Ch. 8: Sampling Distributions

Question 1

Look at picture in camera roll of symbols we will be using for Chapter 8 and know each of them and write on cheat sheet.

Answer 1

Okay

Question 1

• The Sampling Distribution of the Sample (proportion/mean) is for large populations, but finite (aka we can count).
• The Sampling Distribution of the Sample (proportion/mean) is for infinite population.

Answer 1

• mean
• proportion

Question 2

If we graph population and it looks roughly like a bell-curve, we call it _________ ___________.
If it looks like anything except a bell-curve, we call it ______-_______ _______.

Answer 2

normal population; non-normal population

Question 3

If we have normal distribution, we can use what to find probabilities?

Answer 3

z-tables

Question 4

taking a sample where everyone has the same probability of getting selected.

Answer 4

random sample

(ex: you have 1 million Auburn residents, and you take a random sample of 100,000 residents)

Question 5

T or F: If you find that sample mean (x with a line over top) = $50,000, then you can say that population mean (M) is equal to this.

Answer 5

FALSE.

This isn’t true. Our goal is to find a POINT ESTIMATE of population mean. (This is what we will do in this Chapter)

Question 6

Since sample mean (x with a line over top) does NOT equal population mean (M), we will start taking more samples from sample mean again. This is called _______ ________.

Answer 6

sampling distribution

(then, we can take the mean of this sample mean (aka Mx). This number would be our point estimate for our population mean. Look at camera roll for pic of this better explained)

Question 7

What is the formula for population mean?

Answer 7

M = Mx

(aka population mean = mean of sample mean. Use this formula when our population is finite (can count), very large population, and when our goal is to find a point estimate for population mean)

Question 8

Look over example in camera roll that starts with population = 3 elements.

Answer 8

Okay (2 pics total)

Question 9

Look at formulas to know page in camera roll.

Answer 9

Okay

Question 10

Population Standard deviation > standard deviation of sample mean. Why is this?

Answer 10

Because when we have a large population, population standard deviation is going to be more, and when we take sample, population is going to be small, which means standard deviation of sample mean will be small.

Question 11

the probability distribution of the population of the sample means obtainable from all possible samples of size n from a population of size N

Answer 11

sampling distribution of the sample mean (x with a line over top)

(Know this definition!)

Question 12

Look over example 8.1 (Car Mileage Case) in camera roll.

Answer 12

Okay

Question 13

Basic Properties:
1. In many situations, the distribution of the population of all possible sample means looks roughly like a ______ curve.
2. If the population is normally distributed, then for any sample size n the population of all possible ______ _______ is also normally distributed.
3. The mean, Mx, of the population of all possible sample means is equal to ____.
4. The standard deviation, σx, of the population of all possible sample means is less than _____.

Answer 13

1. normal (bell-shaped)
2. sample means (know this one)
   (aka if our population, N, is normally distributed, then no matter what our sample size is (could be 2,3, or whatever), our sampling distribution shape will be normal)
3. M (population mean)
4. σ (population standard deviation)

Question 14

T or F: If the population is normally distributed, then sampling distribution is also normally distributed.

Answer 14

True (basic property #2)

Question 15

If the population of individual items is normal, then the population of all sample means is also normal. Even if the population of individual items is not normal, there are circumstances when the population of all sample means is normal. What theorem is this?

Answer 15

Central Limit Theorem
(when the population is not normally distributed; will look into this later… population won’t always be normal all the time)

Question 16

The Empirical Rule holds for the sampling distribution of the sample mean:
1. ____% of all possible sample means are within (plus or minus) one standard deviation σx of M.
2. ____% of all possible observed values of x are within (plus or minus) two σx of M.
3. ____% of all possible observed values of x are within (plus or minus) three σx of M.

Answer 16

1. 68.26%
2. 95.44%
3. 99.73%

Question 17

What is the formula for the variance of the sampling distribution of x?

Answer 17

= σ^2 / n

(in camera roll)

Question 18

The variance of the sampling distribution of x (with a line over top) is:
1. ______ proportional to the variance of the population.
2. ______ proportional to the sample size.

Answer 18

1. directly
2. inversely

Question 19

What formula do you use to find standard deviation of the sample mean?

Answer 19

σ / square root of n

(look at camera roll for this and write on cheat sheet the stuff about directly proportional and inversely proportional on cheat sheet)

Question 20

Our purpose of sampling distribution of sample mean (x with line over top) is to tell us how accurate the sample mean is likely to be a ____ ____ of population mean.

Answer 20

point estimate

(We want M to be as near as possible to sample mean (x with line over top)).

Question 21

If we take a large sample size, we are (more/less) likely to obtain sample mean (x with line over top) near the population mean.

