STAT1008 Study Guide - Final Guide: Central Limit Theorem, Confidence Interval, Statistical Hypothesis Testing
Distribution of a Sample Proportion
• The standard error for
• The larger the sample size, the smaller the SE.
Sufficiently Large n
• The larger the sample size, the more like a normal distribution it becomes. A normal
distribution is a good approximation as long as np10 and n(1 - p) 10.
Central Limit Theorem for
• ~ N (
• A normal distribution is a good approximation as long as np 10 and n(1-p) 10.
6.2 Confidence Interval For a Single Proportion
• When doing inference, we don’t know p so substitute p̂ as it is our best guess.
• Provided the sample size is large enough so that np̂ ≥ and n(1-p̂≥ , a confidence interval
can be computed by *
Margin of Error
• If we want to estimate a population proportion to within a desired ME, we should select a
sample of size
• Neither p or p̂ is known in advance. To be conservative, use p = 0.5.
• N ≈ (1/ME)2
6.3 Test For a Single Proportion
Hypothesis Testing
• For hypothesis testing, we want the distribution of the sample proportion assuming H0 is true.
• To test Ho: p=po:
Test for a Single Proportion
•
• If npo 10 and n(1-po) 10 , then the p-value can be computed as the area in the tail(s) of a
standard normal beyond z.
6.4 Distribution of a Sample Mean
Standard Error of Sample Means
• The SE for =
• The larger the sample size (n), the smaller the SE.
Central Limit Theorem for Sample Means
• If n is sufficiently large:
• A normal distribution is usually a good approximation, as long as n30.
The Distribution of Sample Means Using the Sample Standard Deviation
• Usuall, e do’t ko the populatio SD
, so estimate it with the sample SD, s.
n=z* /ME
( )
2
p(1-p)
SE
=
p
0
(1
-
p
0
) / n
z=p-p0
p0
(1
-p0
) /
n
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Document Summary
Sufficiently large n: the larger the sample size, the smaller the se, the larger the sample size, the more like a normal distribution it becomes. A normal: the standard error for is (cid:4666)(cid:2869) (cid:4667) distribution is a good approximation as long as np 10 and n(1 - p) 10. Central limit theorem for : ~ n (, (cid:4666)(cid:2869) (cid:4667) (cid:4667, a normal distribution is a good approximation as long as np 10 and n(1-p) 10. can be computed by * (cid:4666)(cid:2869) (cid:4667) If we want to estimate a population proportion to within a desired me, we should select a sample of size n = z * /me. )2 p(1- p: neither p or p is known in advance. For hypothesis testing, we want the distribution of the sample proportion assuming h0 is true: to test ho: p=po: Test for a single proportion p - p0 z = p0 (1- p0 ) / n standard normal beyond z.
