Chapter 7: Margin Of Error & Confidence Intervals

Question 1

What do descriptive statistics do?

Answer 1

They DESCRIBE our data, study, standard deviation, mean, etc. They DESCRIBE something.

This is what we’ve been learning about thus far, in chapters 1-6.

Question 2

What do inferential statistics do?

Answer 2

With inferential statistics something about the sample infers/tells us something about the population.

This is what chapter 7 is about and pretty much everything else we’ll be learning about in this class.

Question 3

True or false:

The distribution of means (estimates) from many different samples (sampling distribution) is symmetric and approximates a normal curve/distribution.

Answer 3

True

Question 4

A sampling distribution is a distribution of:

A: Scores
B: Sample means (estimates)

Answer 4

B: Sample means (estimates)

• The distribution of sample means (estimates) from many different samples

Question 5

True or false:

The standard error is the standard deviation of the estimates from many random samples (sampling distribution) (i.e., average amount of sampling error, or expected amount by which an estimate is “off”).

Answer 5

True

She might also refer to standard error as:
* The average amount of sampling error
* The average distance between the sample mean and population mean
* The average amount of sampling error across random samples

Question 6

If we have many samples (sampling distribution):

A: Standard error = deviation score of the many sample means (sampling distribution)
B: Standard error = standard deviation of the many sample means (sampling distribution)
C: Standard error = sampling error of the many sample means (sampling distribution)

Answer 6

B: Standard error = standard deviation of the many sample means

She might also refer to standard error as:
* The average amount of sampling error
* The average distance between the sample mean and population mean

Question 7

True or false:

If we only have one sample: We cannot estimate the standard error by using the central limit theorem.

Answer 7

False!

If we only have one sample: We CAN also estimate standard error by using the central limit theorem.

Question 8

Estimated Standard Error Calculation/Formula:

Answer 8

σx̄ = σ ÷ √N sx̄ = s÷ √N

Since it’s unlikely that we’ll know the population standard deviation, we’ll likely use the second formula to find the sample standard error/standard deviation, which will give us information about the population.

s = sample standard deviation
N = sample size

• Standard error is influenced by the sample size and the population standard deviation or the sample standard deviation
• The sample standard deviation is a good estimate of the population standard deviation

Question 9

Rule Of Thumb For A Normal Curve:

____% of the observations in a normal distribution fall
within ± 1 (a rough value) standard deviation of the
mean, and roughly _____% are within ± 2 (a rough value)
standard deviations.

A: 50% / 85%
B: 68% / 95%
C: 65% / 95%

Answer 9

B: 68% / 95%

More accurately, 95% of observations from a normal distribution fall within ± 1.96 standard deviations of the distribution’s center.

• This is more accurate than saying ± 2
• 1.96 can be understood as a z-score. It’s the margin of error. It can ONLY be used for normal distributions!
• Observations refers to sample means not individual scores
• We can apply this theory (population distribution) to sampling distributions - as long as it’s a normal distribution.

Question 10

True or false:

The standard error/standard deviation of a sampling distribution is the true standard error.

A: True
B: False

Answer 10

B: False

It is an ESTIMATED standard error

Question 11

Sampling Distribution And Margin Of Error:

Define margin of error:

Answer 11

The margin of error is a statistic expressing the amount of random sampling error in the results of a survey.

The larger the margin of error, the less confidence one should have that a poll result would reflect the result of a census of the entire population.

It is essentially half the confidence interval

Question 12

Sampling Distribution And Margin Of Error:

Margin of error formula:

A: 1.96 x standard error
B: 1.96 x 2
C: 1.96 x 1
D: 1.96 x sample mean

Answer 12

A: 1.96 x standard error

Example:
N = 1071
Standard error = 0.34
1.96 x 0.34 = .067

Interpretation:
95% of samples with N = 1071 have estimates within ± 1.96 × .034 = .067 of the unknown population mean. So 95% of the estimates will fall between -0.67 and +0.67 of the unknown population mean.

