Question

In a previous? year, 59?% of females aged 15 years of age and older lived alone. A sociologist tests whether this percentage is different today by conducting a random sample of 800 females aged 15 years of age and older and finds that 448 are living alone. Is there sufficient evidence at the alpha equals0.01 level of significance to conclude the proportion has? changed?

Answer 1

Solution:

   We are given that : In a previous year, 59?% of females aged 15 years of age and older lived alone.

That is : p = 0.59

Sample size = n = 800

Number of females aged 15 years of age and older are living alone = x = 448

Thus sample proportion of females aged 15 years of age and older are living alone is :

Level of significance = 0.01

Claim : The percentage of females aged 15 years of age and older lived alone is different or changed today.

Thus we use following steps:

Step 1) State H0 and H1:

Step 2) Find test statistic value:

Formula:

Step 3) Find z critical value:

Level of significance

This is two tailed test, thus find Area =

Look in z table for Area = 0.0050 or its closest area and find z value:

Area 0.0050 is in between 0.0049 and 0.0051, Thus we look for both area and find both z values:

z value for area 0.0051 is -2.57 and z value for area 0.0049 is -2.58

Thus their average is = ( -2.57 + - 2.58 ) / 2 = -2.575

Thus z critical value = -2.575

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

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

Since z test statistic value = -1.73 is neither < -2.575 , nor   > 2.575 , we fail to reject H0 at 0.01 level of significance.

Step 5) Conclusion: Since we failed to reject H0 , there is not sufficient evidence to support the claim that : The percentage of females aged 15 years of age and older lived alone is different or changed today.