Chapter 9 Sampling distributions. Parameter vs Statistic In each of the following determine which...

Chapter 9

Sampling distributions

Parameter vs Statistic

In each of the following determine which bold face # is a parameter or a statistic.

During the first quarter, Mr. Pines was planning on giving out on average about 20 baseball cards per student. After surveying 25 students at random there were actually about 32 cards per student given out.

A Parameter is the number that goes with the population,the number that we are expecting from prior studies or given information.

A Statistic is the number obtained from a sample.

Parameter vs Statistic

In each of the following determine which bold face # is a parameter or a statistic.

On a certain airline flight last week, 7% of the 195 passengers were selected for random security screening. Typically this same flight randomly selects 11% of its passengers.

A Parameter is the number that goes with the population,the number that we are expecting from prior studies or given information.

A Statistic is the number obtained from a sample.

Ch2 Vs Ch9

The Chapter 2 Test scores had a mean of 73 and a standard deviation of 9

Ch2 Question: What is the probability that a randomly chosen test score was greater than 88?

Ch9 Question: What is the probability that 5 randomly chosen test scores have a mean greater than 88?

Do you see the difference?

Symbols

The German Tank Problem


As a group you will need to come up with a “Statistic” for estimating the number of tanks.

You will be given 5 sample tank #’s.

Using these numbers only you will need to estimate the total number of tanks.

As a group you will need to come up with a “Statistic” for estimating the number of tanks.

You will be given 5 sample tank #’s.

Using these numbers only you will need to estimate the total number of tanks.


Your main goal is to create a “formula”

Your formula should give you a number greater or equal to the max tank number in the sample.

5 students will choose a tank number from the bag.

The closest group to the actual number of tanks will earn 5 cards each, 2nd will earn 3 cards, and 3rd will receive 1 card each.

Your main goal is to create a “formula”

Your formula should give you a number greater or equal to the max tank number in the sample.

5 students will choose a tank number from the bag.

The closest group to the actual number of tanks will earn 5 cards each, 2nd will earn 3 cards, and 3rd will receive 1 card each.

101 47 65 62 117

31 126 55 23 51

64 41 124 10 62

30 123 14 29 52

80 120 50 49 20

45 94 34 71 29

20 18 69 93 52

20 101 52 76 55

Let’s say that 5 different tanks were captured, how well does your “statistic” do for other samples. Use your statistic to calculate each below. Plot NEATLY on a dotplot. Same rewards as before for the closest.

Let’s say that 5 different tanks were captured, how well does your “statistic” do for other samples. Use your statistic to calculate each below. Plot NEATLY on a dotplot. Same rewards as before for the closest.


There were 127 TanksThere were 127 Tanks


This is the formula the British Mathematicians used.This is the formula the British Mathematicians used.


After 10 samples of 9 tanks.After 10 samples of 9 tanks.

127



127

Vocabulary

μ : this is the mean of a population, it is rarely known.

: this is the mean of a sample— the goal of is to be an estimator of μ.

σ –this is the populations’ standard deviation. s– this is the standard deviation of a sample

—it is meant to be an unbiased estimator of σ.

Means

The mean of the sample means should be close to μ.

The standard deviation of the distribution of sample means is

Central Limit Theorem(CLT)

CLT states that large samples become closer to the Normal Distribution

In this class, if a sample is 30+ we can assume the distribution is Normal by the CLT

Let’s practice

There are 200 boys in the freshman class with a mean height of 64” and a standard deviation of 3 inches. Assume that the heights are normally distributed. What is the probability that a randomly chosen boy will be 67 inches tall or more?

Same scenario as above. What is the probability that 10 randomly chosen boys will average 67 inches tall or more?

What is the mean and standard deviation of 20 kids? Of 50 kids?

Practice!!

Which question from the previous example would be most affected if the data is non-normal? Why?

a) A sample of 10 freshman

b) A sample of 20 freshman

c) A sample of 50 freshman

9.2 Sample proportions

Let’s sample for a proportion---roll 2 dice and tell me the proportion of times you get a 7. Chart the distribution of Is the mean of this distribution close to P. What is P in

this situation? Some facts about the distribution of .

It’s distribution is close to normal in shape if you have enough samples.

