SELF TEST EIGHT: HYPOTHESIS TESTING

SELF TEST EIGHT: HYPOTHESIS TESTING
1. List the formal steps in hypothesis testing.
In questions 2 - 7, state the appropriate null and alternative hypotheses and find the critical z value
for the test. In each case, assume that the population standard deviation is known.
2. A Belchfire dealership will be feasible in Kasuthville if average family income in K-ville is at least
$60,000 a year. We wish to survey a sample of K-ville families to determine the feasibility of placing a
dealership there; we will test at α = 0.05.
3. We are arranging a lab test for our new drug, and we wish to be able to certify, at 8% significance, that
capsules on average contain no more than 5 parts per billion of impurities.
4. Letter-Rip is an imitation of the famous Assimilate spreadsheet; Letter-Rip’s distributor claims that on
average Letter-Rip recalculates spreadsheets in the same time as Assimilate. We have a set of
spreadsheets for which the average recalculation time with Assimilate is known to be 12 seconds. We
would like to test LR’s claim at 2% significance (α = 0.02).
5. Engineers believe that a synthetic oil will reduce engine wear. With existing oils, bearing surfaces wear
down by 0.5 mm in 500 hours of operation. State the appropriate null and alternative hypotheses to test
whether the synthetic reduces wear; we want α = 0.1.
6. Pop’s Deli sells hamburger guaranteed to be no more than 10% fat. Pop wants to set up an ongoing test
procedure to ensure that he meets his guarantee, but since lean meat is expensive, Pop wants to be sure
that he doesn't have less than 10% fat either. Pop wants to test at 10% significance (α = 0.1).
7. To be successful, a new TV show must capture at least 30% of its prospective market. To test a new
show being offered, a TV network will show it to a sample audience and observe the proportion in the
sample who like the show. The network wants to conduct its test at 2% significance (α = 0.02).
8. We want to test whether our Yuppie’s-n-cream ice cream is on average at least 34% butterfat, so we
have H 0 : μ ≥ 34% with H 1 : μ < 34%. For all such ice cream the standard deviation of butterfat content
= 1%, and the distribution among packaged quarts is known to be normal. We have a sample of 23
quarts of Y-n-c, and we wish to test at α = 0.05. In our sample, the mean butterfat content is 33.7%.
What is the p-value of the test Should we reject H 0 Bonus: What value of butterfat content would
cause us to reject H 0
9. West Coast Wood is producing chopsticks for export to Japan; their customer specifies that the
chopsticks are to average 17 cm. in length. Over time, West Coast has found that the length of their
chopsticks is normally distributed with σ = 0.75 cm. From a new lot of chopsticks, a sample of 22 are
drawn. We will run a two-tailed hypothesis test on them with α = 0.1. If⎯x = 17.2, does WCW need to
adjust their machinery What is the p-value of this test
10. Senator Fudd thinks it may be worth making a run for the Presidency if he can believe at 5%
significance that at least 35% of the voters favor him already. So the senator's political pollsters
conduct a survey of 1,132 registered voters to test the null hypothesis H 0 : π ≥ 0.35. If in the sample
only 33% per cent favor Fudd, will running for President be worthwhile What are the standard error,
the z value and the p-value of the test
11. The Pay-less tax preparation firm claims to have an error rate of no more than 2%. To test this claim, a
consumer information organization selects a sample of 600 tax returns prepared by Pay-less and
examines them for errors; they will use the result to test H 0 : π ≤ 0.02 with H 1 : π > 0.02, and they wish to
use α = 0.01. In the sample from Pay-less, the investigators find 21 incorrect returns. Should they
accept or reject H 0 What is their conclusion about Pay-less’s claim

