Answer the following problems showing your work and explaining (or
analyzing) your results.
1. Explain Type I and Type II errors. Use an example if needed.
A type I error is the mistake of rejecting the null hypothesis when it is actually true.
A type II error is the mistake of not rejecting the null hypothesis when it is actually
false.
For example, consider a claim that a drug increases hair growth in men. Here the
null hypothesis would be that the drug has no effect on hair growth. A type I error
would occur if we erroneously reject the null hypothesis and conclude that the drug
has an effect, when in fact it does not. A type II error would occur if we conclude the
drug does not work when in fact it does.
2. Explain a one-tailed and two-tailed test. Use an example if needed.
In a one-tailed test, we are testing a claim that a parameter is “greater than” or “less
than” a given value. For example, the claim that the average height of adult males in
a population is greater than 58 inches would be tested with a one-tailed test. In a
two-tailed test, we are testing a claim that a parameter is different from a given
value, either significantly more or less than a given value. For example, testing the
claim that a drug can alter the 50/50 chance of conceiving a baby girl/boy would be
tested with a two-tailed test.
3. Define the following terms in your own words.
 Null hypothesis
The null hypothesis is the statement that a population parameter is equal to a given
value. Generally we are checking to see if something causes a change to this
parameter.

P-value
The P-value is the probability of getting a test statistic at least as extreme as the
critical value, that is, the value set by the confidence level.

Critical value
The value of the test statistic set by the significance or confidence level. This is the
point in the probability distribution such that the area beyond equals the
significance level.

Statistically significant
This phrase indicates that the probability of the event occurring by chance is less
than the prescribed significance level.
4. A homeowner is getting carpet installed. The installer is charging her for 250 square
feet. She thinks this is more than the actual space being carpeted. She asks a second
installer to measure the space to confirm her doubt. Write the null hypothesis
Ho and the alternative hypothesis Ha.
H0: The actual space is equal to 250 sq. ft.
H1: The actual space is less than 250 sq. ft.
5. Drug A is the usual treatment for depression in graduate students. Pfizer has a new
drug, Drug B, that it thinks may be more effective. You have been hired to design the
test program. As part of your project briefing, you decide to explain the logic of
statistical testing to the people who are going to be working for you.

Write the research hypothesis and the null hypothesis.
H0: The effectiveness of drug B is equal to the effectiveness of drug A.
H1: The effectiveness of drug B is greater than the effectiveness of drug A.

Then construct a table like the one below, displaying the outcomes that would
constitute Type I and Type II error.
Actual Effectiveness of Drugs
No difference between A and B
Predicted
effectiveness
No difference
between A
and B
B is better
than A

No error
Type II error: Claim B is better,
but there is really no difference.
Could result in wasted costs
developing and marketing drug B.
B is better than A
Type I error: Do not detect a
difference, but B is really
better than A. Could result in
lost profit.
No error
Write a paragraph explaining which error would be more severe, and why.
Both errors can be severe. If the drug is a very important product in the
marketplace, then perhaps the Type I error is more severe. This is because the Type
I error may go forever unknown, as they abandon research on a drug that would
have been a huge profit center for the company. On the other hand, the Type II
error would likely result in more money spent readying the drug for the market.
Then its lack of increased effectiveness could well be caught in subsequent trials.
Even if not, it would probably have reasonable sales at first until it’s lack of
effectiveness was eventually borne out from increased statistics.
6. Cough-a-Lot children’s cough syrup is supposed to contain 6 ounces of medicine per
bottle. However since the filling machine is not always precise, there can be
variation from bottle to bottle. The amounts in the bottles are normally distributed
with σ = 0.3 ounces. A quality assurance inspector measures 10 bottles and finds
the following (in ounces):
5.95 6.10 5.98 6.01 6.25 5.85 5.91 6.05 5.88 5.91
Are the results enough evidence to conclude that the bottles are not filled
adequately at the labeled amount of 6 ounces per bottle?
a) State the hypothesis you will test.
H0: The mean volume of the bottles is equal to 6 ounces.
H1: The mean volume of the bottles is not equal to 6 ounces.
b) Calculate the test statistic.
The test statistic for the mean is the z score. This is defined as z =
m-m
s2 N
where m
is the mean of N samples,  is the known standard deviation of the population
mean, and  is the population mean being tested. Evaluating this for the data above,
we find z = -0.1160 .
c) Find the P-value.
P = 0.9077
d) What is the conclusion?
We do not reject the null hypothesis. (Although we did not specify a significance
level, the P value is very large, and no reasonable significance level would be
satisfied). Therefore, we do not have sufficient evidence to conclude that the
bottles are inadequately filled.
7. Calculate a Z score when X = 20, μ = 17, and σ = 3.4.
z = 0.8824
8. Using a standard normal probabilities table, interpret the results for the Z score in
Problem 7.
PRIGHT = 0.1888
P2-SIDED = 0.3776
This means that there is an 18.88% probability that a value of 20 or greater would
occur by chance, and a 37.76% chance that a value differing by 3 or more from 17
would occur by chance.
9. Your babysitter claims that she is underpaid given the current market. Her hourly
wage is $12 per hour. You do some research and discover that the average wage in
your area is $14 per hour with a standard deviation of 1.9. Calculate the Z score and
use the table to find the standard normal probability. Based on your findings, should
you give her a raise? Explain your reasoning as to why or why not.
12 -14
= -1.052
1.9
At a rate of $12/hr, she is being underpaid by slightly more than 1 standard
deviation from the typical babysitter. At this level, there are only 14.63% of the
babysitters that are paid less than she is. I would give her a raise.
z=
10. Tutor O-rama claims that their services will raise student SAT math scores at least
50 points. The average score on the math portion of the SAT is μ = 350 and σ = 35.
The 100 students who completed the tutoring program had an average score of 385
points. Is the average score of 385 points significant at the 5% level? Is it significant
at the 1% level? Explain why or why not.
H0: the average score of tutored students is 399
H1: the average score of tutored students is less than 399.
z=
385- 399
2
= -4.000
P = 3.167 ´10-5
35 / 100
Since the P value is less than our significance level of 0.01, we have sufficient
evidence to reject the null hypothesis. We can therefore conclude that Tutor-Orama’s claim is false at the 99% confidence level.