DID YOU DO BETTER THAN EXPECTED ON EXAM 1?

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25% 25%25%25%1. Yes2. No3. I don’t know yet.

UPCOMING WORK

Quiz #4 in class this Thursday

HW #8 due next Sunday

Part 3 of Data Project due April 2nd in class

POSSIBLE TESTS

One-proportion z-test

Two-proportion z-test

One-sample t-test for mean

Two-sample t-test for differences of means

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EXAMPLES OF ONE-PROPORTION TEST

Everyone (100%) believes in ghosts

More than 10% of the population believes in ghosts

Less than 2% of the population has been to jail

90% of the population wears contacts

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EXAMPLES OF TWO-PROPORTION TESTS

Women believe in ghosts more than men

Blacks believe in ghost more than whites

People who have been to jail believe in ghosts more than people who haven’t been to jail

Women smoke more than men

Women use facebook in the bathroom more than men

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EXAMPLES OF ONE-SAMPLE T-TEST

All Priuses have fuel economy > 50 mpg

Ford Focuses get 5 mpg on average

The average starting salary for ISU graduates >$100,000

The average cholesterol level for a person with diabetes is 240.

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EXAMPLES OF TWO-SAMPLE T-TEST

The MPG for the Prius is greater than the MPG for the Ford Focus

ISU male graduates have a greater starting salary than women

The cholesterol levels are the same for people with and without diabetes.

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CHAPTER 18Sampling Distribution Models

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TWO-TYPES OF SAMPLING DISTRIBUTION: Proportion: parameter is p

Examples: p=proportion of people who have believe in ghosts p=proportion of cars made in Japan p=proportion of internet sales that are shipped on time

Mean: parameter is μ Examples:

μ= Average level of monoxide emitted from a car μ=Average payoff in a game of craps μ=Average starting salary for ISU graduate

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MODELING THE DISTRIBUTION OF SAMPLE PROPORTIONS

What proportion of the population believes in ghosts? Sample ten people, randomly

Sampling Distribution is the distribution you would get if you repeatedly sample the population. It’s a theoretical distribution.

What does the distribution look like?

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SAMPLING DISTRIBUTION OF PROPORTIONS -WHAT DOES IT LOOK LIKE?

The histogram of the sample proportions 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.

Unimodal, symmetric, and centered at p.

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A NOTE ON THE STANDARD DEVIATION

The standard deviation of a sample, s, is just the square root of the variance and is measured in the same units as the original data.

The standard deviation of a sampling distribution, is σ(p)=

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s

y y 2n 1

pq

n

SAMPLING DISTRIBUTION FOR A PROPORTION - PROBLEM

From past experience, I have found that 60% of my students believe in ghosts.

I sample ten students from Fall 2012.

What’s the probability that the sampled proportion of students, who believe in ghosts, is greater than 90%?

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THE SAMPLING DISTRIBUTION MODEL FOR A PROPORTION (CONT.)

Provided that the sampled values are independent and the sample size is large enough, the sampling distribution of is modeled by a Normal model with Mean: Standard deviation: .

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p̂

( )p̂ p

( )ˆ pqSD p n

WHAT IS THE MEAN AND STANDARD DEVIATION OF THE PROPORTION OF THE POPULATION THAT BELIEVES IN GHOSTS?

1. Mean = 0.6 SD= 0.6*0.4

2. Mean = 0.6 SD= sqrt(0.6*0.4)

3. Mean = 0.6 SD= sqrt(0.6*0.4/10)

4. Mean = 0.6 SD= 0.6*0.4/10

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MODELING THE DISTRIBUTION OF SAMPLE PROPORTIONS (CONT.)

A picture of what we just discussed is as follows:

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WHAT’S THE Z-SCORE FOR A PROPORTION OF 0.90?

1. Z=(0.90-0.60)/0.1552. Z=(0.60-0.90)/ 0.1553. Z=(0.90- 0.155)/ 0.64. Z=(0.155 -0.90)/0.6

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WHAT’S THE PROBABILITY THAT THE SAMPLED PROPORTION OF STUDENTS WHO BELIEVE IN GHOSTS IS GREATER THAN 90%?

1. 0.97382. 1-0.97383. 0.94. 0.6

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SAMPLING DISTRIBUTIONS – PROPORTION VS. MEAN

But which Normal model? For proportions, the sampling distribution is

centered at the population proportion. For means, it’s centered at the population mean.

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BUT WHICH NORMAL? (CONT.)

SD yn

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The Normal model for the sampling distribution of the mean has a standard deviation equal to

where σ is the population standard deviation.

SAMPLING DISTRIBUTION FOR A MEAN - PROBLEM

From past experience, the average starting salary of an ISU graduate is $52,000 with a SD of $5,000

I survey 100 graduates.

What’s the probability that their average salary is less than $40,000?

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WHAT IS THE Z-SCORE FOR $40,000?

1. Z=(40,000-52,000)/5,000/1002. Z=(40,000-52,000)/5,000/1003. Z=(40,000-52,000)/ sqrt(25,000,000/100)4. Z=(40,000-52,000)/sqrt(5,000/100)

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WHAT’S THE PROBABILITY THAT THE SAMPLE’S AVERAGE SALARY IS LESS THAN $40,000?

