Download 3-5-13 Worksheet - Iowa State University

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Stat 226 SI:
3/5/13
Supplemental Instruction
Iowa State University

Carly
Stat 226
(Various)
3/5/13
Exam 2 Material Review
o Chapter 13: “Samples and Surveys”
 Accuracy vs. Precision
 The meaning of 𝑋̅ and how it is different from 𝑥̅ and μ
 Sampling Strategies
 SRS
 Voluntary
 Convenience
 Stratified
 Cluster
 Bias in sampling
 Shape, center, spread in histograms
o Chapter 14: “Sampling Dist. of the Sample Mean and the Central Limit Theorem”
 Shape, center, spread for the dist. of 𝑋̅
 3 Cases
 X~N(μ, σ2)
2
𝜎
o 𝑋̅~N(𝜇, ( 𝑛) )

√
X is symmetrically distributed and bell-shaped (but not normal)
2
𝜎
o 𝑋̅~approx. N(𝜇, ( 𝑛) )

√
X is not normal, nor symmetric (skewed, multimodal, etc.)
2
𝜎
o 𝑋̅~approx. N(𝜇, ( 𝑛) ) if n is sufficiently large (n ≥ 30)
√


o Use of the CLT
Standard Deviation of 𝑋̅  “Standard Error”
Generic Problems for Sampling Distribution of 𝑋̅
1.) Variable X is normally distributed with a mean of 50 and standard deviation of 6.
a.)
What can we say about the shape of the sampling distribution of 𝑋̅?
𝑋̅ is normally distributed because X is normally distributed.
b.)
Do we need to use the CLT to comment on the shape of the dist. of 𝑋̅?
No, we do not need to use the CLT because X follows a normal dist.
What are the mean and standard error of the sampling distribution of 𝑋̅?
𝜇𝑋̅ = 𝜇 = 50
𝜎
6
SE(𝑋̅) = 𝜎𝑋̅ = 𝑛 = 𝑛
(A value for “n” was not given.)
√
√
2.) Variable X is not normally distributed, but it is symmetric and bell-shaped. It has a
mean of 50 and a standard deviation of 6.
c.)
1060 Hixson-Lied Student Success Center  515-294-6624  [email protected]  http://www.si.iastate.edu
a.)
What can we say about the sampling distribution of 𝑋̅?
𝑋̅ is approximately normally distributed.
b.)
Does the standard error of 𝑋̅ change if we go from a normal sampling
distribution of 𝑋̅to one that is only approximately normal? If so, how?
𝜎
No, the standard error is still equal to 𝜎𝑋̅ = 𝑛.
√
c.)

Is the mean for a distribution of X different from the mean of a sampling
distribution of 𝑋̅? If so, when?
Regardless of the shape of the distribution of X, the sample mean will
follow a normal distribution with a mean 𝜇𝑋̅ = 𝜇, according to the CLT.
A farmer in Iowa owns an apple orchard. He claims that the number of apples per tree on
his apple orchard is normally distributed with a mean of 100 apples and a standard
deviation of 20 apples. Assume a sample size of 300.
What is the shape of the distribution of X?
X is normally distributed.
What is the shape of the sampling distribution of 𝑋̅?
𝑋̅ is normally distributed because X is normally distributed.
𝑋̅
X
Mean
Standard Deviation
Mean
μ = 100
σ = 20
𝜇𝑋̅ = 𝜇 = 100
Standard Error
𝜎𝑋̅ =
𝜎
√𝑛
=
20
√300
= 1.155
Can we make conclusions about the probability distribution of X?
Yes, because X is normally distributed.
What is the probability of “obtaining” a single tree with more than 115 apples?
P(X > 115)
= 1 – P(X < 115)
𝑥− 𝜇
115− 100
= 1– P( 𝜎 <
)
20
= 1 – P(Z < 0.75)
= 1 – 0.7734
= 0.2266
= 22.66%
What is the probability of obtaining a sample mean greater than or equal to 103 apples?
P(𝑋̅ ≥ 103)
= 1 – P(𝑋̅ ≤ 103)
= 1 – P(
𝑥̅ − 𝜇
𝜎
√𝑛
≤
103− 100
= 1 – P(Z ≤ 2.60)
= 1 – 0.9953
= 0.0047
= 0.47%
20
√300
)