Download AP Statistics: Chapter 19 – Confidence Intervals with Proportions

AP Statistics: Chapter 19 – Confidence Intervals with Proportions
Name_____________________
1.
In a national survey, respondents were asked if they smoke, to which 191 responded “yes” and 1005
responded “no.” Obtain a 90% confidence interval for the proportion of American adults who
smoke.
2.
An anthropologist wants to estimate the proportion of individuals in a tribe in the Philippines who
die before reaching adulthood. For a sample of families who had children born between 1960 and
1970, it was found that 15 of 50 children died before reaching adulthood. Obtain a 95%
confidence interval for the corresponding population proportion.
3.
For every car passing a certain location during a particular 10-hour period, it was noted whether
the driver was wearing a seatbelt. For the 2550 cars observed, 1848 drivers were wearing
seatbelts. Construct a 99% confidence interval for the population proportion of drivers who wear
seatbelts.
4.
A random sample is taken of students at a large university to determine the proportion who own
automobiles. If 34% of the students in the sample own automobiles, find a 99% confidence
interval for the proportion of all students owning automobiles under the following conditions:
a)
sample size is 50
b)
sample size is 100
c)
sample size is 400
5.
Out of a random sample of 400 individuals who are registered and planning to vote in a mayoral
election, 160 plan on voting for Jones and 240 for Smith. Compute a 99% confidence interval for
the proportion of votes that Jones will receive. Do you think that Jones will lose the election?
6.
In problem 5, suppose that 16 out of 40 sampled individuals plan to vote for Jones. Again, find the
99% confidence interval and, if possible, predict the winner. How does the result here compare to
the result in problem 5?
7.
A city council votes to appropriate funds for a new civic auditorium. The mayor of the city
threatens to veto this decision unless it can be shown that a majority of the citizens would use it
at least twice a year. The council commissions a poll to be taken of city residents. Out of a
random sample of 400 residents, 230 plan to use the facility at least twice a year, if it is built.
Find a 98% confidence interval for the proportion of all residents of the town who would use the
proposed auditorium at least twice a year. Interpret the interval and advise the mayor.
8.
An experiment was conducted to study the effect on pregnancies of exposure to a certain toxic
substance. Of 100 pregnant rats that were exposed to a certain dose of the substance, 20
suffered spontaneous abortions (miscarriages).
a)
Calculate a 90% confidence interval for the population proportion of pregnant rats that
would suffer a spontaneous abortion in this situation.
b)
Explain what is meant when we say that we have 90% confidence in this interval.
c)
What must the sample size be to obtain a confidence interval about half as wide as the one
obtained in part a?
9.
Refer to problem 2. Suppose that, before obtaining the sample, the anthropologist wanted to
determine how large a sample would be necessary in order to estimate the population proportion to
within .07 with 95% confidence. Calculate the sample size that will ensure at least this degree of
accuracy.
10. It is desired to estimate the proportion of traffic deaths in Florida last year that were alcoholrelated. Determine the necessary sample size, if we want our estimate of this proportion to be
accurate to within .06 with confidence .90.
11.
A study is planning for estimating the proportion of married women with living parents in the
United States who live in the same state as their parents. Approximately how large a sample is
needed to estimate this proportion correct to within the following:
a)
.10 with probability .95
b)
.05 with probability .95
c)
.05 with probability .99
d)
.01 with probability .99
Compare sample sizes for parts a and b, b and c, c and d.
KEY:
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
11)
(.14228, .17712)
(.17298, ,42702)
(.70192, .74749)
a)
(.16744, ,51256)
b)
(.21798, .46202)
c)
(.27899, .40101)
(.33691, .46309); Predict Jones to lose election. Entire interval is below .5.
(.20048, .59952); Jones might win. Has a larger margin of error, smaller n.
(.5175, .6325); The mayor should support the new auditorium because we can be 98% confident that
the true proportion of all town residents who will use the auditorium at least twice a year lies between
.5175 and .6325 (more than half).
a) (.13421, .26579)
b) The 90% confidence is in the method that if we take many, many samples of 100 pregnant rats and
create a 90% CI for each, 90% of the intervals will capture the true population proportion of pregnant
rats that would suffer spontaneous abortion.
c)
The new sample size must be 4 times the original.
n should be at least 165.
n should be at least 188.
a) n should be at least 97.
b) n should be at least 385.
c) n should be at least 664.
d) n should be at least 16,590.
a and b: smaller margin of error needs larger sample
b and c: higher confidence requires larger sample
c and d: smaller margin of error needs larger sample