Flashcards in Lecture 6 - Determining the Sample Size Deck (22):

1

## With sampling, what is the really general statement about sample size?

### There is a number out there that is the proper sample size. Some managers have standards, but really it comes down to statistical methods to calculate the sample size

2

## What are the statistical concerns with sample size?

### Is sample size statistically sufficient, and can we be confident about accuracy of our statistical estimates

3

## What are the managerial concerns with sample size?

### Do managers feel confident about results?

4

##
What underlies the statistical sample size? what is the formula?

Will be on quiz

###
Sampling distribution underlies the determination of sample size.

Standard error of mean, which is the error associated with the estimation of a population mean

σx = σ/√n

σx = standard error/deviance of the mean

σ = the standard deviation of the original distribution

n = the sample size

√n = Root of the sample size

5

## Desired precision influences the sample size. So what does precision depend on?

###
Precision depend on the size of the confidence interval.

So if you are 99% confident rather than 95, then the range is bigger and the precision is less.

6

## If desired precision is +- 25, what is total and half precision?

###
Desired: +- 25

total precision is 25*2 = 50

Half (H) = 25

7

## Know formulas

### in visual folder

8

##
What are the z-score values for:

95%

99%

?

###
the z-score values for:

95%: z=1.96 CI

99%: z=2.58 CI

?

9

##
What is σ^2?

Do we usually know it?

### This is the population variance, and we normally don't know this value. Just σ is the population standard deviation

10

## If population variance is unknown, what do you do?

###
1.

Conduct a pilot study using a small sample. Calculate s (sample standard deviation) from your little pilot study, and just assume that σ = s

2.

If you can't do pilot study due to budget or complexity, than use rule of thumb such as rating scale variance. So if you have a 4 point scale, range of variances 0.7-1.3, or if 10 point scale use 3.0-7.0

To be safe use the higher end of the range as population variance.

11

##
If you don't know the population variance and we are bound to a rating scale of 1 - 7, how would we calculate σ^2 before finding the minimum required sample size to be statistically sufficient?

We have CL of 95%

We choose desired precision to be +- 1

Say our result is that x̄ is 3.25, what is our min and max true value?

###
So CL of 95% = z=1.96

95% of all possible samples will provide a mean within my error margin.

We choose H to be 1.

Then because scale 1-7, we pick σ^2 of 4 to be safe. Because 7 scale points has variance range of 2.5-4.0

So n=(1.96/1)*4

n=15.3

Say our result is that x̄ is 3.25, what is our min and max true value?

A:

So since H=1, our min true value is going to be 2.25, and max true value is going to be 4.25

12

## Here's another way to calculate population variance (σ^2) if it's unknown. So if range is known, divide the range by ___ to calculate the population standard deviation

###
Here's another way to calculate population variance (σ^2) if it's unknown. So if range is known, divide the range by ___ to calculate the population standard deviation

So if σ = (max-min)/6

13

##
So to find population variance by the whole dividing by 6 method, find n, by finding σ^2 with this example:

You want to find how many dollars people spend on a visit to a theme park.

CL = 95%

Desired precision is +- 25

After conducting the study, x̄ is $35, Sx = $60. Based on these numbers, is the result more or less precise?

Also, assume that by talking with vendors the max you can spend is $450

Find the new H, and is this more precise or less?

###
n = (z/H)^2 * σ^2

σ = (max-min)/6

Min: 0 (don't have to spend at park, not including initial ticket)

Max: $450 (can find this by talking with vendors, or using theme park data)

σ= (450 - 0)/6 = 75

σ^2 = 5625

z = 1.96 (CL: 95%)

H = 25

n = (1.96/25)^2 * 5625

n = 35

After conducting the study, x̄ is $35, Sx = $60. Based on these numbers, is the result more or less precise?

We found that the variance is less than we guessed, which is good! This means that we were being more precise by guessing a higher variance.

H = z * σ∨x̄ (variation of x̄)

σ∨x̄ = s/√n

so 1.96 (60/√35) = 20

H=20,

So min true value is $15 and max is $55. This is more precise than what we initially put as H=25

14

## What is the formula for finding H?

###
H = z * σ∨x̄ (variation of x̄)

15

## What is the population proportion?

###
So it is π, and you can estimate it by:

1. using past studies and published data

2. Conduct a pilot study

3. Using judgment to create rand of possible estimates

4. Worst-case scenario, just assume π = 0.5

16

## When is π(1-π) maximized? What does this tell us about standard of population proportion?

###
so by plugging in numbers,

0.1 = 0.09

0.2 = 0.16

0.3 = 0.21

0.4 = 0.24

0.5 = 0.25 ***

0.6 = 0.24

0.7 = 0.21

0.8 = 0.16

0.9 = 0.09

So π=0.5 gives us the highest number, which means it will give us highest n. So this is the safest assumption, and relied on as last resort if you can't find π any other way.

17

## If desired precision is +- 2 percentage points, what is H?

### H=0.02

18

##
For canadian election results, our desired preciseness is +-3.5%

CL= 99%

Using proportion formula, what is required sample size?

###
For canadian election results, our desired preciseness is +-3.5%

CL= 99%

Using proportion formula, what is required sample size?

Assuming π=0.5

(2.58/0.035)^2 (0.5 (1-0.5)) = 1359

so required n = 1,358

19

## What do you always do with your decimal values with n?

### Always round n up, even if say 150.2 would be 151 = n

20

## If more than one parameter is estimated in a given study, what do you do for choosing a sample size?

### Calculate the desired sample size for each parameter and choose the largest number as a sample size

21

##
Does population size enter the calculate of sample size?

What does it depend on however from population?

### No it doesn't. You can have population size of 1,000,000 and still have sample size of 16. It all depends on population variability.

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