Hypothesis Tests for 1 sample Proportions
7. Blinking timers. Many people have trouble programming their VCRs, so a company has developed what it
hopes will be easier instructions. The goal is to have at
least 96% of customers succeed. The company tests the
new system on 200 people, of whom 188 were
successful. Is this strong evidence that the new system
fails to meet the company's goal? A student's test of
this hypothesis is shown below. How many mistakes
can you find?
H 0 : pˆ  0.96
H A : pˆ  0.96
SRS, 0.96(200) > 10
188
 0.94 ; SD  pˆ  
200
0.96  0.94
z
 1.18
0.017
 0.94  0.06 
200
 0.017
P = P(z > 1.18) = 0.12
There is strong evidence that the new system does not
work.
8. Got milk? In November 2001, The Ag Globe Trotter
newsletter reported that 90% of adults drink milk. A
regional farmers' organization planning a new
marketing campaign across its multi-county" area polls
a random sample of 750 adults living there. In this
sample, 657 people said that they drink milk. Do these
responses provide strong evidence that the 90% figure
is not accurate for this region? Correct the mistakes
you find in a student's attempt to test an appropriate
hypothesis.
H 0 : p  0.9
H A : p  0.9
SRS, 750 > 10
657
 0.876 ; SD  pˆ  
750
0.876  0.94
z
 2
0.012
 0.88  0.12 
750
 0.012
P = P(z > -2) = 0.977
There is more than a 97% chance that the stated
percentage is correct for this region.
9. Dowsing. In a rural area only about 30% of the wells
that are drilled find adequate water at a depth of 100
feet or less. A local man claims to be able to find water
by "dowsing"—using a forked stick to indicate where
the well should be drilled. You check with 80 of his
customers and find that 27 have wells less than 100
feet deep. What do you conclude about his claim? (We
consider a P-value of around 5% to represent strong
evidence.)
a) Write appropriate hypotheses.
b) Check the necessary assumptions.
c) Perform the mechanics of the test. What is the Pvalue?
d) Explain carefully what the P-value means in this
context.
e) What's your conclusion?
10. Autism. In the 1980s it was generally believed that
autism affected about 5% of the nation's children.
Some people believe that the increase in the number of
chemicals in the environment has led to an increase in
the incidence of autism. A recent study examined 384
children and found that 46 of them showed signs of
some form of autism. Is this strong evidence that the
level of autism has increased? (We consider a P-value
of around 5% to represent strong evidence.)
a) Write appropriate hypotheses.
b) Check the necessary assumptions.
c) Perform the mechanics of the test. What is the Pvalue?
d) Explain carefully what the P-value means in this
context.
e) What's your conclusion?
f) Do environmental chemicals cause autism?
13. Pollution. A company with a fleet of 150 cars found
that the emissions systems of 7 out of the 22 they
tested failed to meet pollution control guidelines. Is
this strong evidence that more than 20% of the fleet
might be out of compliance? Test an appropriate
hypothesis and state your conclusion. Be sure the
appropriate assumptions and conditions are satisfied
before you proceed.
14. Scratch and dent. An appliance manufacturer
stockpiles washers and dryers in a large warehouse for
shipment to retail stores. Sometimes in handling them
the appliances get damaged. Even though the damage
may be minor, the company must sell those machines
at drastically reduced prices. The company goal is to
keep the level of damaged machines below 2%. One
day an inspector randomly checks 60 washers and
finds that 5 of them have scratches or dents. Is this
strong evidence that the warehouse is failing to meet
the company goal? Test an appropriate hypothesis and
state your conclusion. Be sure the appropriate
assumptions and conditions are satisfied before you
proceed.
15. Twins. Some doctors suspect that young mothers have
fewer multiple births. In 2001 a national vital statistics
report indicated that about 3% of all births produced
twins. Data from a large city hospital found only 7 sets
of twins were born to 469 teenage girls. Does that
suggest that mothers under age 20 may be less likely to
have twins? Test an appropriate hypothesis and state
your conclusion. Be sure the appropriate assumptions
and conditions are satisfied before you proceed.
16. Football. During the 2000 season, the home team won
138 of the 240 regular season National Football
League games. Is this strong evidence of a home field
advantage in professional football? Test an appropriate
hypothesis and state your conclusion. Be sure the
appropriate assumptions and conditions are satisfied
before you proceed.
21. Dropouts. Some people are concerned that new
tougher standards and high stakes tests adopted in
many states may drive up the high school dropout rate.
The National Center for Education Statistics reported
that the high school dropout rate for the year 2000 was
10.9%. One school district, whose dropout rate has
always been very close to the national average, reports
that 210 of their 1782 students dropped out last year. Is
their experience evidence that the dropout rate may be
increasing? Explain.
