Download Section 8.2 Testing a Proportion problems

Section 8.2 Testing a Proportion
Statistics
Informal Significance Testing
P20.) A 1997 article reported that two-thirds of teens in grades 7 – 12 want to
study more about medical research. You wonder if this proportion still
holds today and decide to test it. You take a random sample of 40 teens
and find that only 23 want to study more about medical research.
a.) What is the standard (the hypothesized value, p0, of the population
proportion, p0)?
b.) What is the sample proportion, p̂ ?
c.) Use Display 8.9 on page 492 to determine whether the result is
statistically significant. That is, is there evidence leading you to
believe that the proportion today is different from the proportion in
1997?
P21.) A student took a 40-question true-false test and got 30 answers correct.
The student says, “That proves I was not guessing at the answers.”
a.) What is the standard, p0?
b.) What is the sample proportion, p̂ ?
c.) Use Display 8.9 on page 492 to determine whether the result is
statistically significant.
d.) Does the answer to part c prove that the student was not guessing?
P22.) This year, 75% of the seniors wanted extra tickets for their graduation
ceremony. To anticipate whether there might be a change in that
percentage next year, you took a random sample of 40 juniors and found
that 32 would want extra tickets.
a.) What is the standard, p0?
b.) What is the sample proportion, p̂ ?
c.) Use Display 8.9 on page 492 to determine whether this result is
statistically significant.
d.) Is this statistical evidence of a change?
The Test Statistic
P23.) For the situation in P20, what value of the test statistic should the junior
class use to test whether there is statistical evidence of a change?
P24.) Forty-five dogs and their owners, chosen at random, were photographed
separately. A judge was shown a picture of each owner and pictures of
two dogs and asked to pick the dog that went with the owner. The judge
was right 23 times. What value of the test statistic should be used to test
whether the judge did better than could reasonably be expected just by
guessing?
P-Values
P25.) Find the P-value for Miguel and Kevin’s test statistic, -3.16, from pages
493-494. Write a sentence explaining what this P-value means in the
context of their situation.
P26.) Suppose you re-create the chimpanzee experiment in the example on
page 494 with an apparatus that tests whether cats can select the rake that
results in the food. You test 50 randomly chosen cats and find that 28
select the rake that pulls in the food. Find the P-value for this test and
interpret it.
P27.) According to the US Census Bureau, about 69% of houses across the
country are occupied by their owners. Your class randomly samples 50
houses in your community and finds that 30 houses are occupied by the
owners. Follow these steps to determine whether your community differs
from the nation as a whole as to the percentage of houses that are owneroccupied.
a.) State the null hypothesis to be tested. What is the alternative
hypothesis?
b.) Find the value of the test statistic.
c.) Find the P-value for this test and explain what it means.
d.) Write a conclusion, based on your analysis, in the context of the
problem.
P28.) Which of statements A – E is the best explanation of what is meant by the
P-value of a test of significance?
A. Assuming that you had a random sample and the other conditions for
a significance test are met, the P-value is the probability that H0 is
true.
B. Assuming that H0 is true, the P-value is the probability of observing a
value of a test statistic at least as far out in the tails of the sampling
distribution as is the value of z from your sample.
C. The P-value is the probability that H0 is false.
D. Assuming that the sampling distribution is normal, the P-value is the
probability that H0 is true.
E. Assuming that H0 is true, the P-value is the probability of observing
the same value of z that you got in your sample.
Critical Values and Level of Significance
P30.) Use Table A on page 824 (or any other z table) to answer these questions.
a.) What critical value is associated with a significance level of 0.12?
b.) What significance level is associated with critical values of z* of
±1.73?
P31.) Use Table A on page 824 (or any other z table) to answer these questions.
a.) What level of significance is associated with critical values of z* of
±2.576?
b.) What critical values are associated with a significance level of 2%?
The Formal Language of Tests of Significance
P32.) Your null hypothesis is that spinning a coin is fair. A class spun quarters
500 times and got 194 heads. Carry out the four steps in a test of the null
hypothesis that spinning a quarter is fair. (1. Give the name of the test and
check the conditions for its use; 2. State the hypotheses, defining any
symbols; 3. Compute the test statistic, z, and find the critical values, z*,
and the P-value; 4. Write a conclusion)
P33.) Suppose that, in a random sample of 500 bookstores across the US, 265
also sell DVDs. Carry out the four steps in a test of the null hypothesis
that half the bookstores in the US sell DVDs.
P34.) State an appropriate null hypothesis for each of these situations.
a.) You wonder if a student is guessing on a multipl-choice test of 60
questions where each question has five possible answers.
b.) You wonder if there is an equal proportion of male and female
newscasters on local television stations.
c.) You wonder if people who wash their cars once a week are most
likely to wash them on Saturday.
P35.) The US 2000 Census found that 4% of all households were
multigenerational (consisting of three or more generations of parents and
their children). You want to test the null hypothesis that this percentage
is the same this year as it was in 2000. You write the null hypothesis as
p = 0.04. Which of these statements best describes what p stands for?
A. The proportion of all households that were multigenerational in
2000.
B. the proportion of all households that are multigenerational this year.
C. the proportion of multigenerational households in the sample in 2000
D. the proportion of multigenerational households in the sample this
year.
E. the proportion fo all multigenerational households in both 2000 and
this year.
Types of Errors
P36.) Hila is rolling a pair of dice to test whether they land doubles 1/6 of the
time. She does not know it, but the dice are actually fiar. She will use a
significance level, α, of 0.05. Hila rolls the dice 100 times and gets
doubles 22 times.
a.) What conclusion should Hila come to?
b.) Did Hila make an error? If so, which type?
P37.) Jeffrey and Taline each want to test if the proportion of adults in their
neighborhood who have graduated from high school is 0.94, as claimed in
the newspaper. Jeffry takes a random sample of 200 adults and uses α =
0.05. Taline takes a random sample of 500 adults and uses α = 0.05.
Suppose the newspaper’s percentage is acutally right.
a.) Is it possible for Jeffrey or Taline to make a Type I error? If so, who
is more likely to do so?
b.) Is it possible for Jeffrey or Taline to make a Type II error? If so, who
is more likely to do so?
P38.) Jeffrey and Taline each want to test if the proportion of adults in their
neighborhood who took chemistry in high school is 0.25, as claimed in
the newspaper. Jeffrey takes a random sample of 200 adults and uses α =
0.05. Taline takes a random sample of 500 adults and uses α = 0.05.
Suppose the newspaper’s percentage is acutally wrong.
a.) Is it possible for Jeffrey or Taline to make a Type I error? If so, who
is more likely to do so?
b.) Is it possible for Jeffrey or Taline to make a Type II error? If so, who
is more likely to do so?
One-Sided Tests of Significance
P42.) In a poll of 1000 randomly sampled adults in the US, 46% said they were
satisfied with the quality of K – 12 education in the nation. Does this
imply that less than a majority of adult residents are satisfied with the
quality of education? Answer this question by working through the steps
in parts a – d.
a.) Verify that the conditions for the test are satisfied.
b.) What is the null hypothesis? What is the most appropriate alternative
hypothesis? Explain your reasoning.
c.) Calculate a test statistic and the corresponding P-value.
d.) Write a conclusion in the context of the original question.