It’s mean is close to P It’s standard deviation gets smaller as the sample size

gets larger.

9.2 sample proportions

The mean of the distribution of is P (the often unknown parameter)

The standard deviation is There are two rules that you need to know involving

proportions. They are sometimes called the rule of thumb 1 and the rule of thumb 2

1. N ≥ 10n…. The overall population needs to be 10 times the sample size---this ensures that the issue of replacement is not violated so that the standard deviation will work.

2. np≥10 and n(1-p)≥10 ---this ensures that there is a big enough sample size for the distribution to be normal.

Means vs Proportions

Practice 9.2

Suppose a large candy machine has 15% orange candies. Imagine taking an SRS of 25 candies from the machine and observing the sample proportion p-hat of orange candies.

(a) What is the mean and standard deviation of the sampling distribution of p-hat?

(b) Check the 10% condition and check to see if the Normal condition is met.

(c) If the sample size was 75 instead of 25, how would this change the sampling distribution of p-hat?

Practice 9.2 continued

(d) Taking an SRS of 75 candies, what is the probability of obtaining a sample with more than 21% orange candies?

Practice

A polling organization asks an SRS of 1500 first year college students whether they applied for admission to any other college. In fact, 35% of all first-year students applied to colleges besides the one they are attending. What is the probability that the random sample of 1500 students will give a result within 2 percentage points of this true value?

Different Wording in this problem.

This is what a typical “AP Test” FRQ may look like.This is what a typical “AP Test” FRQ may look like.

What sample percent of binge drinking would have the same chance of happening on the other end of the normal curve?What sample percent of binge drinking would have the same chance of happening on the other end of the normal curve?

The answer means that if the professor took many samples on different days he would get a sample of 39.3% binge drinkers or lower about 7% of the time.(In statistics 7% is not considered to low a percent)

Similar idea as previous slide

Rene has a mean score of 71 on his tests this year and a StDev of 6.(we will assume his scores follow a normal dist)

Rene scores an 80% on his next test. Should the teacher be surprised? Should the teacher accuse him of cheating? Explain.

To speed things up lets use a calculator.

normalcdf(.80,1000,71,6) = .0668

To speed things up lets use a calculator.

normalcdf(.80,1000,71,6) = .0668

This means that we could expect Rene to score an 80 or higher on a test about 7% of the time.

This means that we could expect Rene to score an 80 or higher on a test about 7% of the time.

The teacher should not accuse him of cheating or be surprised.The teacher should not accuse him of cheating or be surprised.

p = .0668 is about 7% which is about how often you can expect Rene to score 80 or higher if his mean is in fact 71 with a stdev of 6.p = .0668 is about 7% which is about how often you can expect Rene to score 80 or higher if his mean is in fact 71 with a stdev of 6.

MECHANICS

Just Kidding, I drew this one

MECHANICS

CLEAN IT UP!!!!

Poker Chip Activity83 Students drew 20 poker chips randomly and found the % of RED chips in their sample

83 Students drew 20 poker chips randomly and found the % of RED chips in their sample

There were 200 total poker chips in the bagThere were 200 total poker chips in the bag

Can you make an estimate of P? The true % of RED poker chips in the bag.

Can you make an estimate of P? The true % of RED poker chips in the bag.

There were exactly 100 RED poker chips P = .50There were exactly 100 RED poker chips P = .50

Poker Chip ActivitySome samples were far from P.Some samples were far from P.

Poker Chip ActivitySome samples were far from P.Some samples were far from P.But if you take enough samples

you will start to see the distribution center itself on P.

But if you take enough samples you will start to see the distribution center itself on P.

This is why in this chapter that we use p-hat as an unbiased estimator for P.

This is why in this chapter that we use p-hat as an unbiased estimator for P.

This is called a sampling distribution…which will have a symmetric shape, a mean of p-hat, and stdev based on the sample size n

This is called a sampling distribution…which will have a symmetric shape, a mean of p-hat, and stdev based on the sample size n

#1 MC on HW Explained

#1 MC on HW Explained

Samples of size 2 taken from the population of previous slideSamples of size 2 taken from the population of previous slide

Sampling with Quarters

There are 405 quarters which has a mean year of 1994 and a standard deviation of 11.53.

The distribution has a strong skew to the left.

