Chapter 7: Margin Of Error & Confidence Intervals Flashcards
What do descriptive statistics do?
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.
What do inferential statistics do?
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.
True or false:
The distribution of means (estimates) from many different samples (sampling distribution) is symmetric and approximates a normal curve/distribution.
True
A sampling distribution is a distribution of:
A: Scores
B: Sample means (estimates)
B: Sample means (estimates)
- The distribution of sample means (estimates) from many different samples
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”).
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
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)
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
True or false:
If we only have one sample: We cannot estimate the standard error by using the central limit theorem.
False!
If we only have one sample: We CAN also estimate standard error by using the central limit theorem.
Estimated Standard Error Calculation/Formula:
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
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%
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.
True or false:
The standard error/standard deviation of a sampling distribution is the true standard error.
A: True
B: False
B: False
It is an ESTIMATED standard error
Sampling Distribution And Margin Of Error:
Define margin of error:
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
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
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
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
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
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.
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.
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
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.
What is a 95% Confidence Interval?
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.
What is the 95% Confidence Interval formula?
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
A 95% confidence interval tells us that 95% of samples would give us such an interval.
A. Yes
B. No
B. No - each sample mean will have its own confidence interval
We are 95% confident that a 95% confidence interval includes the sample mean
A. Yes
B. No
B. No - We are 95% confident that a 95% confidence interval includes the POPULATION mean
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
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.
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
B: A smaller standard error provides a smaller margin of error
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
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
What are Critical Values?
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.
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
A: The amount of “stretch” in the t distribution increases as the sample size gets smaller
- See slide 36 for a visual of this
