8.2 Sampling Distributions – Sample Means
1. Sampling Distributions: Review
There are a huge number of sample means you could get when you sample from a population. The set of all sample means for sample sizenmakes the sampling distribution of sample means.
Today we’re going to look at the sampling distribution of sample means. Later we’ll look at the sampling distribution of sample proportions.
So, here’s a critical question:
What is the mean and standard deviation of the
sampling distribution of sample means?
This is a HUGELY important question, because knowing this will tell us how accurate our sample mean is relative to the population mean!!! (do you see why?)
2. TheMost Important Rule in all of Statistics
Assume you take a sample of size n from a population of mean μ and standard deviation σ.
Let= the mean of the sampling distribution of sample means
Let = the standard deviation of the sampling distribution of sample means
Then FOUR things are true:
1) . Always.
* The sample means are centered at the population mean. Useful! Tells us we’re in the ballpark with a sample mean. But maybe thestandard deviation is really big, so that the sample means are really spread out? Then our sample mean could be way off…
2) .
-Always true if the population is infinite.
-Approximately true if the population is finite and n ≤ 0.10N (the sample size is less than 10% of the population).
*A stunner! The sample means are less variable than the population itself(do you see that in the formula?), and the larger your sample size, the less variable the means are (do you see that??)
3) When the population distribution is normal, then the sampling distribution of is automatically normal for any size n.
*Extremely nice if we’re a normal population. We can use this to figure out how accurate our sample mean is if our population is normal, as you will see. But what if it isn’t normal??
4) (The Central Limit Theorem) When n is sufficiently large (in practice,
n > 30), then the sampling distribution of is well approximated by a normal curve, even when the population isn’t normal.
*COOL!! This is the real deal in statistics. This theorem is a TEN in importance in statistics, as it will allow us to use z-scores to figure out how close our sample mean is, regardless of the shape of the population!!!
8.2 Sampling Distributions #2 – Sample Means, Continued
1. Review
In the sampling distribution of sample means,
1) . Always.
2) if n ≤ 0.10N (the sample size is less than 10% of the population).
3) If the population is Normal, then the sampling distribution of is automatically Normal.
4) If n > 30, then the sampling distribution of is approximately Normal, even if the population isn’t Normal.
2. HOT DOGS!
Suppose a hot dog company produces hot dogs with of fat per hot dog, with .
a. Describe the sampling distribution of sample means of sample size 36.
Hint: Center, shape, spread!
b. Suppose we sample 36 hot dogs. What’s the chance that we get a sample mean that’s greater than 19g of fat?
Answer: So we want to find . Not P(x > 19)!!
Remember that , and .
Check assumptions: (1) n > 30. √ Yes. What does that tell us?
(2) n ≤ 0.1N. √ Yes. What is N? What’s does that tell us?
= 6
(3) random sample. √ (so that it’s representative)
So we find the z-score for 19g, and find the area to the right.
6) = 0.0000.
Basically no chance! WOW!
c. A different company with claims that their hot dog hasof fat and no more. We do a sample and find that g. What can we make of their claim?
Answer: This is how it works.
Weassume that and then under those conditions find . These are the hot dogs that will disprove their claim!!
If the company’s claim is correct that , then the following will have to be true:
and
So:
= (how rare is a z of 2.4?)
= 1 – 0.9918 = What do we make of their claim?