evidence, they want to estimate the IQ scores of three-year-old
children born to
mothers who where on this medication during pregnancy.
28. Previous studies suggest that the SD of IQ scores of three-year-old chilren is 18
points. How many such children should the researchers sample in order to obtain a
90% confidence interval with a margin of error less than or equal to 4 points?
29. An inventor has developed a new, energy-efficient lawn mower engine. He claims
that the engine will run continuously for 5 hours (300 minutes) on a single gallon of
regular gasoline. From his stock of 2000 engines, the inventor selects a simple
random sample of 50 engines for testing. The engines run for an average of 295
minutes,
with a population standard deviation of 20 minutes
. Test the null hypothesis
that the mean run time is 300 minutes against the alternative hypothesis that the
mean run time is not 300 minutes. Use a 0.05 level of significance. (Assume that run
times for the population of engines are normally distributed.)
30. Bon Air Elementary School has 1000 students. The principal of the school thinks
that the average IQ of students at Bon Air is at least 110. To prove her point, she
administers an IQ test to 20 randomly selected students. Among the sampled
students, the average IQ is 108 with a sample standard deviation of 10. Based on
these results, should the principal
accept
or reject her original hypothesis? Assume
a significance level of 0.01. (Assume that test scores in the population of engines
are normally distributed.)
31. A particular brand of tires claims that its deluxe tire averages 50,000
miles before it
needs to be replaced. From past studies of this tire, the standard deviation is known
to be
8,000
. A survey of owners of that tire design is conducted. From the 28 tires
surveyed, the mean lifespan was 46,500 miles with a standard deviation of
9,800
miles. Using
α=0.05
, is the data highly inconsistent with the claim?
32. A Nissan Motor Corporation advertisement read, “The average man’s I.Q. is 107.
The average brown trout’s I.Q. is 4. So why can’t man catch brown trout?” Suppose
you believe that the brown trout’s mean I.Q. is greater than four. You catch 12 brown
trout. A fish psychologist determines the I.Q.s as follows: 5; 4; 7; 3; 6; 4; 5; 3; 6; 3;
8; 5. Using
α=0.05, c
onduct a hypothesis test of your belief.
33.
The television habits of 30 children were observed. The sample mean was found
to be 48.2
hours per week, with a standard deviation of 12.4 hours per week.
Using
α=0.05, t
est the claim that the standard deviation was at least 16 hours per
week.
34.
After many years of teaching, a statistics pro- fessor computed
the variance of the
marks on her final exam and found it to be
a
2
= 250. She recently made changes to the
way in which the final exam is marked and wondered whether this would result in a
reduction in the variance. A random sample of this year’s final exam marks are listed
here. Can the pro- fessor infer at the 10% significance level that the
variance has
decreased?
57
92 99 73 62 64
75 70
88
60
35. The US Department of Energy reported that 51.7% of homes were heated by natural
gas. A random sample of 221 homes in Kentucky found that 115 were heated by
natural gas. Does the evidence support the claim for Kentucky at the α=0.05 level in
Kentucky?
36.
A number of restaurants feature a device that allows credit card users to swipe their
cards at the table. It allows the user to specify a percentage or a dollar amount to leave
as a tip. In an experiment to see how it works, a random sample of credit card users
was drawn. Some paid the usual way, and some used the new device. The percent left
as a tip was recorded and listed below. Can we infer that users of the device leave
larger tips
at the α=0.05 level
? (
𝜎
1
= 𝜎
2
)
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