Statistical Analysis Chapter 5 Central Limit Theorem Dr. Roderick Graham Fashion Institute of Technology.

Statistical Analysis – Chapter 5“Central Limit Theorem”

Dr. Roderick GrahamFashion Institute of Technology

The Central Limit Theorem

In chapter 4, we looked at drawing samples from a BINOMIAL POPULATION

Now, we are concerned with predicting means for sampling distributions (not individual values)

We will call the distributions of these samples “x-bar distributions ”

X

Equations for x-bar distributions….(when we want to know information about our sample drawn from a population)

X

nX /

X

Xz

Mean of X-bar distribution

Standard Deviation of X-bar distribution(in SPSS called “Standard Error”)

Z- score (in this case, you must use standard deviation of X-Bar)

With these equations, you can use the normal table to predict SAMPLE SAMPLE DISTRIBUTION VALUESDISTRIBUTION VALUES.

The large histogram is the distribution of individual values.

The small x-bars are distributions of sample averages for the individual values

If we collect enough sample averages, the averages will be normally distributed, and look like a normal curve

Because of this, distribution, we can make predictions

Taking Samples from Populations

This is Parent’s Age from the data we used for our project. µ = 44.74. Let’s imagine that this is the true population value. (The N is large, so we can assume this.)

Taking Samples from Populations

Because the true population is normal, any sample we draw will have approx. the same mean as the true population, and a stand. dev. that decreases as the N of our sample increases.

Taking Samples from PopulationsSample N = 92 = 44.86 .715

Sample N = 266 = 44.33 .453

Sample N = 502 = 44.46 .326

Pop. Mean = 44.74

XX

X

X

X

X

Means of sample close to means of population

Standard deviation changes as N changes

The Central Limit Theorem

For sample sizes with n > 30, the sample distribution (x-bar distribution) (x-bar distribution) will be normally distributed with mean µ and standard deviation σ/√n.

Note: If we assume that the population is normally distributed, the sample size is not important.

The Central Limit Theorem

Because of the central limit theorem, you can use the normal table for any random sample over 30 respondents to find the probability of sample averages!

The main idea behind the problems we will work today is that we are trying to understand the probabilities of getting sample values using the z scores and the normal table.

When solving x-bar problems…

X

nX /

X

Xz

The mean of the distribution is the mean of the true population

The n is the total number of respondents in your sample.

You use this z – score to find percentages on the normal table.

Rounding Technique for the Remainder of the Course1. Work in three decimal places for the entire

problem

2. Round your final answers to two decimal places

3. Z-scores are always presented in two decimal places (because on the normal table chart, z-scores are up to two decimal places)

Sample Problems… Let’s do this problem (problem 5.1) I’ll do A, and

then I’ll get volunteers to help us set up B and C as a class.

A machine cuts pieces of silk material to an average length of 1000 mm with a standard deviation of 12 mm. Between what two lengths would we expect to find the middle 95% of all sample averages?

a)n = 36b)n = 144c)n = 576

Sample ProblemsAnswer to problem 5.1

1. Calculating the standard deviation for each N. 2. Find the z-score associated with 47.5%. This because we need

95%, and we need 47.5% on the positive side 0, and 47.5% for the negative side of 0. The z-score for 47.5% is +/-1.96.

3. Then you use the z score formula to solve for X-bar.

a. Between 996.08 mm and 1003.92 mmb. Between 998.04 mm and 1001.96 mmc. Between 999.02 mm and 1000.98

THIS PROBLEM SHOWS THAT THE LARGER OUR SAMPLE SIZE, THE MORE ACCURATE OUR PREDICTIONS BECOME!

Sample Problems….(5.8, p. 157)A nationwide marketing study concluded that the average age of horror film moviegoers is 17.4 years old with a standard deviation of 2.7 years.

a. Assuming a normal distribution, what percentage of horror film moviegoers nationwide would you expect to be over 18 years old? (hint: when we assume a normal distribution, we don’t worry about the N of the sample)

b. If we take random samples of 81 horror film moviegoers nationwide and calculate the sample average for each sample, what percentage of sample averages would you expect to be over 18 years? (hint: now we are not assuming a random sample, and we are given an N)

Sample Problems… Let’s do problem 5.12

A survey indicated the average yearly salary of entry-level women managers to be µ = $56,700 with σ = $7,200.

a) Assuming a normal distribution, what is the probability a woman manager’s entry level salary will exceed $58,000?

b) What is the probability a random sample of 50 women managers will yield an average entry-level salary exceeding $58,000?

d) Assuming n = 42, with what probability can we assert a sample average will fall within $1500 of µ = $56700?