Lecture 6: Statistical Inference/Estimation

Question 1

Given n = 250 and ý = 27, what can be say according to the LLN?

Answer 1

sample size of 250 means its closer than if we had 200 or 20

Question 2

Given n = 250 and ý = 27, what can be say according to the CLT?

Answer 2

Sampling distribution of the sample mean should be approximately normal in shape (functional form), mode=median=mean of the sampling distribution is in the middle and equal to the population mean; standard deviation of sampling distribution is sigma/squared n

Question 3

Why is CLT cool?

Answer 3

As n gets larger, the sapling distribution approaches normality, even if the pdf of X in the population is not normal

Question 4

Point estimate

Answer 4

a single number, calculated from a set of data, that is the best single guess for the population parameter

Question 5

Point estimator

Answer 5

a sample statistic that predicts the
value of that parameter; can think of this as the
equation used to produce the point estimate

Question 6

Problems with a point estimate?

Answer 6

Maybe high probability of being highly wrong

Question 7

What problem do you still have even with CLT and LLN?

Answer 7

Though we know the outcome of increasing sample size, is it clear how good of a guest our sample statistics are relative to the population parameter?

Question 8

Interval estimate

Answer 8

consists of a range of numbers around the point estimate, within which the parameter is believed to fall

Question 9

What is another word for the interval estimate?

Answer 9

the confidence interval

Question 10

What does interval estimate allow us to do?

Answer 10

Gauge accuracy of a point estimate using probability

Question 11

Why are we able to gauge the accuracy of a point estimate using probability?

Answer 11

We know the probability of the population parameter falling in a given interval

Question 12

What is confidence interval based on? (2)

Answer 12

A point estimator and the spread of the sampling distribution of that estimator

Question 13

What assumption must be met for you to use confidence intervals?

Answer 13

that the sampling distribution is approximately normal

Question 14

How do you construct a confidence interval?

Answer 14

1. Make sure the sampling distribution is approximately normal
2. Adding to and subtracting from the point estimate some multiple of its standard error (i.e. a z-score)

Question 15

Why is knowing “If we know the parameters of a population, specifically the mean and standard deviation, then we can predict the chance that a given sample of size n will have a sample mean within a certain distance of the population” useful?

Answer 15

We can reverse it to find confidence interval

Question 16

What is the implication of a 95% confidence?

Answer 16

5% chance that the sample mean does not fall within the interval you get a mean that range does not include the true population parameter

Question 17

Is it true that there’s a 95% chance that the mean will fall within the interval?

Answer 17

No, it either does or it does not. Rather, 95% of possible means will contain the population mean

Question 18

How do we reverse the logic?

Answer 18

given a sample of size n and a sample standard deviation and mean, we predict the chance that the unknown population mean is within a certain distance of the sample mean

Question 19

Once a sample mean is calculated, if the sample
mean does fall within the interval
µ−1.96σÝ and µ+1.96σŶ, then…

Answer 19

the interval from Ý−1.96σ Ý and Ý +1.96σ Ý
contains μ

Question 20

What is the equation for confidence interval?

Answer 20

ci= Ý ± z ˆσý

Question 21

For confidence interval equation - how do we get z?

Answer 21

Usually not given z. instead start with desired confidence interval, then select appropriate z-score.

Question 22

Confidence coefficient

Answer 22

The probability that the interval estimate contains the parameter. Typical confidence coeffieincets are .99 and .95

Question 23

What drives the z-score that you use for confidence interval calculations?

Answer 23

the confidence coefficient

Question 24

If you decide on confidence coefficient of .95, what z-score would you use?

Answer 24

1.96 (0.025 of the distribution falls outside this z-score)

Question 25

When calculating confidence interval for a mean, we do not have the σ (standard error of the population). What to do?

Answer 25

If the sample has 30 observations or more, can substitute in s (standard deviation for the sample)

Question 26

When should you be skeptical if a paper estimates the standard error? (2)

Answer 26

If they do it with less than 30 observation If they have 30 or more observations but there is a lot of skew

Question 27

If you have 30 observations but a lot of skew, can you still estimate the standard error?

Answer 27

You need 100 or more observations

Question 28

What is the interpretation of an estimated standard error of 5?

