In a random sample of 1066 U. S. adults, 580 think that the U. S. government

is doing too little to protect the environment. In another random sample of

1038 U. S. adults taken 10 years ago, 514 think that the U. S. government is

doing too little to protect the environment. At a = 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 has not changed?

Resource: Two Samples, Proportions

Step-by-step explanation:

This is a test of 2 population proportions. Let 1 and 2 be the subscript for the adults who think that the U. S. government is doing too little to protect the environment presently and the adults who think that the U. S. government is doing too little to protect the environment 10 years ago. The population proportions would be p1 and p2 respectively.

p1 - p2 = difference in the proportion of adults who think that the U. S. government is doing too little to protect the environment presently and the adults who think that the U. S. government is doing too little to protect the environment 10 years ago.

The null hypothesis is

H0 : p1 = p2

p1 - p2 = 0

The alternative hypothesis is

Ha : p1 ≠ p2

p1 - p2 ≠ 0

it is a two tailed test

Sample proportion = x/n

Where

x represents number of success

n represents number of samples

For the present,

x1 = 580

n1 = 1066

p1 = 580/1066 = 0.54

For 10 years ago,

x2 = 514

n2 = 1038

P2 = 514/1038 = 0.5

The pooled proportion, pc is

pc = (x1 + x2) / (n1 + n2)

pc = (580 + 514) / (1066 + 1038) = 0.52

1 - pc = 1 - 0.52 = 0.48

z = (p1 - p2) / √pc (1 - pc) (1/n1 + 1/n2)

z = (0.54 - 0.5) / √ (0.52) (0.48) (1/1066 + 1/1038) = 0.04/0.02178551744

z = 1.84

Since it is a two tailed test, the curve is symmetrical. We will look at the area in both tails. Since it is showing in one tail only, we would double the area

From the normal distribution table, the area above the test z score in the right tail 1 - 0.967 = 0.033

We would double this area to include the area in the left tail of z = - 1.84 Thus

p = 0.033 * 2 = 0.066

Alpha = 0.1

Since alpha, 0.1 > p value 0.066, we would reject the null hypothesis.

Therefore, at a = 0.10, we can 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 has not changed.