1. Stats: Modeling the World
Chapter 18
Sampling Distribution Models

2. Bias and Variability

3. The Fundamental Theorem of Statistics: AKA The Central Limit Theorem
Let’s imagine that we ask people “Do you believe in ghosts?”
-In 2005 one poll 40% of 889 adults said yes, in another during the same year 48% of 809 adults said yes.
Why the difference?
How can surveys taken at the same time with the same question get different results?

4. The Fundamental Theorem of Statistics: AKA The Central Limit Theorem
The answer is at the heart of statistics!
Different samples will lead to different proportions!
Can we predict how much a sample proportion will vary?
YES!

5. The Fundamental Theorem of Statistics: AKA The Central Limit Theorem
Imagine we take all possible random samples of 808 people that were asked the ghost question-
Where would the center be???
Well, we don’t know! – but we do know that the center will be the true proportion – let’s call it p
For argument’s sake let’s call p = 0.45
What would be the shape of the histogram?

6. The Fundamental Theorem of Statistics: AKA The Central Limit Theorem
It would look like this:
We call this a sampling distribution of proportions and each sampling
proportion is called (p=hat) This is a normal curve with a
and a standard deviation of

7. The Fundamental Theorem of Statistics: AKA The Central Limit Theorem
For our example:
and
Remember the Rule?
This 95% of the data is within two standard deviations or 2(1.75%)=3.5%
and the poll which gave us 48% belief in ghosts is consistent with our guess of 45%
The 3.5% is called the sampling error (not really an error but the variability from one sample to another – better term: sampling variability

8. The Fundamental Theorem of Statistics: AKA The Central Limit Theorem
What Assumptions do we have to make to do all of this?
INDEPENDENCE: the sampled values must be independent
SAMPLE SIZE: the sample size, n, must be large enough
We check these by checking 3 conditions
Randomization: was the data collected representative of the population
10% condition: The sample size, n, must be no larger than 10% of the population
Success/Failure condition: np and nq need to be at least 10

9. The Fundamental Theorem of Statistics: AKA The Central Limit Theorem: Example
The candy company that makes M & M’s claims that 10% of the candies are green. In a really large bag of 500 M & M’s a class of statistics students found that 12% were green. Is this unusual?
Step 1:Check Conditions
Randomization: It would be reasonable to assume the candies were placed in the bag randomly
10% condition: 500 M & M’s is less than the population of M & M’s
Success/Failure: np=500(0.10)=50
and nq = 500(0.90)=450 are both greater than 10

10. The Fundamental Theorem of Statistics: AKA The Central Limit Theorem: Example
The candy company that makes M & M’s claims that 10% of the candies are green. In a really large bag of 500 M & M’s a class of statistics students found that 12% were green. Is this unusual?
So,
By the Rule, 1.49 is within 2 standard deviations, so this result is not unusual.

11. The Fundamental Theorem of Statistics: AKA The Central Limit Theorem: Example
Suppose 13% of people are left-handed. At a particular college lecture hall there are 200 seats with built in desks, 15 of these of these desks are built for left-handed students. In a class of 90 students, what is the probability that there will not be enough seats for the left-handed students?
PLAN: I want to find out the probability that in a group of 90 students, more than 15 will be left handed (15/90 = 16.7%).
So my sample size is 90, the proportion of lefties in a population is 13%, and I want to find P(L > 16.7%)

12. The Fundamental Theorem of Statistics: AKA The Central Limit Theorem: Example
Suppose 13% of people are left-handed. At a particular college lecture hall there are 200 seats with built in desks, 15 of these of these desks are built for left-handed students. In a class of 90 students, what is the probability that there will not be enough seats for the left-handed students?
Check Conditions:
Independence: It is reasonable to assume that left-handedness is independent
Randomization: The 90 students in the class can be thought of as a random sample
10% condition: 90 students is certainly less than 10% of all studnets
Success/Failure: np = 90(0.13)=11.7 and nq = 90(0.87) = 78.3
which are both larger than 10

13. The Fundamental Theorem of Statistics: AKA The Central Limit Theorem: Example
Suppose 13% of people are left-handed. At a particular college lecture hall there are 200 seats with built in desks, 15 of these of these desks are built for left-handed students. In a class of 90 students, what is the probability that there will not be enough seats for the left-handed students?
OK—
Conclusion: There is about a 14.5% chance that there will not be enough seats for the left-handed students in the class.

