CAS MA 113 Study Guide - Quiz Guide: Central Limit Theorem, Statistical Parameter, Point Estimation
9 May 2018
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Study Guide for MA113 Exam 3
This study guide covers topics necessary for exam 3 of MA 113, with 7 example problems. It is
meant as a supplement to homework and textbook reading.
Part 1) Confidence Intervals
In the previous sections of the course, we calculated distributions for and using the Central
Limit Theorem. To do this, we needed to know the population parameters μ, σ, or p.
Now, we wish to estimate μ, σ, or p. We can do this by creating confidence intervals using our
sample estimators. Note that confidence intervals represent a range of likely values for
parameters. Since population parameters are fixed values and not distributions, the probability a
parameter is inside a certain confidence interval is either 0% or 100%, and nowhere in between.
For example, if we created the 95% CI for μ given by:
95% CI for μ: (10, 16)
It would be incorrect to say “The probability that μ is between 10 and 16 is 95%.” Instead we
would say “We are 95% sure our confidence interval for μ is correct.” We could also say “We
are 95% confident that μ is in the interval (10,16).” We replace the word “probable” with
“confident” to show this is not a true probability problem. This is an important distinction
showing that the confidence interval is the part that varies, not the population parameter. If we
were to take a large number of different samples from our population and calculate a confidence
interval for each, we would expect around 95% of those intervals to correctly encapsulate μ.
The center of our confidence interval for either μ or p is called the “point estimate.” The distance
between the point estimate and the upper or lower bound is called the “margin of error.” The
values we use from either the t-table, z-table, or chi square-table are called “critical values.”
Formulas:
Confidence Interval for p:
If we meet the conditions:
(a) n ≥ 10
Confidence Interval for μ:
If we meet the conditions:
(a) population is normally distributed or
(b) n ≥ 30
Confidence interval for :
,
)
If we meet the condition:
(a) sample is normally distributed
Note: to find the CI for σ, simply take the square root of the above equation.
To find the correct critical value we must use the appropriate tables, which will be provided for
you on the exam. Note that we use a different table for each confidence interval: The z-table is
used for p, the t-table for μ, and the -table for . To look up values from the z-table, you
must find the appropriate percentile of the z-value for the confidence level that is given to you.
For example, to calculate a 95% CI for p, you would look up the z-value for the .975 percentile,
which is z = 1.96, NOT the z-value for the .95 percentile. The graph below illustrates why this is
the case. (In this example 1 – α = .95)
To look up critical values in the t-table and chi square-table, you also need to know the degrees
of freedom. For both tables, DF = n – 1. For example, a sample with 30 respondents has 30 – 1 =
29 degrees of freedom. Once you find DF, you then identify the correct percentile for the t-value
or chi square-value in the same way you would do for a z-table. If the table does not list the
required DF, pick the value that closest matches your DF.
Note: For the -table, it is necessary to look up two different critical values to compute both the
lower bound and upper bound of the confidence interval. This is because the graph is not
symmetric, unlike the standard normal or t – distribution.
You can decrease the size of the confidence interval (and thus the margin of error) by either
decreasing your confidence level or increasing the sample size.
What Is Actually True
What We Believe
Part 2) Type I and Type II errors
Even when we do not know the true population parameters, we often wish to test an idea that a
parameter is equal to a certain value. This idea is called a “hypothesis.” To test to see if our
hypothesis is correct, we create a confidence interval for our sample. If the hypothesis is inside
our CI, we believe it is a plausible value. If the hypothesis is outside the CI, we do not believe
the hypothesis is true (This is called “rejecting” the hypothesis.)
Anytime we make a claim about a value we do not really know, we open ourselves up to possible
errors.
When we reject a hypothesis (the value falls outside our CI) it is possible that the hypothesis
was actually true. This is a Type I error. Also known as a “false discovery,” since we believed
that we had discovered an error in the hypothesis. The probability we make a type I error is given
by α, which is 1 – confidence level.
When we fail reject a hypothesis (the value falls inside our CI), it is possible that the hypothesis
was actually false. This is a Type II error. Also known as a “missed discovery,” since we have
missed out on the chance of proving the hypothesis wrong. The probability we make a type II
error is given by β, which you will not be expected to calculate or know for this course.
Note: It is not yet important for you to memorize the terminology about hypothesis. If all you
want to remember about type I and II errors is “false discover” vs. “missed discovery,” you will
be fine on the test. I have simply laid out this new terminology to make telling the difference a
bit easier.
Hypothesis is True
Hypothesis is False
Do Not Reject Hypothesis
(Inside CI)
We are correct
P = 1 - α
Type II error
“Missed Discovery”
P = β
Reject Hypothesis
(Outside CI)
Type I error
“False Discovery”
P = α
We are correct
P = 1 - β
Note: It is not possible to make both a type I and type II error at the same time. Which type of
error you could make is determined by whether or not you keep the hypothesis.