Answer 21

more

(The higher the sample size (n), the better accuracy of the sample mean)) write this on cheat sheet; know this

Question 22

Look over example 8.2 in camera roll. If room on cheat sheet, def write down!

Answer 22

Okay

Question 23

T or F: The Central Limit Theorem deals with what happens when our population is not normal.

Answer 23

True

Question 24

Central Limit Theorem: 1. T or F: Still have Mx=M and σx = σ/square root of n. 2. _____ correct if infinite population. 3. _____ correct if population size N finite but much larger than sample size n. 4. But if population is non-normal, what is the shape of the sampling distribution of the sample mean? - The sampling distribution is ________ ________ if the sample is large enough (n > or equal to 30), even if the population is non-normal). This is the Central Limit Theorem

Answer 24

1. True 2. Exactly 3. Approximately 4. approximately normal (look in camera roll for this written better and visual (2 pics total))

Question 25

When sample size (n) is greater than or equal to ______, our sampling distribution graph will look like a normal graph.

Answer 25

30 (Then, we can find the probabilities using the z-distribution)

Question 26

- If population is normal sample size can be _______ size, and sampling distribution will be normal. - If population is non-normal, sample size has to be at least _____. Only then will sampling distribution be normal.

Answer 26

- any - 30

Question 27

The Central Limit Theorem (cont.): The (larger/smaller) the sample size n, the closer the sampling distribution of the sample mean is to being normal.

Answer 27

larger (In other words, the larger n, the better the approximation)

Question 28

T or F: If n is at least 30, it will be assumed that the sampling distribution of x (with line over top) is approximately normal.

Answer 28

True (Need normal to find probabilities. If it's not normal we will not be able to find probabilities).

Question 29

Look at example 8.3 in camera roll. Write part a on cheat sheet if room.

Answer 29

Okay

Question 30

To calculate probability of sample mean steps: 1. Calculate Mx and standard deviation of x using formulas. 2. Convert the given sample mean (x with line over top) to z-form. 3. Use the z-table to find probability.

Answer 30

Fax (look at example 8.5 if still need help with this in notes)

Question 31

What is the formula to convert the sample mean (x with a line over top) to z-form in order to find probability?

Answer 31

z = x (with line over top) - Mx (x with line over top) / σx (x with line over top) REMEMBER: To find σx (x with line over top), you do σ/ square root of n. Don't just plug in the standard deviation given into the formula to convert to z-form.

Question 32

Population Proportions → when we have a large ________ population. 1. P = stands for... 2. p with a hat on top = stands for... 3 Mp with a hat on top= stands for...

Answer 32

infinite 1. P = population proportion 2. p with a hat on top = sample proportion 3. Mp with a hat on top = mean of sampling distribution of sample proportion (look in camera roll for pic of this)

Question 33

What is the formula for calculating sample proportion?

Answer 33

p with a hat on top = x / n x = something we're interested in n = total sample size (ex: what proportion of fruit loop cereal is red? So you would do red / n, and whatever that equals will be our sample proportion)

Question 34

A fraction of the population that has certain characteristics

Answer 34

population proportion (ex: red fruit loops, proportion of 1,000 people that have blue eyes)

Question 35

For population proportions: We can use our probability or z-formula only if sample size is large enough. To figure out if sample size if large enough, two criteria need to be met:

Answer 35

1. n x p > or equal to 5 AND 2. n(1-p) > or equal to 5 If these two criteria are met, we have a normal distribution.

Question 36

If sample size is large enough after we have met the 2 criteria, then we can say: 1. Mp (p with a hat on top) = _____. 2. σp (p with a hat on top) = _____.

Answer 36

1. P (mean of sample proportion is same as population proportion) 2. square root of: P(1-P) / n

Question 37

For finding probabilities of sample proportion, we will use what formula to convert to z-form?

Answer 37

z = p (with hat on top) - Mp (with hat on top) / σp (with hat on top) (Remember, first see if sample size is large enough, and then we can continue) (look in camera roll for pic of this formula)

Question 38

Look at cheese example in camera roll.

Answer 38

Okay

Question 39

What are the steps to find the probability of population proportions?

Answer 39

1. Find out if sample is large enough (use the 2 criteria) 2. If so, calculate Mp (with a hat on top) and σp (with a hat on top). 3. Use these to convert p(with a hat on top) into z-form using z converting formula. 4. Then, use z-table to find probability.

Question 40

What is the formula to calculate variance of the sampling distribution of the sample proportion p (with a hat on top)?

Answer 40

Variance (aka σ2p (p with a hat on top)) = P(1-P) / n (look in camera roll for pic of this formula; 8.19 example)