Math 160 Chapter 17 Lecture Notes Fall 2015

8/18/2019 Math 160 Chapter 17 Lecture Notes Fall 2015

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Chapter 17: Sampling Distribution Models

Distribution of the Sample Proportion

Example: In a sample of 98 statistics students at Golden West College,

9 students failed the course. So, we would say the sample proportion of

statistics students at GWC who failed is ˆ p = 9/98 " 0.0918. Or we

could say that, in the sample of 98 students, 9.18% of the students

failed. We can use this sample proportion to estimate the proportion of

all statistics students who will fail at Golden West College.

If we surveyed every possible sample of size 98 statistics students at

GWC, we could find the proportion ____ for each sample who did not pass the course. Then we could see how much those sample proportions

varied.

Rather than showing real repeated samples, imagine what would happen

if we were to actually draw many samples.

Now imagine what would happen if we looked at the sample proportions for these samples.



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The histogram we’d get if we could see all the proportions from all

possible samples is called the sampling distribution of the proportions.

What would the histogram of all the sample proportions look like?

We would expect the histogram of the sample proportions to center at

the true proportion, p, in the population.

As far as the shape of the histogram goes, we can simulate a bunch of

random samples that we didn’t really draw.

It turns out that the histogram is unimodal, symmetric, and centered at

p.More specifically, it’s an amazing and fortunate fact that a Normal

model is just the right one for the histogram of sample proportions.

Modeling how sample proportions vary from sample to sample is one of

the most powerful ideas we’ll see in this course.

A sampling distribution model for how a sample proportion varies

from sample to sample allows us to quantify that variation and how

likely it is that we’d observe a sample proportion in any particular

interval.

To use a Normal model, we need to specify its mean and standard

deviation. We’ll put !, the mean of the Normal, at p.



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Describe the Sampling Distribution of a Sample Proportion

When describing the sampling distribution of a sample proportion, we

will identify three things:

(1) shape of the distribution

(2) the mean and

(3) the standard deviation.

When working with proportions, knowing the mean automatically gives

us the standard deviation as well—the standard deviation we will use is:

So, the distribution of the sample proportions is modeled with a

probability model that is

Shape:________ Mean:_________ Std Dev:________



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Because we have a Normal model, for example, we know that 95% of

Normally distributed values are within two standard deviations of the

mean.

So we should not be surprised if 95% of various polls gave results that

were near the mean but varied above and below that by no more than

two standard deviations.

This is what we mean by sampling error. It’s not really an error at all,

but just variability you’d expect to see from one sample to another. A

better term would be sampling variability.

The Normal model gets better as a good model for the distribution of

sample proportions as the sample size gets _______________.Just how big of a sample do we need? This will soon be revealed…

Assumptions and Conditions

Most models are useful only when specific assumptions are true.

There are two assumptions in the case of the model for the distribution

of sample proportions:

1.The Independence Assumption: The sampled values must be

independent of each other.

2. The Sample Size Assumption: The sample size, n, must be large

enough.

Assumptions are hard—often impossible—to check. That’s why we

assume them. Still, we need to check whether the assumptions are

reasonable by checking conditions that provide information about the

assumptions.



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The corresponding conditions to check before using the Normal to

model the distribution of sample proportions are the Randomization

Condition, the 10% Condition and the Success/Failure Condition.

1. Randomization Condition: The sample should be a simple randomsample of the population.

2.10% Condition: the sample size, n, must be no larger than 10% of

the population.

3. Success/Failure Condition: The sample size has to be big enough so

that both np (number of successes) and nq (number of failures) are atleast 10.

…So, we need a large enough sample that is not too large.



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Example 1: According to a study done by the Gallup organization, 82%

of Americans are satisfied with the way things are going in their lives.

Suppose a random sample of 100 Americans are asked, “Are you

satisfied with the way things are going in your life?” Describe the

sampling distribution of ˆ p, the proportion of Americans who are

satisfied with the way things are going in their life.

Practice 1: According to credicard.com, the proportion of adults who

do not own a credit card is 0.29. Suppose a random sample of 500

adults is asked, “Do you own a credit card?” Describe the sampling

distribution of ˆ p, the proportion of adults who own a credit card.



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Compute Probabilities of a Sample Proportion

Provided that the sampling distribution of ˆ p is normal and that the

sample size is less than or equal to105% of the population size, we can

find the mean and standard deviation of ˆ p. So, we can use the normal

curve to compute probabilities involving sample proportions.

Example 1 (continued ): According to a study done by the Gallup

organization, 82% of Americans are satisfied with the way things are

going in their lives. Suppose a random sample of 100 Americans is

asked, “Are you satisfied with the way things are going in your life?”

Would it be unusual for a survey of 100 Americans to reveal that 75 or

fewer are satisfied with the way things are going in their life? Why?



)

Example 2: According to credicard.com, the proportion of adults who

do not own a credit card is 0.29. Suppose a random sample of 500

adults is asked, “Do you own a credit card?”

a) What is the probability that in a random sample of 500 adults

between 25% and 30% do not own a credit card?

b) Would it be unusual for a random sample of 500 adults to result in

125 or fewer who do not own a credit card? Why?



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Distribution of the Sample Mean

Like any statistic computed from a random sample, a sample mean also

has a sampling distribution. We can use simulation to get a sense as to

what the sampling distribution of the sample mean might look like…

Let’s start with a simulation of 10,000 tosses of a die. A histogram ofthe results is:



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Averaging More Dice

What the Simulations ShowAs the sample size (number of dice) gets larger, each sample average is

more likely to be closer to the population mean. So, we see the shape

continuing to tighten around 3.5

And, it probably does not shock you that the sampling distribution of a

mean becomes Normal



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Central Limit Theorem



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Example 3: A simple random sample of size n = 20 is obtained from a population with µ = 64and " =17

a) What must be true regarding the distribution of the population in

order to use the normal model to compute probabilities involving the

sample mean? Assuming this condition is true, describe the sampling

distribution of x .

b) What is P(65.2 " x < 67.3)?



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What can go wrong?

Don’t confuse the sampling distribution with the distribution of the

sample.

When you take a sample, you look at the distribution of the values,

usually with a histogram, and you may calculate summary statistics.

The sampling distribution is an imaginary collection of the values that a

statistic might have taken for all random samples—the one you got andthe ones you didn’t get.

Example 4: In 2000, the mean ACT Math score was 20.7. If ACT Math

scores are normally distributed with a standard deviation of 5, answerthe following questions.

a) What is the probability that a randomly selected student has an ACT

Math score 18 or less? Draw and shade!



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Example 4: (cont.) In 2000, the mean ACT Math score was 20.7. If

ACT Math scores are normally distributed with a standard deviation of

5, answer the following questions.

b) What is the probability that a random sample of 10 ACT test takers

has a mean math score of 18 or less? Draw and shade!

c) What is the probability a random sample of size 15 will have a mean

math score 3 points within the mean? Draw and shade!

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Math 160 Chapter 17 Lecture Notes Fall 2015

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