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Problem Set II: Probability, Sign Test, Z-test and the
Three T-tests
1. A pharmaceutical company would like to evaluate its newest weight-loss drug on women.
Below are the weights of 12 women before and after having used the diet pill for three months.
Using an alpha level of .05, test the hypothesis that the diet pill affects weight. Assume a
normal distribution.
before
145
160
140
134
170
134
150
120
150
130
135
140
after
140
158
141
132
160
129
130
131
120
122
121
136
a. Which test is appropriate?
b. Report the null hypothesis:
c. Report the alternative hypothesis:
d. Find the critical value:
e. Calculate the obtained statistic:
f. Make a decision:
g. What does your decision mean?
2. For the past 8 days I have been recording the time it takes a train to arrive at my local station.
My data are below (in minutes). On a particular day, when I fear I may be running late to an
important appointment, I have the option to take the train or I could spend money on a cab. I
reach the station and wait for the train. Construct the 95% confidence interval of my wait
estimate. Would you wait or take a cab at this point? Justify your answer using the confidence
interval you calculated.
x = {5, 7, 12, 14, 4, 6, 6, 7}
3. The Hershey company has been receiving complains about the number of almonds in their
chocolate almond bars. Customers believe Hershey’s bars contain fewer almonds than they’ve
previously contained because the company is trying to spend less money on their product. In a
review of the company’s records from over the course of Hershey’s history, it was documented
that each bar of candy contains a mean of 13 almonds with a standard deviation of 4 almonds.
A sample of 49 of last month’s chocolate bars has been collected, and has a mean of 9 almonds.
Test the hypothesis ( = .05) that Hershey’s has been skimping on almonds and thus there are
fewer almonds in recent chocolate bars than in previous years.
a. Which test is appropriate?
b. Report the null hypothesis:
c. Report the alternative hypothesis:
d. Find the critical value:
e. Calculate the obtained statistic:
f. Make a decision:
g. What does your decision mean?
4. I believe that Zinc helps reduce the duration of a cold. On average, in the US, a cold lasts 7.5
days. I would like to ask a group of 14 friends to use Zinc for 1 year, and document the length
of their colds (assuming they all become ill at least once that year). Using an alpha level of .01,
test the hypothesis that Zinc reduces the duration of a cold, using the data below:
Name
Perle
Benjamin
Rosie
Greg
Stephen
Matt
Joanna
Duration of Cold
(days)
3
8
12
5
4
8
17
Name
Diane
Max
Kamil
Briana
Rita
Sibel
Stavros
Duration of Cold
(days)
5
9
5
6
11
7
6
a. Which test is appropriate?
b. Report the null hypothesis:
c. Report the alternative hypothesis:
d. Find the critical value:
e. Calculate the obtained statistic:
f. Make a decision:
g. What does your decision mean?
5. Assume I toss a fair coin exactly 17 times. Find the probability of the following outcomes:
a. 4 tails
b. 17 tails
c. 5 heads
d. 17 heads
e. 13 or more tails
f. 5 or less tails
Now assume I toss an unfair coin (45% baseline probability for Heads) 17 times as well:
a. 4 tails
b. 17 tails
c. 5 heads
d. 17 heads
e. 13 or more tails
f. 5 or less tails
6. I walk into the Psych department and notice that there is a box of pens and pencils being given
away for free. They are all Brooklyn College-brand pens and pencils so they all have the same
size and shape—the only way to tell whether it’s a pen or a pencil is to write with it. It turns
out, however, that they do vary in color. The breakdown in the box is as follows:
4 red pens
2 red pencils
2 blue pens
8 blue pencils
9 black pens
Evaluate the probability for each of the following, assuming you are starting with a fresh box
for each question. (assume WITHOUT replacement for multiple reaches):
Please report all probabilities to FOUR decimal places.
a. Reaching in once and pulling out a red writing utensil?
b. Reaching in once and pulling out a blue pen?
c. Reaching in twice and pulling out a red utensil followed by a blue utensil?
d. Reaching in three times and pulling out three blue pencils in a row?
e. Reaching in 7 times and pulling out the following sequence:
Red pen, Blue pencil, Blue Pencil, Red Pen, Blue Pencil, Blue Pencil
f. Reaching in once and pulling out either a red or a blue pen?
g. Reaching in once and pulling out either a black pen, a blue pencil, or a red utensil?
h. Reaching in once and pulling out either a red utensil or a pen?
i. Reaching in twice and pulling out first a black pen and then either a blue utensil or a pencil?
7.
Below are pupil sizes from two samples: 8 men and 8 women (measured in millimeters). Test
the hypothesis that there is a difference in pupil sizes between males and females, using an
alpha level of .01.
males
7.7
7.5
6.9
7.8
7.7
6.9
4.5
7.5
females
8.9
8.7
9.0
8.8
7.9
8.8
9.0
9.0
a. Which test is appropriate?
b. Report the null hypothesis:
c. Report the alternative hypothesis:
d. Find the critical value:
e. Calculate the obtained statistic:
f. Make a decision:
g. What does your decision mean?
8.
I believe that Echinacea helps reduce the duration of a cold. On average, in the US, a cold lasts
7.5 days, with a standard deviation of 2.5 days. I would like to ask a group of 25 friends to use
Echinacea for 1 year, and document the length of their colds (assuming they all become ill at
least once that year). If I plan to use an alpha level of .05 for this analysis, what is the
maximum cold duration average I would have to observe in my sample in order to still reject
the null hypothesis?
9.
A scientist would like to determine whether athletes are generally different in height than
non-athletes. He asks 10 individuals (a sample of 5 athletes and a sample of 5 non-athletes) to
record their heights (the table below contains known information about this data set). Using
an alpha level of .05, test the hypothesis that athletes and non-athletes have different heights.
𝑥
s
Athlete
61”
3.4”
Non-athlete
64”
1.4”
a. Which test is appropriate?
b. Report the null hypothesis:
c. Report the alternative hypothesis:
d. Find the critical value:
e. Calculate the obtained statistic:
f. Make a decision:
g. What does your decision mean?
10. In a specific population, there are just as many females as males. Forbes magazine publishes
an article that 11 out of 15 of the wealthiest individuals in this population are male. I would
like to test the hypothesis that the wealthiest individuals in this population are male (using
alpha=.01).
a. Which test is appropriate?
b. Report the null hypothesis:
c. Report the alternative hypothesis:
d. Find the critical value:
e. Calculate the obtained statistic:
f. Make a decision:
g. What does your decision mean?
11. A feline researcher would like to evaluate the effect of catnip on motor activity in kittens. He
puts 5 kittens in a playpen and counts how many times each kitten swats at a hanging toy
placed in the middle of the pen. He then exposes the kittens to catnip and again lets them play
in the pen and counts the number of swats. Test the hypothesis that catnip has an effect on
motor activity in kittens using an alpha of .01. Assume a normal distribution.
No catnip
5
7
6
8
1
catnip
8
9
8
8
0
a. Which test is appropriate?
b. Report the null hypothesis:
c. Report the alternative hypothesis:
d. Find the critical value:
e. Calculate the obtained statistic:
f. Make a decision:
g. What does your decision mean?