Sampling Distribution

Chapter 4

The probability distribution of a statistic is called sampling distribution.

A statistic (e.g. sample mean, sample standard deviation) is a random variable whose value depends only on the observed sample and may vary from sample to sample.

The sampling distribution of a statistic will depend on the size of the population, the size of the sample, and the method of choosing the sample.

Basic Concepts

The standard deviation of the sampling distribution is called the standard error of the statistic. It tells us the extent to which we expect the values of the statistic to vary from different possible samples.

The probability distribution of the sample mean is called the sampling distribution of the mean.

Basic Concepts

Basic ConceptsConsider 4 observations making up the population values of a random variable X having the probability distribution

Note that =E(X)=3/2 and 2=Var(X)=5/4.

Suppose we list all possible samples of size 2, with replacement, and for each sample compute for the value of the sample mean, .

3,2,1,0,41)( xxf

X

Basic ConceptsNo. Sample No. Sample1 0, 0 0 9 2, 0 1.02 0, 1 0.5 10 2, 1 1.53 0, 2 1.0 11 2, 2 2.04 0, 3 1.5 12 2, 3 2.55 1, 0 0.5 13 3, 0 1.56 1,1 1.0 14 3, 1 2.07 1, 2 1.5 15 3, 2 2.58 1, 3 2.0 16 3, 3 3.0

XX

X X

Sampling Distribution of the Sample Means0 0.5 1.0 1.5 2.0 2.5 3.0

P( ) 1/16 2/16 3/16 4/16 3/16 2/16 1/16XX

Theorems

1. If all possible random samples of size n are drawn with replacement from a finite population of size N with mean and standard deviation , then the sample mean will have a mean and variance given by:

)(XE nXVar2

)(

.

Theorems2. If all possible random samples of size n are

drawn without replacement from a finite population of size N with mean and standard deviation , then the sample mean will have mean and variance given by:

)(XE

.

.1

)(2

N

nNn

XVar

4.2 Central Limit Theorem

If is the mean of a random sample of size n taken from a (large or infinite) population with mean and variance 2, then the sampling distribution of is approximately normally distributed with mean and variance

when n is sufficiently large. Hence, the limiting form of the distribution of

X

X)(XE

nXVar

2

)(

nXZ

/

4.2 Central Limit TheoremNote

The normal approximation in the theorem will be good if n30 regardless of the shape of the population.

If n

ExampleSuppose a team of biologists has been studying a particular fishing pond. Let x represent the length of a single trout taken at random from the pond. Assume xhas a normal distribution with a mean= 10.2 in. and standard deviation = 1.4 in.

(a) What is the probability that a single trout taken at random from the pond is between 8 and 12 inches?

(b) What is the probability that the mean length of 5 trout taken at random is between 8 and 12 inches?

ExampleSolution:

(a) What is the probability that a single trout taken at random from the pond is between 8 and 12 inches?

Given: X~N(10.2, 1.42))128( XP

4.1

2.10124.1

2.108 ZP

29.157.1 ZP

)57.1(29.1 ZPZP

0582.09015.0 8433.0

Example

Solution:b. What is the probability that the mean length of 5

trout taken at random is between 8 and 12 inches?Given: X~N(10.2, 1.42)

)128( XP

5/4.12.1012

5/4.12.108 ZP

87.251.3 ZP

)51.3(87.2 ZPZP

0001.09979.0 9978.0

ExampleAssume that the mean systolic blood pressure of normaladults is 120 millimeters of mercury (mm Hg) and thestandard deviation is 5.6. Assume the variable is normallydistributed.

a. If an individual is selected, find the probability that theindividuals pressure will be between 120 and 121.8mm Hg.

b. If a sample of 30 adults is randomly selected, find theprobability that the sample mean will be between 120and 121.8 mm Hg.

Example

Solution:a. If an individual is selected, find the probability that

the individuals pressure will be between 120 and121.8 mm Hg.

Given: X~N(120, 5.62)

)8.121120( XP

6.5

1208.1216.5120120 ZP

32.00 ZP

)0(32.0 ZPZP

5000.06255.0 1255.0

Example

Solution:b. If a sample of 30 adults is randomly selected, find

the probability that the sample mean will bebetween 120 and 121.8 mm Hg.

Given: X~N(120, 5.62)

)8.121120( XP

30/6.51208.121

30/6.5120120 ZP

76.10 ZP

)0(76.1 ZPZP

5000.09608.0 4608.0

4.3 Sampling from the Normal Distribution

The t-distributionIf and are the mean and variance, respectively, of a random sample of size n taken from a population which is normally distributed with mean and variance 2, then

is a random variable having the t-distribution with degrees of freedom v=n-1.

Notation:

X 2S

nsXT

/

1~ nvtT

4.3 Sampling from the Normal Distribution

Comparison between t-distribution and the standard normal distribution

1. Both are symmetric about zero.2. Both are bell-shaped, but the t-distribution is more

variable.i) t-values depend on the fluctuation of two values: andii) z-values depend only on the change of from sample to

sample

3. When the sample size is large i.e. n30, the t-distribution can be will approximated by the standard normal distribution.

X 2SX

4.3 Sampling from the Normal Distribution

Area under the curveJust like any continuous probability distribution, the probability that a random sample produces a t-value falling between any two specified values is equal to the area under the curve of the t-distribution between any two ordinates corresponding to the specified values.

Notation: is the value leaving an area of in the right tail of the t-distribution. That is, if then is such that .

Since the t-distribution is symmetric about zero, .

t)(~ vtT

)( tTPt

tt 1

Example

1. Find the following values on the t-table:a) t0.025 when v=14b) t0.99 when v=10

2. Find k such that P(k

Example3. A manufacturing firm claims that the batteries used in their electronic games will last an average of 30 hours. To maintain this average, 16 batteries are tested each month. If the computed t-value falls between -t0.025 and t0.025, the firm is satisfied with its claim. What conclusion should the firm draw from a sample that has a mean of 27.5 hours and standard deviation of 5 hours? Assume the distribution of battery lives to be approximately normal.