Inference for Proportions: Confidence Intervals and Hypothesis Tests, Exams of Probability and Statistics

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Chapter 7 Inference for Proportions This chapter presents the z procedures for one-sample and two-sample inference about pop- ulation proportions. The procedures are approximate, based on the large-sample normal approximation. 7.1 Inference for a population proportion Example:

(a) What is the current unemployment rate in U.S.? (b) What is President Clinton’s approval rating? (c) What proportion of students in Math 243 will be awarded Grade A? (d) · · ·

Let p be an population proportion of successes, and pˆ the sample proportion of an SRS with sample size n. Recall

• Mean of pˆ = p
• Standard deviation of pˆ: σpˆ =

√p(1p) n

• √pˆ (^) p^ −(1^ pp) n

≈ N (0, 1)

• Standard error of pˆ: σˆpˆ =

√ (^) pˆ (1 pˆ) n

• √ (^) pˆpˆ(1^ − −^ p pˆ) n

≈ N (0, 1)

Confidence interval for p Assumptions:

• The data are an SRS
• Population size ≥ 10 × sample size
• n pˆ, n(1 − pˆ) ≥ 10

An approximate level C C.I. for p is (p.490)

p ˆ ± z

√ (^) pˆ(1 − pˆ) n ,

where z is the upper (1 − C)/ 2 critical value of Z. Or,

p ˆ ± margin of error,

where margin of error= zσ ˆpˆ.

Example 7.1 Do you approve or disapprove of the way Bill Clinton is handling his job as president? A Gallup poll conducted November 20-22, 1998 found that 66% of 1,015 adults interviewed approved Clinton’s job performance. The Gallup poll claims that “For results based on the total sample of adults nationwide, one can say with 95% confidence that the margin of sampling error is no greater than +/ − 3 percentage points.”. Explain.

Choosing the sample size The margin of error is

m = z

√

p ˆ(1 − pˆ) n Sample size for desired margin of error: The level C confidence interval for p will have a margin of error approximately equal to a specified value m when the sample size is

n =

( (^) z m

) 2

p (1 − p )

where p is a guessed value for the sample proportion.

• Conservative sample size: n =

( (^) z m

) 2

(1/4).

The margin of error will be less than or equal to m.

Example 7.2 Find the sample size needed if the margin of error of the 95% confidence interval is

(a) m = 1% (b) m = 2% (c) m = 3% (d) m = 3%, p =. 3

Hypothesis tests for p Null hypothesis: H 0 : p = p 0 z-test statistic: z = √ pˆ^ −^ p^0 p 0 (1 − p 0 ) n Alternative P-value Ha : p > p 0 P (Z ≥ z) Ha : p < p 0 P (Z ≤ z) Ha : p 6 = p 0 2 P (Z ≥ |z|)

Assumptions:

• The data are an SRS
• Population size ≥ 10 × sample size
• n p 0 , n(1 − p 0 ) ≥ 10

Example 7.

Newt Gingrich, you don’t need to resign!

The GOP had 228 seats of 435 house representatives before the midterm election, and holds 223 seats after the election. Is the decline of GOP’s house representatives statistically sig- nificant?

Example 7.4 The English mathematician John Kerrich tossed a coin 10,000 times and obtained 5067 heads.

(a) Is this significant evidence at the 5% level that the coin is not balanced? (b) Find a 95% confidence interval for the probability that Kerrich’s coin comes up heads.

7.2 Comparing Two Proportions Suppose we have two independent samples.

Pop’n Sample Sample Pop’n prop’n size prop’n 1 p 1 n 1 pˆ 1 2 p 2 n 2 pˆ 2

The sampling distribution of pˆ 1 − pˆ 2 :

• The mean of pˆ 1 − pˆ 2 is p 1 − p 2
• The standard deviation of pˆ √ 1 − pˆ 2 is p 1 (1 − p 1 ) n 1 +^

p 2 (1 − p 2 ) n 2

• The standard error of pˆ 1 − pˆ 2 is SE = σˆpˆ 1 pˆ 2 =

√

p ˆ 1 (1 − pˆ 1 ) n 1 +

pˆ 2 (1 − pˆ 2 ) n 2

• pˆ^1 −^ pˆ^2 − SE^ (p 1 −^ p^2 ) ≈ N (0, 1)

Confidence intervals for p 1 − p 2 :

p ˆ 1 − pˆ 2 ± z SE

where SE = ˆσpˆ 1 pˆ 2 =

√

p ˆ 1 (1 − pˆ 1 ) n 1 +

pˆ 2 (1 − pˆ 2 ) n 2

Example 7.5 A study is made to determine if a cold climate results in more students being absent from school during a semester than for a warmer climate. Two groups of students are selected at random, one group from Vermont and the other group from Georgia. Of the 300 students from Vermont, 64 were absent at least 1 day during the semester, and of the 400 students from Georgia, 51 were absent 1 or more days. Find a 90% confidence interval for the difference between the fractions of students who are absent in the two states.

Recall: √ p^ ˆ^1 −^ pˆ^2 −^ (p^1 −^ p^2 ) p 1 (1 − p 1 ) n 1 +^

p 2 (1 − p 2 ) n 2

≈ N (0, 1)

Significance tests for p 1 − p 2 Null hypothesis: H 0 : p 1 = p 2 z statistic: z = √ pˆ^1 ^ pˆ^2 p ˆ(1 − pˆ)

( 1

n 1 +^

n 2

) ,

where pˆ is the pooled sample proportion

p ˆ = n^1 n^ pˆ^11 ++^ nn^22 pˆ^2 = count of successes in both samples n 1 + n 2

Alternative P-value Ha : p 1 > p 2 P (Z ≥ z) Ha : p 1 < p 2 P (Z ≤ z) Ha : p 1 6 = p 2 2 P (Z ≥ |z|)

Example 7.6 Does taking aspirin regularly help prevent heart attacks? A double-blind ran- domized comparative experiment assigned 11,037 male doctors to take aspirin group and another 11,034 to take a placebo. After 5 years, 104 of the aspirin group and 189 of the control group had died of heart attacks. Is this difference large enough to convince us that aspirin works?