self-test 8, page 2
12. Suppose that a null hypothesis to be tested is true. Then the probability of a rejection error (Type I
error) is _________.
13. If the null hypothesis is true, then β = _________.
14. Other things being equal, as α increases, β ___________.
15. For each of the following situations, state what a Type I error is, what a Type II error is and the likely
costs associated with each.
a) An earthquake may have damaged the grandstands at Candlestick Park; engineers will inspect the
stadium seating with the hypothesis that the stadium has been damaged.
b) Betsy the Buyer conducts a hypothesis test in deciding whether to buy a truckload of potatoes; her
hypothesis is H 0 : μ ≥ 8 oz.
c) Blue Moon Nightclub is willing to advertise in the Charlotte Daily Mullet Wrapper if they can be
reasonably sure that at least 20% of the newspaper’s readers are under 35 years of age.
d) Safety personnel are examining the remains of an airplane crash; their initial hypothesis is that the
crash resulted from pilot error. If they’re convinced that was the cause, they’ll drop their
investigation.
16. In the following sketch, we are testing the hypothesis H 0 : μ ≥ 170. We will reject the null hypothesis
whenever ⎯x < 135. On the sketch, indicate the area that represents α and the area that represents β.
Show what will happen if the significance level is reduced; if it is increased.
Calculating Beta
45 60 75 90 105 120 135 150 165 180 195 210 225 240
17. In designing a hypothesis test, if the cost of a Type I error is very high and that of a Type II error low,
we would prefer to have a __________ significance level, since the probability of a Type I error is
_______.
18. Find the critical z values(s) for each of the following pairs of hypotheses and state the decision rule:
a) H 0 : μ ≤ 80 vs. H 1 : μ > 80, α = 0.05
b) H 0 : π = 0.65 vs. H 1 : π ≠ 0.65, α = 0.05
c) H 0 : μ ≥ 112 vs. H 1 : μ < 112, α = 0.1
d) H 0 : μ = 9000 vs. H 1 : μ ≠ 9000, α = 0.01
19. Find the critical t value(s) for each of the following pairs of hypotheses:
a) H 0 : μ ≤ 43 vs. H 1 : μ > 43 with n = 18 and α = 0.05.
b) H 0 : μ = 57 vs. H 1 : μ ≠ 57 with n = 20 and α = 0.01.
c) H 0 : μ ≥ 29 vs. μ < 29 with n = 28 and α = 0.025.
d) H 0 : μ = 2001 vs. H 1 : μ ≠ 2001 with n = 12 and α = 0.05.

self-test 8, page 3
20. ART engineers are trying to rearrange their production line, and they need to be sure that on average it
takes no more than 12 minutes for assembly operation A. It is firmly believed that for processes of this
type, the distribution of assembly times is normal. Thus, the engineers will test μ ≤ 12 min. with H 1 : μ
> 12 min., and it is decided to conduct the test at 5% significance (α = 0.05). The engineers select a
sample of 21 workers and have them perform the operation; for the workers in the sample⎯x = 12.19
min. with s = 0.5 min. Should the engineers accept or reject the null hypothesis
21. The Dean of Students at an urban university believes that students commute an average distance of 22
miles and that commuting distance is approximately normally distributed.
a) State the appropriate null and alternative hypotheses for testing the Dean's belief.
b) The null hypothesis will be tested with a sample of 65 students; find the critical t values for tests at
10% and 1% significance.
c) In the sample ⎯x = 24 miles and s = 6 miles. Does this result confirm the Dean's belief
22. In performing hypothesis tests about population means, the z distribution should be used in either of the
following cases:
a) whenever____________ is known and __________, regardless of sample size;
b) or when ______ and the sample size __________, regardless of the shape of the population.
23. The t distribution should be used in testing hypotheses about population proportions if the population
standard deviation is ___________. For small samples we must also believe that the population is
__________ distributed.
24. The z distribution is always used in testing hypotheses about ___________.
25. A sample of 23 records are drawn from a stockbroker's accounts and used to test the hypothesis that the
average account is at least $28,000. The hypothesis is tested using the sample results ⎯x = $27,450 and s
= $3,129; it is known that the distribution of all accounts is normal. What is the appropriate sampling
distribution, or test statistic, to use in this test
26. An academic advisor in the College of Business asserts that at least 40% of all students take Stats I
more than once. This proposition is to be tested by drawing a sample of 200 student records and
observing how many of those in the sample have taken the course more than once. In this test the
appropriate test statistic is the _____________.
27. Economic researchers have begun to look at workers’ expectations for wage increases, but they know
little about the population distribution. The Bush Administration’s Council of Economic Advisors
asserts that the current average expectation is for no more than a $700 increase next year. If this
hypothesis is to be tested by interviewing a sample of 75 workers, the appropriate sampling distribution,
or test statistic, to use is the ______.
28. In the preceding question, suppose that for the sample of 75 workers,⎯x = $800 and s = $500. At 5%
significance, should we accept or reject the null hypothesis implied by the Administration's claim
29. In setting up an hypothesis test, ART's engineers discover that with a 5% significance level (α = 0.05),
they have unacceptably large values for β for some possible values of μ. The engineers would in fact
like to reduce α even further, preferably to 0.01. The only way out of their dilemma is ____________.
30. In designing a hypothesis test, if the cost of a Type I error is very high and that of a Type II error low,
we would prefer to have a __________ significance level, since the probability of a Type I error is
_______.
31. What three factors must be taken into account in designing a hypothesis test

