1. EQ: How are sample means distributed?
Chapter 9 Sample means
EQ: How are sample means distributed?

2. C. All people living in the state of Colorado.
1. A large simple random sample of people aged nineteen to thirty living in the state of Colorado was surveyed to determine which of two MP3 players just developed by a new company was preferred. To which of the following populations can the results of this survey be safely generalized?
A. Only people aged nineteen to thirty living in the state of Colorado who were in this survey.
B. Only people aged nineteen to thirty living in the state of Colorado.
C. All people living in the state of Colorado.
D. Only people aged nineteen to thirty living in the United States.
E. All people living in the United States.

3. 2. The buyer for an electronics store wants to estimate the proportion of defective wireless game controllers in a shipment of 5,000 controllers from the store’s primary supplier. The shipment consists of 200 boxes each containing 25 controllers. The buyer numbers the boxes from 1 to 200 and randomly selects six numbers in that range. She then opens the six boxes with the corresponding numbers, examines all 25 controllers in each of these boxes, and determines the proportion of the 150 controllers that are defective. What type of sample is this? A. Biased random sample B. Nonrandom sample C. Simple Random sample D. Stratified random sample E. Cluster random sample

4. Process of Inference Population Sample Parameter Statistic measure
estimate
Parameter
Statistic

5. Population proportion Population standard deviation
Key Terms
Parameter: a number that describe the population.
Statistic: a number that describes a sample.
Want to estimate
Statistic
Parameter
Population proportion
Population mean
Population standard deviation

6. We want unbiased estimators

7. Sampling Distribution: Means
- Take a random sample of 10 heights from the list of heights of individuals taking AP Statistics. - Find the mean of your 10 heights - Plot your x-bar on dotplot 1

8. Sampling Distribution: Means
- Take a random sample of 25 heights from the list of heights of individuals taking AP Statistics. - Find the mean of your 25 heights - Plot your x-bar on dotplot 2

9. Sampling Distribution: Means
The mean of the population is –
The standard deviation of the population is –
What do you notice of the mean and standard deviation of the two sampling distributions?

10. The Sampling Distribution of
When we choose many SRSs from a population, the sampling distribution of the sample mean is centered at the population mean µ and is less spread out than the population distribution. Here are the facts.
Sampling Distribution of a Sample Mean
as long as the 10% condition is satisfied: n ≤ (1/10)N.

11. Sampling From a Normal Population
Sampling Distribution of a Sample Mean from a Normal Population

12. What we should have learned:
When samples are taken from a normal distribution:
The sample means will be approximately Normally distributed
Sample size doesn’t matter but must be from a SRS

13. Reel Talk Fishing Tourney
Boat #
# of fish caught 1 25 2 7 3 22 4 23 5 16 6 19 18 8 9 20 10 21 11 17 12 13 27 14 26 15 24
Reel Talk Fishing Tourney
The data to the left is for the number of fish caught by the 20 boats in the Reel Talk Fishing Tournament:
Take a random sample of 8 boats and find the average number of fish caught.
Take 2 more random samples of 8 boats and record the average number of fish caught
Plot your 3 x-bar vales on the dotplot on the board

14. The Central Limit Theorem
Most population distributions are not Normal. What is the shape of the sampling distribution of sample means when the population distribution isn’t Normal?
It is a remarkable fact that as the sample size increases, the distribution of sample means changes its shape: it looks less like that of the population and more like a Normal distribution!
When the sample is large enough, the distribution of sample means is very close to Normal, no matter what shape the population distribution has, as long as the population has a finite standard deviation.

15. The Central Limit Theorem
As the previous example illustrates, even when the population distribution is very non-Normal, the sampling distribution of the sample mean often looks approximately Normal with sample sizes as small as n = 25.
Normal/Large Condition for Sample Means
The central limit theorem allows us to use Normal probability calculations to answer questions about sample means from many observations even when the population distribution is not Normal.

16. What we should have learned:
When samples are taken from ANYdistribution:
According to the Central Limit Theorem, the sample means will be approximately
Normally distributed
As long as n is large and from a SRS

17. Be Careful
The reason we can use a normal distribution here is because we are taking a sample
If we were just looking for the probability that 1 boat would be at a certain value we can’t use a normal probability model for non-normal data
In this example (and many of the examples we are going to do) we take a sample and then take a look at the likelihood of getting a certain mean for that sample.

18. Conditions for Sampling Means
1) Either sample is: - From a normal distribution OR - a large sample (typically more than 30) 2) Sample is taken using SRS

19. Young women’s height
The height of young women follows a Normal distribution with mean mu=64.5 in and standard deviation = 2.5 inches.
Find the probability that a randomly selected woman is taller than 66.5 inches. Show work.
Find the probability that the mean height of an SRS of 10 young women exceeds 66.5 in. Show work. (mean, st.d.)

20. Servicing AC (CLT)
Your company has contracted to perform preventive maintenance on thousands of air-conditioning units in a large city. Based on service records from the past year, the time (in hours) that a technician requires to complete the work follows a strongly right-skewed distribution with mu=1 hour and st.d=1 hour. In the coming week, your company will service and SRS of 70 a/c units in the city. You plan to budget on average of 1.1 hours per unit for the tech to complete the work. Will this be enough?
Problem: What is the probability that the average maintenance time x bar for 70 units will exceed 1.1 hours? Not normal but passes CLT = There is a 20% chance that the techs will NOT complete there work in the budget time. You will have decide if to take the risk or budget more time.