17. WebZine. A magazine is considering the launch of an
online edition. The magazine plans to go ahead only if
it's convinced that more than 25% of current readers
would subscribe. The magazine contacts a simple
random sample of 500 current subscribers, and 137 of
those surveyed expressed interest. What should the
company do? Test an appropriate hypothesis and state
your conclusion. Be sure the appropriate assumptions
and conditions are satisfied before you proceed.
22. Acid rain. A study of the effects of acid rain on trees in
the Hopkins Forest shows that of 100 trees sampled, 25
of them exhibited some sort of damage from acid rain.
This rate seemed to be higher than the 15% quoted in a
recent Environmetrics article on the average proportion
of damaged trees in the Northeast. Does the sample
suggest that trees in the Hopkins Forest are more
susceptible than the rest of the region? Comment, and
write up your own conclusions based on an appropriate
confidence interval as well as a hypothesis test. Include
any assumptions you made about the data.
18. Seeds. A garden center wants to store leftover packets
of vegetable seeds for sale the following spring, but the
center is concerned that the seeds may not germinate at
the same rate a year later. The manager finds a packet
of last year's green bean seeds and plants them as a
test. Although the packet claims a germination rate of
92%, only 171 of 200 test seeds sprout. Is this evidence
that the seeds have lost viability during a year in
storage? Test an appropriate hypothesis and state your
conclusion. Be sure the appropriate assumptions and
conditions are satisfied before you proceed.
23. Lost luggage. An airline's public relations department
says that the airline rarely loses Passengers' luggage. It
further claims that on those occasions when luggage is
lost, 90% is recovered and delivered to its owner
within 24 hours. A consumer group who surveyed a
large number of air travelers found that only 103 of
122 people who lost luggage on that airline were
reunited with the missing items by the next day. Does
this cast doubt on the airline's claim? Explain.
19. Women executives. A company is criticized because
only 13 of 43 people in executive-level positions are
women. The company explains that although this
proportion is lower than it might wish, it's not
surprising given that only 40% of all their employees
are women. What do you think? Test an appropriate
hypothesis and state your conclusion. Be sure the
appropriate assumptions and conditions are satisfied
before you proceed.
24. TV ads. A start-up company is about to market a new
computer printer. It decides to gamble by running
commercials during the Super Bowl. The company
hopes that name recognition will be worth the high cost
of the ads. The goal of the company is that at least 40%
of the public recognize its brand name and associate it
with computer equipment. The day after the game, a
pollster contacts 420 randomly chosen adults, and finds
that 181 of them know that this company manufactures
printers. Would you recommend that the company
continue to advertise during Super Bowls? Explain.
20. Jury. Census data for a certain county shows that 19%
of the adult residents are Hispanic. Suppose 72 people
are called for jury duty, and only 9 of them are
Hispanic. Does this call into question the fairness of
the jury selection system? Explain.
Hypothesis Tests for 1 sample Proportions
Answers
7. 1) Use p in hypotheses, not p̂ .
2) The question was about failing to meet the goal,
so HA should be p< 0.96.
3) Did not check 0.04(200) = 8.
.96 .04 
 0.014
200
0.876  0.9
5) z is incorrect; should be z 
 1.43
0.014
4) 188/200 = 0.94: SD( p̂ ) =
6) P = P(z < -1.43) = 0.076
7) Since p-value is greater than alpha (0.05), we will
fail to reject that at least 96% of customers succeed
programming their VCRs.
8. 1) Use p in hypotheses, not p̂ .
2) The question asks, "not accurate," so HA should
be two sided: p  0.9.
3) The correct conditions are SRS, (0.9)(750) > 10,
and (0.10)(750) > 10.
.9 .1
 0.011
750
0.876  0.9
 2.18
5) 2 is incorrect; should be z 
0.011
4) p = 657/750 = 0.876; SD( p̂ ) =
6) P = 2P(z < -2.18) = 0.029
7) Since p-value is less than alpha (0.05), we will
reject that the proportion of adults who drink milk
here is 90%.
9. a) H0: p = 0.30; HA: p > 0.30
b) Possibly an SRS; we don't know if the sample is
less than 10% of his customers but could be viewed
as less than 10% of all possible customers; (0.3)(80)
> 10 and (0.7)(80) > 10. Wells are independent only
if customers don't have farms on the same
underground springs.
c) z = 0.73; P-value = 0.232
d) If his dowsing is no different from standard
methods, there is more than a 23% chance of seeing
results as good as those of the dowsers, or better, by
natural sampling variation.
e) Since p-value is greater than alpha (0.05), we will
fail to reject that the dowser's chance of finding
water is the same as normal drilling.
