unit 10: inference for proportions

Question 1

For what values of n and p does the normal approximation to the distribution of pˆ work best? For
what values of n and p is the normal approximation very poor?

Answer 1

The normal approximation is best when n is large and p = 0.5. The normal approximation is worst
when n is small and p is close to 0 or 1.

Question 2

In words, what is the meaning of SE(ˆp1 − pˆ2)?

Answer 2

• SE(ˆp1 − pˆ2) is the standard error of the difference in sample proportions, which is an estimate of
  the standard deviation of the sampling distribution of pˆ1 − pˆ2.
• It is a measure of the variability in pˆ1 − pˆ2 if samples were to be repeatedly drawn from the populations.

Question 3

What is the difference in meaning of the symbols pˆ and p?

Answer 3

• pˆ is the sample proportion, whereas p represents the true proportion for the entire population.
• Once the sample is drawn, we will know the value of pˆ, but p is typically an unknown value that we are trying to estimate.

Question 4

Would it ever make sense to test H0: pˆ = 0.25?

Answer 4

No, this would never make sense. We test hypotheses about parameters, and not about statistics

Question 5

pˆ is an unbiased estimator of p.

Answer 5

True, since E(ˆp) = p.

Question 6

When n < 30 we should use the t distribution when calculating confidence intervals for p

Answer 6

False.

The t distribution never arises in inference for proportions.

Question 7

The true distribution of pˆ is based on the binomial distribution

Answer 7

True. pˆ =X/n where X has a

binomial distribution with parameters n and p

Question 8

The true standard deviation of the sampling distribution of pˆ depends on the value of p.

Answer 8

True.

Question 9

The sampling distribution of pˆ is perfectly normal for large sample sizes.

Answer 9

False. It’s an approximation, but it can be a very good approximation.

Question 10

The normal approximation to the sampling distribution of pˆ works best when we have a large
sample size and p = 0.5.

Answer 10

true

Question 11

The sampling distribution of pˆ becomes more normal as p tends to 1.

Answer 11

false. the sampling list becomes strongly skull as p approaches 1

Question 12

The sampling distribution of p is approximately normal for large sample sizes

Answer 12

False. p is a

parameter, not a statistic, and as such does not have a sampling distribution.

Question 13

All else being equal, the value of SE(ˆp) decreases as the sample size increases.

Answer 13

true

Question 14

In repeated sampling, exactly 95% of 95% confidence intervals for p will capture p.

Answer 14

False, since we are using a normal approximation and not an exact procedure. (But if the sample size is very large then the true percentage will be very close to 95%.)