1. A sample of beers that were bought at a particular bar were measured for their volume. Test
whether the standard deviation of the pours is greater than .2 ounces.
Data Set A: 15.7, 15.8, 16.0, 15.5, 15.7, 15.9, 16.3, 15.1, 15.4, 15.9, 16.0, 15.9,
16.1, 15.8, 15.5, 15.4, 15.8, 15.7, 16.2, 15.6
a. State the null and alternative hypothesis
H0: σ = .2
H1: σ > .2
b. Using the critical value method, what is/are the critical value(s), and which distribution
is being used?
Note: a mistake, the level of significance was not specified – and it needs to be. So I
will assume an α=.01.
The distribution being used is χ2. This is a one tail (right) test. Since n=20, the degrees
of freedom is 19. The critical value is 36.19
c. Using the critical value method, what is the result (statistically) of the hypothesis test
and why?
You need to calculate s (sample standard deviation). Use 1 Var Calc on the calculator.
It will find that s = .294. Now use the formula for the χ2 which is it is equal to
(n -1)s2/σ2.
So now calculate 19*.2942/.22 = 41.14
Note, you should not use a rounded s estimate (the .294), use the actual value stored in
the calculator.
Since the test statistics (41.14) is more extreme ( > ) than the critical value 36.19, we
reject the null and accept the alternative
d. Draw a diagram to represent the previous test.
rejection
0
36.19 41.14
crit. Test
value stat
e. State, in English, the result of the hypothesis test
There is sufficient evidence to reject the hypothesis that the standard deviation of the
pour of beers is .2 ounces. And we accept the alternative that the standard deviation is
greater than .2 ounces.
2. What kind errors can be made when doing hypothesis testing, and how do we control those
errors?
There are two types of errors
Type I error is when you reject the null hypothesis when it IS true.
Type II error is when you fail to reject the null hypothesis when it is not true.
In statistics, you choose explicitly the probability of Type I error. This is called the “level of
significance” and is represented by an “α”. This means that in doing a hypothesis test, the
probability of rejecting the null hypothesis that is true is α. If you reduce one type of error,
you’ll increase the size of the other. For example, if you make α smaller, you make the Type II
error larger.
3. What are the different probability distributions used in hypothesis testing and under what
conditions are each used?
There are three probability distributions.
There’s the normal distribution where you use z-scores. When z-scores are used, this is the
“standardized” normal distribution which has a mean µ = 0 and standard deviation σ = 1. The
normal distribution is a bell-shaped curve. You use this when testing a mean and the
population standard deviation is known. To use this distribution, the data must be distributed
normally or the sample size must be greater than 30 so you can use the CLT. Normal
distribution is also used in proportions because when you make an assumption of mean, p, you
also form an assumption about the standard deviation. The assumption for proportions, since
the data is a binomial, is that the sample size must be large enough to be distributed normally.
This happens when there are at least 10 observations in each category. Also the size of the
population must be 20 times the size of the sample
There’s the t-distribution. This is used when you are testing a mean, but the population standard
deviation is not known. The t-distribution is also a bell-shaped curve with a mean µ = 0 and
standard deviation σ = 1 though it’s a bit heavier in the tails than the normal distribution.
When using the t-distribution, you need to specific the degrees of freedom. The degrees of
freedom will equal n-1 when testing a mean. When testing the β of a regression, degrees of
freedom will equal n-2. To use the t-distribution, the data must be distributed normally or the
sample size is large enough (>30) to use CLT.
There’s the χ2 distribution. This distribution is used to test whether a standard deviation is
equal to some quantity. The distribution starts at 0 and is skewed right. You must know the
degrees of freedom which are n-1 for tests of a standard deviation. To use this distribution the
data must be distributed normal. CLT doesn’t apply for this distribution. CLT is about the
distribution of 𝑥̅ if the sample size is large enough, not about σ. So CLT can’t be used when
testing σ.