Question

In a previous​ year, 54​% of females aged 15 and older lived alone. A sociologist tests whether this percentage is different today by conducting a random sample of 750 females aged 15 and older and finds that 398 are living alone. Is there sufficient evidence at the alphaequals0.01 level of significance to conclude the proportion has​ changed?

Answer 1

Solution:

Given:

p =0.54

n = 750

x = Number of females aged 15 and older living alone = 398

Level of significance = 0.01

We have to test if there sufficient evidence at the 0.01 level of significance to conclude the proportion has​ changed.

Step 1) State H0 and H1:

Vs

( two tailed test)

Step 2) Test statistic:

where

Thus

Step 3) Critical values:

Level of significance = 0.01

Since this is two tailed test , find

Look in z table for Area = 0.0050 and find corresponding z value.

Area 0.0050 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 )

Step 4) Decision Rule:
Reject null hypothesis ,if z  test statistic value < z critical value= -2.575 or z  test statistic value > z critical value= 2.575 , otherwise we fail to reject H0.

Since z  test statistic value = -0.51 is neither < -2.575 , nor greater than 2.575, we fail to reject H0.

Step 5) Conclusion:

At 0.01 level of significance , there is not  sufficient evidence to conclude that the proportion has​ changed