• The margin of error teaches us more about the population mean

Question 13

True or false:

We can say that 95% of the time, a sample
mean will fall within ± 1.96 standard errors of
the true population mean

Answer 13

True!

Even though we don’t know the population mean we can still use the margin of error to learn
something about the population mean

Question 14

Which of the following is true?

A: The population mean can move and so can the sample mean but 95% of the time the sample mean is within a certain range of the population mean.
B: The population mean never moves, it is constant/fixed. The sample mean can move but 95% of the time it’s within a certain range of the population mean.
C: The population mean can move but the sample mean cannot and 95% of the time the sample mean is within a certain range of the population mean.

Answer 14

B: The population mean never moves, it is constant/fixed. The sample mean can move but 95% of the time it’s within a certain range of the population mean.

Question 15

Which of the following is true:

A: 50% of the time we should draw a sample with a mean higher/lower than that of the full population
B: 95% of the time we should draw a sample with a mean higher/lower than that of the full population
C: 25% of the time we should draw a sample with a mean higher/lower than that of the full population

Answer 15

A: 50% of the time we should draw a sample with a mean higher/lower than that of the full population

BUT…

• The LOWEST the unknown population mean could be is x̄ - (1.96 x σx̄) which is the sample mean/estimate − (1.96 × standard error)
• The HIGHEST the unknown population mean could be is x̄ + (1.96 x σx̄) which is the sample mean/estimate + (1.96 × standard error)
• In real life, we only have a sample mean. We use the sample mean to guess the location of the population mean. Inferential statistics gives us information about the population mean.

Question 16

What is a 95% Confidence Interval?

Answer 16

The 95% confidence interval is formed by the
limits computed as sample mean/estimate ± margin of error

“Confidence” is a long-run idea. You have to
imagine that there are many samples from the
same population with the same sample size

95 out of 100 samples we could potentially
work with will yield confidence intervals that
include the true population mean

In the other 5% of samples, it will not

We are somewhat confident (usually 95%) that an interval or range contains the true mean in the full population.

Question 17

What is the 95% Confidence Interval formula?

Answer 17

95% C.I. = sample mean/estimate ± (1.96 × standard error)

Don’t forget that (1.96 × standard error) = the margin of error

You can calculate a confidence interval for each sample mean

Question 18

A 95% confidence interval tells us that 95% of samples would give us such an interval.

A. Yes
B. No

Answer 18

B. No - each sample mean will have its own confidence interval

Question 19

We are 95% confident that a 95% confidence interval includes the sample mean

A. Yes
B. No

Answer 19

B. No - We are 95% confident that a 95% confidence interval includes the POPULATION mean

Question 20

What factors influence the margin of error:

A: Sample size and sample mean
B: Sample size and population standard deviation or sample standard deviation
C: Sample mean and sample standard deviation

Answer 20

B: Sample size and population standard deviation or sample standard deviation

Since standard error is part of the margin of error formula you can assume that the factors that influence standard error (sample size and population standard deviation or sample standard deviation) will also influence the margin of error.

Question 21

Which of the following is true:

A: A larger standard error provides a smaller margin of error
B: A smaller standard error provides a smaller margin of error

Answer 21

B: A smaller standard error provides a smaller margin of error

Question 22

True or false:

When working with small samples, the margin of error is more accurate if we use a t distribution that is slightly
wider than the normal distribution (1.96) as the
sampling distribution

Answer 22

True!

t distribution = A symmetric distribution that resembles a normal curve, but is wider, especially at small sample sizes.

For example:
* If N=150 the “critical value” should be +- 1.97 S.E. (margin of error)

• If N=30 the “critical value” should be +- 2.04 S.E. (margin of error)
• If N=10 the “critical value” should be +- 2.26 S.E. (margin of error)
• NOTE: Notice how the 1.96 value in the margin of error computation has been replaced by critical values!

