Question

The Environmental Protection Agency wanted to compare the proportion of plants in violation of air quality standards for two different industries: steel and utility. Two independent samples of plants were selected and monitored. The following data was recorded: Steel: n1=150, number of violations=x1=12 Utility: n2=160, number of violations=x2=12

(a) Define the population parameters and set up appropriate null and alternate hypothesis.

(b) Compute the value of the test statistic.

(c) Set up the appropriate rejection region for alpha = 0.01

(d) What is the conclusion of the test?

(e) State the assumptions used for conducting the test as above. Were the assumptions satisfied?

Please explain clearly and very appreciate.

Answer 1

Solution:

    We are given that:

Steel: n₁=150, number of violations=x₁=12

Utility: n₂=160, number of violations=x₂=12

Part a) Define the population parameters and set up appropriate null and alternate hypothesis.

Population Parameters are:

Population proportion of plants in violation of air quality standards for Steel = p₁

Population proportion of plants in violation of air quality standards for Utility = p₂

Null and alternate hypothesis are:

Vs

( This is two tailed test , since it is non directional)

Part b) Compute the value of the test statistic.

Test statistic formula:

where

and

Thus

Part c) Set up the appropriate rejection region for alpha = 0.01

Since this is two tailed test , find

Look in z table for Area = 0.005 and find corresponding z value.

Area 0.005 is in between 0.0049 and 0.0051, and both the area are at same distance from 0.005

thus we look for both area and find both z values.

Area 0.0049 corresponds to -2.5 and 0.08 , thus z= -2.58

Area 0.0051 corresponds to -2.5 and 0.07 , thus z= -2.57

Thus average of both z values is = ( -2.57 + -2.58 ) / 2 = -2.575

Thus critical z value is = -2.575

Since this is two tailed test , there are two z critical values = ( -2.575 , 2.575 )

Thus rejection region is : Reject H0 , if z test statistic value < -2.575   or z test statistic value > 2.575, otherwise we fail to reject H0

Part d) What is the conclusion of the test?

Since z test statistic value = 0.16 is neither < -2.575 , nor > 2.575 , we failed to reject H0.

Thus we conclude that: there is no significant difference between the proportion of plants in violation of air quality standards for two different industries: steel and utility.

Part e) State the assumptions used for conducting the test as above. Were the assumptions satisfied?

Assumptions: Sampling distribution of sample proportions of two population has an approximate Normal distribution if   .

>10

and > 10

both the assumptions are satisfied.