SP 225Lecture 10

The Central Limit Theorem

Differences Between Statistics and Parameters

Population: All People

Parameter: 5 of 15 or 33% wear glasses

Sample: 3 Randomly Selected People

Statistic: 0 of 3 or 0% wear glasses

Methods to for Better Samples

Random sampling makes samples representative

Book term: probability sample EPSEM (Equal probability of selectino

method)

EPSEM Technique

Begin with a list of all population members

Generate random numbers to identify members of the list to be selected in the sample

Do everything possible to get selected members to participate in the survey

Sampling Distributions

The shape, measure of center and measure of variation associated with many sample statistics

Unique and different from the population distribution

Three Separate Distributions

Sample distribution Empirical and known Intended to help learn about the population

Population distribution Empirical and unknown Properties estimated using statistics

Sampling distribution Theoretical or non-empirical Properties well-known and based on probabilities

Population DistributionNumber of Siblings

0

2

4

6

8

10

12

14

16

0 1 2 3 4 5 6

Population Distribution

Population Distribution

Roll 1 Roll 2 Roll 3 Roll 4 Roll 5 Roll 6

Dice Face

Fre

qu

ency

Sample Distribution

600 Rolls of the Die

0

20

40

60

80

100

120

Roll 1 Roll 2 Roll 3 Roll 4 Roll 5 Roll 6

Fre

qu

ency

Sampling Distribution of Mean

3.653.603.553.503.453.403.35

9

8

7

6

5

4

3

2

1

0

C1

Frequency

Histogram of C1

Central Limit Theorem

For normally distributed populations If repeated samples of size N are drawn

from a normal population with mean µ and standard deviation σ, then the sampling distribution σ of sample means will be normal with a mean of µ and a standard deviation of σ/√N

Central Limit Theorem

For any population If repeated samples of size N are drawn

from a normal population with mean µ and standard deviation , then, as N becomes large, the sampling distribution σ of sample means will be normal with a mean of µ and a standard deviation of σ/√N

The Probability Distribution

We can calculate probabilities for sample means using the sampling distribution for sample means

Similar to calculating probabilities for an individual using a population distribution

Use standard deviation of the sampling distribution instead of sampling distribution of the population

Empirical Rule for Data with a Bell-Shaped Distribution (3)

Example Problem (1)

The average GPA at a particular school is m=2.89 with a standard deviation s=0.63. A random sample of 25 students is collected. Find the probability that the average GPA for this sample is greater than 3.0.

Example Problem (2)

The time it takes students in a cooking school to learn to prepare seafood gumbo is a random variable with a normal distribution where the average is 3.2 hours with a standard deviation of 1.8 hours. Find the probability that the average time it will

take a class of 36 students to learn to prepare seafood gumbo is less than 3.4 hours.

Find the probability that it takes one student between 3 and 4 hours to learn to prepare seafood gumbo.

Confidence Interval Mathematical statement that says that the

parameter lies within a certain range of values

The average employee of XYZ Automotive has been employed between 8 and 12 years

95% confident that the mean length of employment at XYZ automotive is between 8 and 12 years

Probability Distribution for Population Mean

95% confident that the mean length of employment at XYZ automotive is between 8 and 12 years

(sample)

Confidence Level

Percent of confidence intervals that contain the population mean over the long run

Probability this confidence interval contains the population mean

95% confident that the mean length of employment at XYZ automotive is between 8 and 12 years

99% confident that the mean length of employment at XYZ automotive is between 8 and 12 years

Confidence Interval Formula

)(..N

ZXic

Confidence Interval Estimate

)1

(..

N

sZXic

Confidence Interval Example

A random sample of 100 television programs contained an average of 2.37 acts of physical violence per program. The standard deviation of the number of acts of violence on television is 3. At the 99% level, what is your estimate of the mean number of acts of violence for all television programs?

Alpha

Percent of confidence intervals that DO NOT contain the population mean over the long run

Probability this confidence interval DOES NOT contain the population mean

Complement of confidence level

Efficiency

Extent to which the confidence interval clusters around the mean

Width of the confidence interval Determined by population standard

deviation and sample size