Chapter 7

Statistical Inference: Estimating a Population Mean

Statistical Inference

Statistical inference is the process of reaching conclusions about characteristics of an entire population using data from a subset, or sample, of that population.

Simple Random Sampling

Simple random sampling is a sampling method which ensures that every combination of n members of the population has an equal chance of being selected.

Figure 7.1 A Table Of Uniformly

Distributed Random Digits 1 6 8 7 0 5 3 4 9 9 2 9 4 8

7 6 4 3 9 0 5 3 6 4 7 3 6 6

5 9 2 6 8 1 8 0 1 8 1 7 1 8

0 4 1 4 5 9 2 0 6 3 2 5 2 7

0 2 6 1 3 2 4 3 8 3 2 8 5 1

4 8 3 3 4 0 2 8 6 5 5 8 0 7

1 0 2 6 6 1 0 1 1 4 6 5 8 3

4 6 3 6 4 8 5 6 2 4 5 4 4 0

5 5 9 9 0 8 6 1 9 1 0 5 4 1

8 3 5 1 5 1 5 8 6 6 1 7 7 1

7 8 1 0 6 5 6 9 1 0 7 1 3 0

2 8 4 1 7 4 2 8 8 9 4 6 9 7

1 3 1 1 4 2 9 4 6 9 8 4 9 5

5 1 6 4 4 8 6 0 3 2 1 2 5 8

5 3 1 0 4 6 9 9 6 1 8 2 8 5

2 9 1 4 9 6 2 8 1 5 4 2 9 0

Sample Data

Sample Population Hrs. of Study Member ID Time(x)

1 1687 20.0 2 4138 14.5 3 2511 15.8 4 4198 10.5 5 2006 16.3. . .. . .. . .49 1523 12.6

50 0578 14.0

Figure 7.2 Bar Chart Showing the Population Study Time Distribution

x

P(x)

1/4 = .25

Study Time

Figure 7.3 Sampling Distribution for for Our Small-Scale Illustration

P( )

15 .167

20 .167

25 .333

30 .167

35 .167

x x

Figure 7.4 Bar Chart Showing the

Sampling Distribution of x

15 20 25 30 35 x

.167

.333

P( )x

Sample Mean Study Time (hrs)

The Sampling Distribution of the Sample Mean

The sampling distribution of the sample mean is the probability distribution of all possible values of the sample mean, , when a sample of size n is taken from a given population.

x

Key Sampling Distribution Properties

• For large enough sample sizes, the shape of the sampling distribution will be approximately normal.

• The sampling distribution is centered on , the mean of the population.

• The standard deviation of the sampling distribution can be computed as the population standard deviation divided by the square root of the sample size.

Figure 7.5 The Shape of the Sampling Distribution When Sample Size is Large (n > 30)

x

Central Limit Theorem

As sample size increases, the sampling distribution of the sample mean rapidly approaches the bell shape of a normal distribution, regardless of the shape of the parent population.

Figure 7.6 Implications of the Central Limit Theorem

n = 2

n = 5

n = 30

x

x x x

x

xx

xx

x

x

The Sampling Distribution of the Sample Mean

x

Population Shapes

Small Samples

In small sample cases (n<30), the sampling distribution of the sample mean will be normal if the shape of the parent population is normal.

Figure 7.7 The Center of the Sampling

Distribution of the Sample Mean

E( ) = xx

Standard Deviation of the (7.1) Sampling Distribution of the

Sample Mean

n

x

Figure 7.8 Standard Deviation of the Sampling Distribution of the Sample Mean

n

x

Figure 7.9 Sampling Distribution of the Sample Mean for Samples of Size n = 2, n = 8, and n = 20 Selected from the Same Population

Population Distribution

Sampling Distribution

n = 2

n = 20

n = 8

x

x

Standard Deviation of the Sampling (7.2) Distribution of the Sample Mean (When sample size is a large fraction of the population size)

1

N

nN

n

x

Interval Estimate of (7.3)

a Population Mean

n

xz

Factors Influencing Interval Width

1. Confidence—that is, the likelihood that the interval will contain m. A higher confidence level will mean a larger z, which, in turn, will mean a wider interval.

2. Sample size, n. A larger sample size will produce a tighter interval.

3. Variation in the population, as measured by. The

greater the variation in the population values, the wider the interval.

Figure 7.10 Intervals Built Around Various Sample Means from the Sampling Distribution

3x2x

4x5x

1x

x

Figure 7.11 Standard Error vs.

Margin of Error

x + z x

Margin of Error

Standard Error

Margin of Error

The margin of error in an interval estimate of measures the maximum difference we would expect between the sample mean and the population mean at a given level of confidence.

Figure 7.12 General Comparison of the t and Normal Distributions

t distribution

Normal distribution

Interval Estimate of (7.4) When s Replaces

x ts

n

Figure 7.13 Comparison of the t and Normal Distributions as Degrees of Freedom Increase

Normal Distribution t with 15 degrees of freedom

t with 5 degrees of freedom

Basic Sample Size Calculator (7.5)

2

E

zn

Sample Size when (7.6) n/N > .05

N

nn

n

1