10. a) H0: p = 0.05; HA: p > 0.05
b) SRS (not clear from information provided), <10%
of all children, (0.05)(384) > 10, and (0.95)(384) >
10.
c) z = 6.28, P = 2 X 10-10
d) If the autism rate has not increased, the chance of
observing at least 46 autistic children in a sample of
384 is 2 x 10-10 (almost 0).
e) Since p-value is less than alpha (0.05), we will
reject that the rate of autism is 5%.
f) We do not know that chemicals cause autism, only
that the rate is higher now than in the past.
13. H0: p = 0.20; HA: p > 0.20. SRS (not clear from
information provided); 22 is more than 10% of the
population of 150; (0.20)(22) < 10. Do not proceed
with a test.
14. H0: p = 0.02; HA: p > 0.02. SRS; less than 10% of all
washers and dryers made by the company;
(0.02)(60) < 10. Do not proceed with a test.
15. H0: p = 0.03; HA: p < 0.03. p = 0.015. One mother
having twins will not impact another, so
observations are independent; not an SRS; sample is
less than 10% of all births. However, the mothers at
this hospital may not be representative of all
teenagers; (0.03)(469) = 14.07 > 10; (0.97)(469) >
10. z = -1.91; P-value = 0.0278. Since p-value is
less than alpha (0.05), we will reject that the rate of
twins born to teenage girls at this hospital is 3%. It is
not clear whether this can be generalized to all
teenagers.
16. H0:p = 0.50; HA: p > 0.50. Results of one game
should not impact another, so games are
independent; data are all results for one season,
which should be representative of all seasons;
sample is less than 10% of all games; (0.50)(240) >
10; (0.50)(240) > 10. z = 2.32; P-value = 0.0101.
Since p-value is less than alpha (0.05), we will reject
that the home team does not have an advantage; they
win more than 50% of games at home.
17. H0: p = 0.25; HA: p > 0.25. SRS; sample is less than
10% of all potential subscribers; (0.25)(500) > 10;
(0.75)(500) > 10. z = 1.24; P-value = 0.1076. Since
p-value is greater than alpha (0.05), we will fail to
reject that 25% of current readers would subscribe;
the company should not go ahead with the WebZine
on the basis of these data.
18. H0: p = 0.92; HA: p < 0.92. Seeds in a single packet
may not be independent of each other. This is a
cluster sample of all seeds in the packet. We may
view this cluster as representative of all year-old
seeds, in which case the sample is less than 10% of
all seeds; (0.92)(200) > 10; (0.08)(200) > 10. z = 3.39; P-value = 0.0004. Since p-value is less than
alpha (0.05), we will reject that these seeds have lost
viability; their germination rate is 92%.
19. H0: p = 0.40; HA: p < 0.40. Data are for all
executives in this company and may not be able to
be generalized to all companies; (0.40)(43) > 10;
(0.60)(43) > 10. z = -1.31; P-value = 0.0955. Since
p-value is greater than alpha (0.05), we will fail to
reject that the proportion of women executives is
40% women in the company in general.
20. H0: p = 0.19; HA: p < 0.19. p = 0.125. z = -1.41; Pvalue = 0.0793. Since p-value is greater than alpha
(0.05), we will fail to reject that Hispanics are
represented in the jury pool is 19% proportion in the
population in general.
21. H0: p = 0.109; HA: p > 0.109. p = 0.118; z = 1.198;
P-value = 0.115. Since p-value is greater than alpha
(0.05), we will fail to reject that the dropout rate is
10.9%.
22. H0: p = 0.15; HA: p > 0.15. p = 0.25; z = 2.80; Pvalue = 0.0026. The 95% confidence interval is
(0.165, 0.335). We must assume the trees sampled
are a SRS of the trees in the area and are less than
10% of all trees in the forest. The results are
generalizable only to the Hopkins forest (or nearby if
the forest is viewed as representative). Since p-value
is less than alpha (0.05), we will reject that the
proportion of trees damaged by acid rain in the
Hopkins forest is 15%.
23. H0: p = 0.90; HA: p < 0.90. p = 0.844; z = -2.05; Pvalue = 0.0201. Since p-value is less than alpha
(0.05), we will reject that the actual rate at which
passengers with lost luggage are reunited with it
within 24 hours is the 90% claimed by the airline.
24. H0: p = 0.40; HA: p > 0.40. p = 0.431; z = 1.29; Pvalue = 0.0977. Since p-value is greater than alpha
(0.05), we will fail to reject that 40% of the public
recognizes the brand; I would not recommend they
continue to advertise during Super Bowls on the
basis of these data.