14. A cable company believes 20% of the subscribers of a cable television company watch the shopping channel at least once a week. The cable company is trying to decide whether to replace this channel with a new local station. A survey of 100 subscribers will be undertaken. The cable company has decided to keep the shopping channel if the sample result is greater than 25%. What is the approximate probability that the cable company will keep the shopping channel? What is the population proportion and standard deviation? Find your z-score and the probability:
p = St Dev=
There is a probability of that the cable company will keep the shopping channel

15. Suppose that 60% of the high school faculty in the state voted in favor of 4 years of math as a graduation requirement. The Californian will be contacting 200 faculty members selected at random as a follow-up. What is the probability that less than half of the faculty members polled voted for the requirement? What is the population proportion and standard deviation? Find your z-score and the probability:
p = St Dev=
The probability that less than half of the faculty members polled voted for the requirement is

16. It is estimated that 48% of all motorists in Brazil use their seat belts. If a police officer observes 400 cars go by in one morning, what is the probability that the proportion of drivers wearing seat belts is between 45% and 55%? What is the population proportion and standard deviation? Find your z-score and the probability:
p = St Dev=
There is a probability that between 45% and 55% of the drivers are wearing their seat belts.

17. Sampling Distributions for Means
Simulation

18. The Central Limit Theorem
The Central Limit Theorem (The Fundamental Theorem of Statistics) says that the sampling distribution for any mean becomes nearly NORMAL as the sample size grows!
The mean =
And the standard deviation =

19. Consider the approximately normal population of heights of male college students with mean of 69 inches and standard deviation of 5 inches. A random sample of 25 heights is obtained.
Find the mean and standard deviation of the average heights for sample sizes of 25.
Find the probability that the sample mean of 25 college male students will be greater than 73 inches. Show your work!

20. The Wechsler Adult Intelligence Scale (WAIS) is a common “IQ Test” for adults. The distribution of WAIS scores for persons over 16 years of age is approximately normal with a mean of 100 and a standard deviation of 15. What is the probability that a randomly chosen individual has a WAIS score of 105 or higher?
The probability that a randomly selected individual has a WAIS score above 105 is

21. The Wechsler Adult Intelligence Scale (WAIS) is a common “IQ Test” for adults. The distribution of WAIS scores for persons over 16 years of age is approximately normal with a mean of 100 and a standard deviation of 15. What is the mean and standard deviation of the sampling distribution of the average WAIS score for a random sample of 60 people? What is the probability that the average WAIS score for a sample of 60 people is 105 or higher?
Mean = St Dev =
The probability the average WAIS score for a sample of 60 people is 105 or higher is
What if this population was NOT normally distributed?!?!?!

22. Within 2 standard deviations of the mean…. 157 to 167 dF
The mean temperature of coffee sold in restaurants is normally distributed with a mean of 162 dF with a standard deviation of 10 dF. Find the mean and standard deviation for the sampling distribution for a random sample of 16 restaurants. Draw this model: What interval contains the middle 95% of the temperatures? What is the approximate probability that the sample of 16 restaurants has a mean temp below 158 dF?
Mean = 162
St Dev =
Within 2 standard deviations of the mean… to 167 dF
The probability mean temp for the 16 restaurants is below 158 dF is is

23. Based on data from the Highway Loss Data Institute, collision repair costs for Honda Odyssey minivans are found to have a distribution that is roughly bell shaped with a mean of $1786 and a standard deviation of $937 Suppose that your next door neighbor just had his Odyssey repaired and the bill was $2300. Is this an unusual repair bill? How do you know?
We can see how unusual this is by looking at its z-score to see how far from the typical repair cost this bill is:
This is less than one standard deviation from the mean, so this is fairly common. About 29% of cars have more expensive repair bills.

24. Based on data from the Highway Loss Data Institute, collision repair costs for Honda Odyssey minivans are found to have a distribution that is roughly bell shaped with a mean of $1786 and a standard deviation of $937 Suppose that you discovered that First OK Honda recently repaired five Odyssey’s and the average repair bill was $2300. How typical is this?
For a sample size of 5… Mean = St Dev =
This is a bit more than one standard deviation from the mean, so this is not extremely rare. More expensive repair bills will happen about 11% of the time with samples of 5 cars.

25. Mean = 3.9 St Dev = Roughly Normal
If we have 160 minutes for 40 songs, each song can last (on average) 4 minutes
The probability that the total amount of time needed exceeds the available airtime is