Question

In a random sample of 1022 U.S. adults, 480 think that the U.S. government is doing too little to protect the environment. In another random sample of 1008 U.S. adults taken 10 years ago, 514 think that the U.S. government is doing too little to protect the environment. At α = 0.10, can you reject the claim that the proportion of U.S. adults who think that the U.S. government is doing too little to protect the environment is less than it was 10 years ago?

Answer 1

Solution:

Here, we have to use two sample z test for the population proportions.

Null hypothesis: H₀: The proportion of U.S. adults who think that the U.S. government is doing too little to protect the environment is same as it was 10 years ago.

Alternative hypothesis: H_a: The proportion of U.S. adults who think that the U.S. government is doing too little to protect the environment is less than it was 10 years ago.

H₀: p₁ = p₂ versus H₀: p₁ < p₂

This is a lower tailed test.

We are given

Level of significance = α = 0.10

The test statistic formula is given as below:

Z = (P1 – P2) / sqrt(P*(1 – P)*((1/N1) + (1/N2)))

Where,

X1 = 480

X2 = 514

N1 = 1022

N2 = 1008

P = (X1+X2)/(N1+N2) = (480 + 514) / (1022 + 1008) = 0.4897

P1 = X1/N1 = 480/1022 = 0.469667319

P2 = X2/N2 = 514/1008 = 0.509920635

Z = (P1 – P2) / sqrt(P*(1 – P)*((1/N1) + (1/N2)))

Z = (0.469667319 – 0.509920635) / sqrt(0.4897*(1 – 0.4897)*((1/1022) + (1/1008)))

Z = -0.04025332 / sqrt(0.4897*(1 – 0.4897)*((1/1022) + (1/1008)))

Z = -1.8140

P-value = 0.0348

(by using z-table)

P-value < α = 0.10

So, we reject the null hypothesis

There is sufficient evidence to conclude that the proportion of U.S. adults who think that the U.S. government is doing too little to protect the environment is less than it was 10 years ago.