So instead of this formula: 95% C.I. = sample mean/estimate ± (1.96 × standard error) where (1.96 × standard error) is = to the margin of error

We’ll use this formula instead: 95% C.I. = sample mean/estimate ± (C.V. × standard error). Now (C.V. × standard error) = to the margin of error

• t distributions are still symmetric distributions, they just have longer tails and are more stretched out

Question 23

What are Critical Values?

Answer 23

Critical value = The number of standard error units above or below the population mean that includes 95% of all sample estimates (the multiplier that determines the 95% margin of error).

Critical values are essentially cut-off values that define a region (margin of error).

We will talk more about critical values in the later lectures.

Critical values (C.V.) change with the sample size.

Question 24

Which of the following is true:

A: The amount of “stretch” in the t distribution increases as the sample size gets smaller
B: The amount of “stretch” in the t distribution
decreases as the sample size gets smaller

Answer 24

A: The amount of “stretch” in the t distribution increases as the sample size gets smaller

• See slide 36 for a visual of this

Question 25

What functions as the “stretch” parameter in the t distribution?

A: Sample size
B: Degrees of freedom
C: Deviation score

Answer 25

B: Degrees of freedom

Recall that the degrees of freedom (N − 1) can be viewed as an adjusted sample size in computing sample variance.

So, the degrees of freedom function as the “stretch” parameter in the t distribution (the lower the N or df, the more the stretch relative to a normal curve)

For example:
* If N=150, df=149 and the “critical value” should be +-1.97 S.E. (margin of error) wider (stretch)

• If N=30, df=29 and the “critical value” should be +-2.04 S.E. (margin of error) even wider (stretch)
• If N=10, df=9 and the “critical value” should be +- 2.26 S.E. (margin of error) even wider still (stretch)

Question 26

True or false:

Confidence Interval formulas will produce and lower and upper bound?

Answer 26

True!

Example:
We are 95% confident that the interval (range) from 3.27 to 3.41 includes the true humor rating mean from the full population.

Step 1 - Set up the formula: 95% C.I. = sample mean/estimate ± (C.V. × standard error)

Step 2 - Plug in the numbers: 3.34 ± (1.962 × .034), where 3.34 is the sample mean (3.27 + 3.41 ÷ 2) and 1.962 (critical value determined by Jamovi) × .034 (sx̄ = s ÷ √N) = the margin of error

Step 3 - Find the lower and upper bounds:
3.34 + .067 = 3.41
3.34 - .067 = 3.27

= [3.27, 3.41] lower and upper bound

• Represented in standard error units

Question 27

What would happen to the width of the confidence interval if a sample of 100 participants gave these statistics instead of a sample of 1071?

A. Increase
B. Decrease

Answer 27

A. Increase

Question 28

What would happen to the width of the confidence interval if the sample standard deviation is 11.1 instead of 1.11?

A. Increase
B. Decrease

Answer 28

A. Increase

Question 29

Margin of Error vs. Confidence Interval:

Answer 29

t Distribution Example:

N = 99
df = 99 - 1 = 98
Mean = 111
Standard deviation = 48.5
Estimated standard error = sx̄ = s ÷ √N = 48.5 ÷ √99 = 4.87
Critical Value (per Jamovi) = +- 1.984 in standard error units

The margin of error = C.V. × standard error
1.984 × 4.87 = 9.66

The confidence interval = sample mean/estimate ± margin of error
111 + 9.66 = 120.66
111 - 9.66 = 101.34

= [101.34, 120.66] lower and upper bound

Interpretation: You can be 95% confident that the interval from 101.34 to 120.66 includes the average Index Of Current Economic Conditions

• You can do ALL of this in Jamovi BUT you should still try to understand it as much as possible and put it on your cheat sheet!

Question 30

The confidence interval tells us the possible location of

A. sample mean
B. population mean

Answer 30

B. population mean