self-test 8, page 4
32. For each of the following hypothesis tests, find the p-value of the test and state whether the null
hypothesis should be rejected at 5% significance level.
a) We are testing H 0 : μ ≥ 20 vs. H 1 : μ < 20. The population standard deviation σ = 2; in a sample of
size 36, the sample mean ⎯x = 19.2.
b) We wish to test whether bottles of Sudso Liquid Detergent contain 40 oz of detergent. The process
standard deviation is 6. In a sample of 28 bottles, the mean content is 42.7 oz.
c) Government investigators wish to determine whether long-haul truckers are exceeding the federal
standard of no more than 92 hours of driving in a two-week period, so H 0 : μ ≤ 92 with H 1 : μ > 92.
The standard deviation of hours driven for all truckers is known to be 5 hours. In a sample of 32
truckers, their logs show that in the previous two weeks they drove an average of 93.3 hours.
d) Nationally, 70% of teenagers eat fast food at least once a week. We’d like to know whether the
proportion is the same in Gopher Prairie. In a sample of 312 teenagers in Gopher Prairie, 65% eat
out at least once a week.

self-test 8, page 5
SELF TEST EIGHT:
1. see notes
2. H 0 : μ ≥ 60,000
H 1 : μ < 60,000; −1.64
3. H 0 : μ ≤ 5
H 1 : μ > 5; +1.41
4. H 0 : μ R = μ L
H 1 : μ R ≠ μ L ; ±2.33
5. H 0 : μ ≥ 0.5, that is, wear is at least .5 mm
H 1 : μ < 0.5; −1.28
6. H 0 : μ = 10%
H 1 : μ ≠ 10%; ±1.64
7. H 0 : π ≥ 0.3
H 1 : π < 0.3; −2.05
8. no, p-value = 0.0749; less than 33.66%
9. no, z=1.25 p-value = 0.211
10. On Fudd’s criteria, he should run. σ p =1.4176 (or 0.01476), z= −1.41, p-value=0.0793
11. reject since p-value = 0.0044; error rate is higher than claimed
12. =α
13. 0
14. decreases
15.
16.
17. low; significance level or α
18. +1.64, reject null if z > 1.64; ±1.96, reject null if z < −1.96 or z > +1.96; −2.33, reject null if z <
−2.33; ±2.58, reject null if z < −2.58 or z > +2.58 .
19. +1.74; ±2.681;−2.052;±2.201
20. reject;t=1.741>t C =1.725
21.
a) H 0 : μ = 22, H 1 : μ ≠ 22;
b) ±1.669 & ±2.655
c) no, reject H 0 with t = 2.687
22.
a) pop. σ is known and the population is normally distributed
b) pop. σ is known; n ≥ 30
23. not known; normally
24. proportions
25. t
26. z
27. t
28. reject; t=1.732>1.666.
29. increase the sample size
30. low; the significance level of the test
32. a. 0.0082; reject H 0 :
b. 0.0174; reject H 0 :
c. 0.0708; do not reject
d. 0.0536; do not reject