Colin Rundel)
Lec 7
February 6, 2013 4 / 26
Confidence intervals Constructing a confidence interval
Example - Relationships
A sample of 50 Duke students were asked how many long term exclusive relationships they have had. The sample yielded a mean of 3.2 and a standard deviation of 1.74. Estimate the true average number of exclusive relationships using this sample.
The approximate 95% confidence interval is defined as
point estimate ? 2 ? SE
x? = 3.2
s = 1.74
s 1.74 SE = = 0.25
n 50
x? ? 2 ? SE = 3.2 ? 2 ? 0.25 = (3.2 - 0.5, 3.2 + 0.5) = (3.15, 3.25)
We are 95% confident that Duke students on average have been in between 3.15 and 3.25 exclusive relationships
Statistics 102 (Colin Rundel)
Lec 7
February 6, 2013 5 / 26
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