Answer 28

This is the standard deviation of the sampling distribution, so on average, a sample mean we would derive from our sample of 900 is 5 bucks away from the true population mean

Question 29

Interpretation of 95% confidence interval and interval ($290.2, $309.8)?

Answer 29

We are 95% confident that the true population parameter (average monthly food expenditure) falls in that interval

Question 30

What happened if you didn't observe the true population parameter in the interval? (2)

Answer 30

1. maybe used inappropriate sampling techniques (want to daycare to get age) 2. maybe got unlucky/randomly picked 100 people really different from the pop.

Question 31

What three components drive the precision of the interval (contained in the formula)?

Answer 31

1. The number of observations (n) 2. z-score/confidence coefficient 3. The s

Question 32

What is the cost of having more confidence?

Answer 32

You get a wider interval and less precision

Question 33

Can you control the s (one of the components that drives the precision of the interval)?

Answer 33

No

Question 34

As the probability of population parameter falling within the interval increase...

Answer 34

the precision of the interval decreases

Question 35

Error probability

Answer 35

the probability that a confidence interval does not contain the population parameter

Question 36

T/F: If confidence interval is (49%, 54%), you can say it will be closer to 54 than 49 and that candidate likely to win, since the majority of values are 50% or greater

Answer 36

FALSE - never can say where within a parameter a confidence interval falls

Question 37

Why can't you say where within a parameter a confidence interval falls?

Answer 37

Saying that would mean you know where the sample mean falls on the x-axis (but the fact you don't know where the sample mean is relative to the population mean is the whole problem!) maybe if had multiple samples coudl triangulate, button if you only have one

Question 38

Interpretation of 52+/-3

Answer 38

This is the confidence interval - does not specify the confidence LEVEL

Question 39

What is the equation to calculate confidence interval for proportions/qualitative data?

Answer 39

point estimate ± z (standard error)

Question 40

When working with proportions, what do we use as the point estimate of the population proportion?

Answer 40

The sample proportion

Question 41

Why is the sample proportion good?

Answer 41

Unbiasd and efficient point estimator of the population proportion

Question 42

How is the proportion treated?

Answer 42

As a special mean

Question 43

What does π represent

Answer 43

The population proportion

Question 44

What does π /the population proportion mean?

Answer 44

It is the mean of the probability distribution having probability π for 1 and (1- π) for 0

Question 45

Equation for the standard deviation of the probability distribution for π?

Answer 45

σ = (sq of)π (1− π)

Question 46

When is the sampling distribution of the sample proportion approximately normal about the parameter pi?

Answer 46

When n is equal or greater than 30

Question 47

What is σˆ/piˆ

Answer 47

estimate standard error of the distribution of al sample proportions

Question 48

What does the "estimate standard error of the distrbution of all sample proportions" measure?

Answer 48

measures how much sample proportion piˆ varies from sample to sample, around the true proportion pi

Question 49

What does "the standard deviation of the probability diststribution" measure

Answer 49

Measures how much individual outcomes (success/failures) vary around the mean proportion pi

Question 50

Do we feel better about estimating standard error for quantiative data or for the binary?

Answer 50

Binary

Question 51

Why do we feel better about estimating standard error for quantiative data or for the binary? (a few reasons)

Answer 51

1. don't have to worry about Jeff Bazos with 1/0 2. limited number of possibilities 3. discrete 4. within a bound range

Question 52

As sample size increases, standard error gets smaller, and the sample proportion...

Answer 52

tends to fall closer to the population proportion

Question 53

As the sample size increases, the precision of the confidence interval

Answer 53

will increase

Question 54

As standard error and standard deviation increase, the precision of the confidence interval...

Answer 54

will decrease

Question 55

If have 99% confidence interval, (.52, .54), interpretation is

Answer 55

99% confident true population parameter falls in the interval (proportion of individuals int the population that voted in the last election is somewhere between .52 and .54)

Question 56

First step when calculating confidence interval?

Answer 56

Determining whether data is qualitative or quantitative

Question 57

when you calculate the sample mean from a sample, what do you not know? (2)

Answer 57

If the sample mean is right or not if the samples you took are representative of the population

Question 58

What is the confidence interval for a mean really predicting?

Answer 58

the chance that the unknown population mean is within a certain distance of the sample mean

Question 59

Dark implication of Thus, with probability .95 a sample mean occurs such that the interval?

Answer 59

there is a probability of .05 that the sample mean does not fall within the interval