A.P. Statistics Lesson 12-1: Inference For a Population Proportion
Conditions for Inference About A Proportion
The data are an __________ from the population of interest.
The population is _______________________________________________ as the sample.
For a test of HO: p = pO, the sample size n is so large that both _____ and __________ are 10 or more.
For a confidence interval, n is so large that both the count of success _____ and the count of failures ______ are
10 or more.
Inference For a Population Proportion
Draw an SRS of size n from a large population with unknown proportion p of successes. An approximate level
C confidence interval for p is
where z* is the upper (1 – C)/2 standard normal critical value.
To test the hypothesis HO: p = pO, compute the z statistic
In terms of a variable Z having the standard normal distribution, the approximate P-value for a test of HO
against
HA: p > pO is P(Z ≥ z)
HA: p < pO
is
P(Z ≤ z)
HA: p ≠ pO
is
2P(Z ≥ |z|)
One Proportion Z-Test Example:
A coin that is balanced should come up heads half the time in the long run. The French naturalist Count Buffon
tossed a coin 4040 times. He got 2048 heads. The sample proportion of heads is
𝑝̂ =
That’s a bit more than one-half. Is this evidence that Buffon’s coin was not balanced?
Solve using a confidence interval.
A.P. Statistics: Introduction to 1-Sample Procedures for Population Proportions
In an attempt to determine if there is a home team advantage, a sports writer takes a sample of 80 games and
notes that in 45 of the 80 games, the home team won the game.
Does this sample give evidence at the 5% level that there is a home team advantage? Show all steps.
Construct a 90% confidence interval for p. What parameter does this interval estimate?
In a recent election, a poll was taken 2 weeks before the election. In a sample of 850 registered voters, 451
indicated that they planned to vote for Candidate A.
Does this poll give evidence at the 5% level that Candidate A will win the majority of the vote? Show all steps.
Construct a 90% confidence interval for p. What parameter does this interval estimate?
Is 0.5 in the interval? Explain why this does not contradict the conclusion of the significance test at the 5%
level?
Suppose the pollsters want the margin of error to be less than 2%. How large must the sample be in order to get
the margin of error to be 3% with 95% confidence?
To test if a pair of dice is fair, a researcher decides to roll the dice 500 times and record the proportion of the
times a 7 is rolled.
What is the smallest proportion of times that a 7 would need to be rolled in order to reject HO: p = 1/6 in favor
of HO: p > 1/6 at the 5% significance level?
What is the smallest proportion of times that a 7 would need to be rolled in order to reject HO: p = 1/6 in favor
of HO: p > 1/6 at the 5% significance level if he rolled the dice 1000 times instead of 500 times?
In 2000, a census was taken of Lincoln. It indicated that 18% of the population was Hispanic. A survey of 400
randomly chosen people in Lincoln was taken this year and it found that 92 out of the 400 were Hispanic.
Does this survey give sufficient evidence at the 5% level that the proportion of Hispanics in Lincoln has
changed since 2000?
Construct a 95% confidence interval for p. How would you use the confidence interval to answer the same
question asked above?