Review problems in sample proportions
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Rest of the slides contain problems
Mainly from Chapter 8
Shall do most of them in class
Please bring tables and calculators to class
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A simple random sample of 100 athletes is
selected from a large high school. In the
sample, there are 15 football players. What is
the estimate of the standard deviation of the
sample proportion of football players?
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A simple random sample of 60 blood donors
is taken to estimate the proportion of donors
with type A blood with a 95% confidence
interval. In the sample, there are 10 people
with type A blood. What is the margin of error
for this confidence interval?
Construct a 97.5% confidence interval. What
is the margin of error of this interval
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A simple random sample of 85 students is
taken from a large university on the West
Coast to estimate the proportion of students
whose parents bought a car for them when
they left for college. When interviewed, 51
students in the sample responded that their
parents bought them a car. What is a 95%
confidence interval for p, the population
proportion of students whose parents bought
a car for them when they left for college?
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A simple random sample of 100 bags of tortilla
chips produced by company X is selected every
hour for quality control. In the current sample,
18 bags had more chips (measured in weight)
than the labeled quantity. The quality control
inspector wishes to use this information to
calculate a 90% confidence interval for the true
proportion of bags of tortilla chips that contain
more than the label states. What is the value of
an estimate of standard error of p̂ ? What is the
90% confidence interval?
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The Information Technology Department at a
large university wishes to estimate p = the
proportion of students living in the
dormitories who own a computer with a 95%
confidence interval. What is the minimum
required sample size the IT Department
should use to estimate the proportion p with
a margin of error no larger than 3 percentage
points?
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The candy company that makes M&M’s claims
that 10% of the M&M’s it produces are green.
Suppose that the candies are packaged at
random in large bags of 200 M&M’s. When we
randomly pick a bag of M&M’s we may
assume that this represents a simple random
sample of size n = 200. Suppose we wish to
test H0: p = 0.10 versus Ha: p ≠ 0.10.
Assuming the null hypothesis what is the sd
of the sample proportion of green M&Ms?
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(a) Suppose that in the randomly selected bag
of M&M’s there are only 12 green M&M’s.
What is the value of the large-sample z
statistic?
(b) What is the value of the corresponding Pvalue?
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A noted psychic was tested for extrasensory
perception. The psychic was presented with 200
cards face down and asked to determine if the
card were one of five symbols: a star, a cross, a
circle, a square, or three wavy lines. The psychic
was correct in 50 cases. Let p represent the
probability that the psychic correctly identifies
the symbol on the card in a random trial. Assume
the 200 trials can be treated as a simple random
sample from the population of all guesses the
psychic would make in his lifetime.
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Suppose you wished to see if there were
evidence that the psychic is doing better than
just guessing. To do this, you test the
hypotheses H0: p = 0.20 versus Ha: p > 0.20.
What is the value of the large-sample z
statistic?
What do we know about the value of the Pvalue for the hypothesis test?
How large a sample n would you need to
estimate p with a margin of error 0.01 with
95% confidence? Use the hypothesized value
p = 0.20 as the value for p*.
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After once again losing a football game to the
college’s archrival, the alumni association
conducted a survey to see if alumni were in
favor of firing the coach. A simple random
sample of 100 alumni from the population of
all living alumni was taken. Sixty-four of the
alumni in the sample were in favor of firing
the coach. Let p represent the proportion of
all living alumni who favor firing the coach.
What is a 99% confidence interval for p?
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Suppose the alumni association wished to see
if the majority of alumni are in favor of firing
the coach. To do this they test the
hypotheses H0: p = 0.50 versus Ha: p > 0.50.
What is the P-value for this hypothesis test?
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An inspector inspects large truckloads of potatoes to
determine the proportion p in the shipment with
major defects prior to using the potatoes to make
potato chips. If there is clear evidence that this
proportion is less than 0.10, she will accept the
shipment. To reach a decision, she will test the
hypotheses H0: p = 0.10, Ha: p < 0.10. To do so, she
selects a simple random sample. 150 potatoes from a
random sample of 3000 potatoes on the truck were
defective.
What is the value of the large-sample z statistic?
What is the P-value for this hypothesis test?
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Define the hypotheses to test, and the significance
level a.
Calculate the value of the test statistic.
Find the P-value based on the observed data.
State the conclusion.
◦ Reject the null hypothesis if the P-value <=a ; if it>a, the
data do not provide sufficient evidence to reject the null.
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A recent study of newborns used umbilical cord
blood to test for 25 hydroxyvitamin D, which is
an indicator of vitamin D status of the baby. It
was reported that 65% of babies tested were
deficient in vitamin D in spite of the fact that the
mothers consumed vitamin D supplements
during pregnancy. A researcher in a northern
region felt that this percentage was too high for
this region because with the reduced hours of
sunshine during winter months pregnant women
tended to use of higher doses of supplements to
compensate. A sample of 125 newborns was
tested, and 72 were declared to be deficient in
vitamin D.
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What would be the appropriate null and
alternative hypotheses that the researcher
should establish for a test of significance?
Based on the data from this region, what is
the value of the appropriate test statistic?
What is the P-value and the conclusion using
α = 0.05?
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A sociologist is studying the effect of having
children within the first two years of marriage
on the divorce rate. Using hospital birth
records, she selects a simple random sample
of 200 couples who had children within the
first two years of marriage. Following up on
these couples, she finds that 80 couples are
divorced within five years. Let p = the
population proportion of couples who had
children within the first two years of marriage
and are divorced within five years.
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Suppose the sociologist wishes to test H0: p =
⅓ versus Ha: p ≠ ⅓. At the 10% significance
level, are the data statistically significant for
testing these hypotheses?