There are 405 quarters which has a mean year of 1994 and a standard deviation of 11.53.

The distribution has a strong skew to the left.

Sampling Distribution

Using your calculator we will simulate taking samples of 3 quarters.

In order to get different results we need to seed our calcs……your#STO>rand

Press the button “MATH” then go to “PRB”, go to 5:randInt(

Type randInt(1,405,3)


Find the 3 quarters that the calculator gave to to you.

Find the mean of these 3 quarters

Round to the nearest year.

We will construct a dotplot.

Population of Quarters

Mean = 1994StDev = 11.53

Dot Plot of n = 3

Mean = 1993.6StDev = 6.43343 Sampling Distribution of 67 samples of size 3


Now a bigger sample should be more accurate to the mean, reduce variability, which should reduce StDev.

Let’s sample for 9 quarters.

Find the mean of your 9 quarters.

Round to the nearest year.

We will construct a dotplot.

Dot Plot of n = 9

Mean = 1994StDev = 3.27263 Sampling Distribution of 67 samples of size 9

Mean = 1994StDev = 11.53

Mean = 1994StDev = 3.27263

Sampling Distribution of 67 samples of size 9

Mean = 1993.6StDev = 6.43343

Sampling Distribution of 67 samples of size 3

POPULATION

The sampling distribution for size 2 would be approximately Normal. The shape would be symmetric and centered around .975. The Standard Deviation would decrease to about .81375

The sampling distribution for size 2 would be approximately Normal. The shape would be symmetric and centered around .975. The Standard Deviation would decrease to about .81375

Vocabulary

Parameter– The “parameter of interest” is a description of the population at hand. Men over 40 Rancho Boys in the freshman class. People who shop at Wal-Mart.

Statistic—a sample of the overall parameter that attempts to describe the parameter. Sample of 20 boys from the freshman class

randomly selected. Sample people as they leave the checkout at

Wal-Mart.

Vocabulary Continued

Bias—how far off from the parameter your statistic is. (Could have high or low bias)

Variability---if you took multiple samples how dispersed are these samples.—see next slides for how it may look.

Vocabulary continued

P: P is a parameter, it is the true population proportion.

: this is the proportion of a statistic involving proportions.

Vocabulary. Know these terms!

Parameter Statistic Sampling Distribution Unbiased Estimator vs Biased Estimator Variability & Bias

Activity for 9.1

Flip a coin 10 times and report your proportion of heads. Chart the distribution of P-hat in this situation. Is P-hat close to the known P of .5?

It is assumed that 20% of all Americans will get cancer. Simulate 20 Americans on a random digit table and decide how you will simulate 20%. Then report what proportion of your 20 American’s will get cancer.

The point for 9.1!!

Here’s the point—if you sample well enough, your mean of your sampling distribution will match exactly the parameter of the population (either P or μ).

Take a sample of numbers like 2,4,6,8 and do all possible combinations of size 2. Chart the work. See if the resulting sampling distribution has a mean of 5, which is the mean of those 4 numbers. TRY IT!!

Relating sample proportions with sample means and binomials What is the probability of rolling two die 500

times and getting more than 100 sevens? Do this with a binomial calculation. Remember

to cover the requirements (assumptions) Do this with a proportion calculation.

Remember to cover the assumptions or requirements.

Do this with a means calculation. Remember to cover the requirements.

Sketch the Normal Curve

Means ProblemStDev = .21457Z > -1.23Calculate p

MAKE IT NEAT!!!!

CH 9 Vocabulary

The sampling distribution of p-hat has a mean equal to the population proportion p.

The sampling distribution of p-hat is considered close to Normal provided that np ≥ 10 and n(1-p) ≥ 10.

Sample means and sample proportions are called unbiased estimators for their corresponding population parameters

Smaller samples have more variability than larger samples

CH 9 Vocabulary

Sampling error is natural variation between samples, is always present, it can be reduced but not eliminated.

Sampling error is smaller when the sample size is larger.

Provided that the population size is greater than the sample size, the spread of a sampling distribution is about the same no matter the population size

When n is large, the sampling distribution of x-bar is approx normal even if the population is not normal

Chapter 9 Sampling distributions. Parameter vs Statistic In each of the following determine which...

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