Question

According to the U.S. Census Bureau, 11% of children in the United States lived with at least one grandparent in 2009 (USA TODAY, June 30, 2011). Suppose that in a recent sample of 1570 children, 224 were found to be living with at least one grandparent. At a 5% significance level, can you conclude that the proportion of all children in the United States who currently live with at least one grandparent is higher than 11%? Use both the p-value and the critical-value approaches. Round your answers for the observed value of z and the critival value of z to two decimal places, and the p-value to four decimal places.

Answer 1

SOLUTION :

From given data,

According to the U.S. Census Bureau, 11% of children in the United States lived with at least one grandparent in 2009 (USA TODAY, June 30, 2011). Suppose that in a recent sample of 1570 children, 224 were found to be living with at least one grandparent. At a 5% significance level, can you conclude that the proportion of all children in the United States who currently live with at least one grandparent is higher than 11%

=

Let

= The Sample proportion of children in the US who currently live with at least 1 grandparent

= 224/1570

= 0.14267

Let

= The Population proportion of children in the US who currently live with at least 1 grandparent

= 0.11

1 - = 1 - 0.11 =0.89

The Hypothesis:

H₀: = 0.11 : The proportion of all children in the US living with at least 1 grandparent is equal to 0.11.

H_a: > 0.11 : The proportion of all children in the US living with at least 1 grandparent is greater than 0.11.

This is a Right tailed Test.

The Test Statistic:

Z = ( - ) /

Z = (0.14267 - 0.11 ) /

Z = 0.03267 / 0.00789662509

Z = 4.13 ( two decimal places )

The p - Value:

The p value (Right tail) for Z = 4.13, is;

p value = 0.0000 ( four decimal places )

The Critical Value:

5% significance level

5/100 = 0.05

= 0.05

/2 = 0.05 /2 = 0.025

The critical value (Right tail)

Z _/2 = +1.96

The Decision Rule:

If Z _/2 > 1.96 Then Reject H₀.

Also If the P value is < 0.025( ), Then Reject H₀

The Decision:

Since Z observed (4.13) is > 1.96, We Reject H₀.

Also since P value (0.00) is < 0.025, We Reject H₀.

The Conclusion:

There is sufficient evidence at the = 0.025 level to conclude that the proportion of all children in the US who currently live with at least 1 grandparent is higher than 0.11.