1. 0.22362. 1-0.22363. .44. .6

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THE FUNDAMENTAL THEOREM OF STATISTICS

The Central Limit Theorem (CLT)The mean of a random sample has a sampling distribution whose shape can be approximated by a Normal model. The larger the sample, the better the approximation will be.

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WHEN TO APPLY THE NORMAL MODEL?

Random Observations

Independent Trials

Each sample is the same size (n)

Sample size is appropriately large

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ASSUMPTIONS AND CONDITIONS (CONT.)1. Independence

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

1. Unbiased2. Representative of the Population

2. 10% Condition: If sampling has not been made with replacement, then the sample size, n, must be no larger than 10% of the population.

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ASSUMPTIONS AND CONDITIONS (CONT.)

1. Large Enough Sample

1. Success/Failure Condition: The sample size has to be big enough so that both and are greater than 10.

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GHOST PROBLEM REVISITED

From past experience, I have found that 60% of my students believe in ghosts.

I sample ten students from Fall 2012.

Can I use the Normal model to approximate a sampling distribution?

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ARE THE CONDITIONS NECESSARY TO USE THE NORMAL MODEL MET?1. Yes, all the conditions are met2. No, the 10% condition is not met3. No, the randomization condition is not met4. No, the success/failure condition is not met5. No, the randomization and success/failure

condition are not met6. No, none of the conditions are met

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0% 0% 0%0%0%0%

HW - PROBLEM 1

State police believe that 40% of the drivers traveling on a major interstate highway exceed the speed limit. They plan to set up a radar trap and check the speeds of 20 cars.

Using the 68-95-99.7 rule, draw and label the distribution of the proportion of these cars the police will observe speeding.

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DO YOU THINK THE APPROPRIATE CONDITIONS NECESSARY FOR YOU ANALYSIS ARE MET?

1. Yes, all the conditions are met2. No, the 10% condition is not met3. No, the randomization condition is not met.4. No, the success/failure conditions are not

met5. No, none of the conditions are met.

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THE REAL WORLD AND THE MODEL WORLD

Be careful! Now we have two distributions to deal with.

The first is the real world distribution of the sample, which we might display with a histogram.

The second is the math world sampling distribution of the statistic, which we model with a Normal model based on the Central Limit Theorem.

STANDARD DEVIATION

ˆpq

SD pn

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Both of the sampling distributions we’ve looked at are Normal. For proportions

For means

SD yn

EXAMPLE – COIN TOSS

You flip a coin 25 times and get a head 18 times (72% of the time).

Is the coin fair?

p=.5 SD=.1

What if you flipped it 64 times? SD=.0625

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HW - PROBLEM 4

The national freshman-to-sophomore retention rate has held steady at 74%.

Acme College has 490 of the 592 freshman return as sophomores.

Does this college have the right to brag that it has an unusually high retention rate?

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CAN THIS COLLEGE BRAG ABOUT ITS RETENTION RATE?

1. Yes, b/c their retention rate is not more than 3 SD above the expected rate.

2. Yes, b/c their retention rate is more than 4 SD above the expected rate.

3. No, b/c their retention rate is not more than 3 SD above the expected rate.

4. No, b/c their retention rate is more than 4 SD above the expected rate.

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STEPS TO CALCULATE THE PROBABILITY FROM A SAMPLING DISTRIBUTION

Calculate mean or proportion from the sample.

Determine if you know the true proportion or mean.

Calculate the standard deviation Determine the ‘standard’ you are comparing

your sample to. Calculate z-score of that ‘standard’ Find percentile of that z-score.

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HW - PROBLEM 5

Just before a referendum on a recycling mandate, a local newspaper polls 358 voters to predict whether the mandate will pass.

Suppose the mandate has the support of 53% of the voters.

What is the probability that the newspaper’s sample will lead it to predict defeat?

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WHAT IS THE PROBABILITY THAT THE NEWSPAPER’S SAMPLE WILL LEAD IT TO PREDICT DEFEAT?1. .12922. 1-.12923. .54. .53

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SAMPLING DISTRIBUTIONS ARE AWESOME!

Sampling distribution models are important becausethey act as a bridge from the real world of

data to the imaginary model of the statistic and

enable us to say something about the population when all we have is data from the real world.

HW - PROBLEM 6 When a truck load of apples arrives at a packing

plant, a random sample of 125 is selected and examined for bruises, discoloration, and other defects.

The whole truckload will be rejected if more than 5% of the sample is unsatisfactory.

Suppose that in fact 9% of the apples on the truck do not meet the desired standard.

What is the probability that the shipment will be accepted anyway.

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WHAT IS THE PROBABILITY THAT THE SHIPMENT WILL BE ACCEPTED ANYWAY?

1. 0.0592. 1-0.0593. 04. 15. .05

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0% 0% 0%0%0%

UPCOMING WORK

Quiz #4 in class this Thursday

HW #8 due next Sunday

Part 3 of Data Project due April 2nd in class