Question

Some years, the Gallup Poll asks respondents how much confidence they have in various American institutions. You may assume the results are based on a simple random sample of 1000 persons each year; the samples are independent from year to year.

a) In 2005, only 41% of the respondents had “a great deal or quite a lot” of confidence in the Supreme Court, compared to 50% in 2000. Is the difference real?

b) In 2005, only 22% of the respondents had “a great deal or quite a lot” of confidence in Congress, whereas 24% of the respondents had “a great deal or quite a lot” of confidence in labor. Is the difference between 24% and 25% real?

Answer 1

Solution

The test procedure for both parts is the same, which is explained in brief below.

Back-up Theory

To test if observed difference in sample proportions reflects real difference, i.e., difference in population proportions

Hypotheses:

Null H₀ : p₁ = p₂   Vs H_A : p₁≠ p₂

Test Statistic:

Z = (p₁cap – p₂cap)/√[pcap(1 - pcap){(1/n₁) + (1/n₂)} where p₁cap and p₂cap are sample proportions, n₁, n₂ are sample sizes and pcap = {(n₁ x p₁cap) + (n₂ x p₂cap)}/(n₁ + n₂).

Distribution, Significance Level, α, Critical Value and p-value:

Under H₀, distribution of Z can be approximated by Standard Normal Distribution

So, given a level of significance of α%, Critical Value = upper (α/2)% of N(0, 1), and

p-value = P(Z > | Zcal |)

Using Excel Functions Statistical NORMSINV and NORMSDIS, Critical Value and p-value can be found.

Decision:

If | Zcal | > Zcrit, or equivalently, p-value < α, H₀ is rejected.

Now, to work out the solution,

Given n₁ = n₂= 1000 for both parts

Part (a)

Calculations:

n1 =	1000
n2 =	1000
x =	410
y =	500
p1cap =	0.41
p2cap =	0.5
pcap =	0.455
Zcal =	-4.041322996
α =	0.05
Zcrit	1.959963985
p-value	0.00005315

Decision:

Since | Zcal | > Zcrit, or equivalently, p-value < α, H₀ is rejected.

Conclusion :

There is enough evidence to conclude that the difference in proportion is real. Answer 1

Part (b)

n1 =	1000
n2 =	1000
x =	220
y =	240
p1cap =	0.22
p2cap =	0.24
pcap =	0.23
Zcal =	-1.062687855
α =	0.05
Zcrit	1.959963985
p-value	0.2879235303

Decision:

Since | Zcal | < Zcrit, or equivalently, p-value > α, H₀ is accepted.

Conclusion :

There is not enough evidence to conclude that the difference in proportion is real, i.e., the observed difference is not real. Answer 2

DONE