Stat 104 – Lecture 21
Sampling Distributions
Quantitative variable
Population Parameters:
μ,σ
known.
Population
Sample
Distribution of
Sample Mean
1
Summary
• Sampling from a population that
follows a Normal Model, μ ,σ .
• Distribution of the sample mean, y
– Shape: Normal model
– Center: μ
– Spread: SD( y ) = σ
n
2
Summary
• Sampling from a population that
does not follow a Normal Model.
• Distribution of the sample mean, y
– Shape: Approximately normal
– Center: μ
σ
– Spread: SD( y ) =
n
3
1
Stat 104 – Lecture 21
Estimation
Quantitative variable
Population Parameter:μ unknown, σ
Population
Sample Sample Mean,
y
4
Example
• For students graduating from an
Iowa high school in 2007, the
distribution of ACT math scores is
approximately normal with mean,
μ = 21.9 and standard deviation,
σ = 4.7.
5
Example
• Twenty five Iowa high school
seniors who intend to take the ACT
exam are randomly chosen and
enrolled in an ACT preparation
course.
6
2
Stat 104 – Lecture 21
Sample Data
20
27
16
23
24
20
15
24
36
21
17
19
20
18
20
31
29
25
20
15
17
22
23
26
27
7
6
4
Count
8
2
12
16
20
24
28
32
ACT Math Score
36
40
8
Sample Mean
•
y = 22.2
• The sample mean is an estimate of the
population mean, μ . The population
consists of all Iowa high school seniors
who take an ACT preparation course.
• The sample mean should be close to
the population mean.
9
3
Stat 104 – Lecture 21
Sampling Distribution
• There is a 95% probability that the
sample mean, y, is within 1.96
standard deviations of the
population mean, μ.
10
Estimation
• We are 95% confident that the
population mean,μ , will be within
1.96 standard deviations of the
sample mean,y .
11
95% Confidence Interval
y − 1.96(SD( y )) to y + 1.96(SD( y ))
⎛
y − 1.96⎜
⎝
σ
n
⎞
⎟
⎠
⎛
to y + 1.96⎜
⎝
σ
n
⎞
⎟
⎠
12
4
Stat 104 – Lecture 21
95% Confidence Interval
• y = 22.2,
• n = 25,
• σ = 4.7
SD( y ) =
σ
n
=
4.7
= 0.94
25
13
Calculation
⎛
y − 1.96⎜
⎝
σ
n
⎞
⎟
⎠
⎛
to y + 1.96⎜
⎝
σ
n
⎞
⎟
⎠
4.7 ⎞
⎛ 4.7 ⎞
⎟ to 22.2 + 1.96⎜
⎟
⎝ 25 ⎠
⎝ 25 ⎠
22.2 − 1.84 to 22.2 + 1.84
20.36 to 24.04
⎛
22.2 − 1.96⎜
14
Interpretation
• We are 95% confident that the
population mean ACT Math score
for Iowa seniors who take a
preparation course is between
20.36 and 24.04.
15
5