To test whether the proportion of adults who are overweight

To test whether the proportion of adults who are overweight differs according to the level of education, the US Department of Agriculture selected two samples from the U.S. population. They found that of 500 adults (n1 = 500) whose highest degree was a graduate degree, 237 were overweight.

a. Estimate the proportion of adults with a graduate degree who were overweight.

A sample of adults with bachelor degrees was also collected. Of 750 adults with BS degrees (n2 = 750), 386 were overweight. Complete the following two-sample hypothesis test of proportions at the 0.10 level of significance:

H₀ : ?₂=?₁ H_a : ?₂??₁

In this two-sample hypothesis test, let the population proportion for adults with a bachelors be ? 2and the population proportion for adults with a graduate degree be ?1. Similarly, let p2 be the sample proportion for those with a bachelors and p1 be the sample proportion for those with a graduate degree.

b. What critical value(s) is (are) used for a standardized test of this hypothesis? Complete the diagram of this standardized test at the right.

c. Calculate the test statistic for this standardized test.?

Solution

Null Hypothesis, There Is No Significance between them Ho: p1 = p2
Alternate Hypothesis, There Is Significance between them H1: p1 != p2
Test Statistic
a)
Sample 1, proportion of adults with a graduate degree who were overweight
X1 =237, n1 =500, P1= X1/n1=0.474

Sample 2, BS degrees
X2 =386, n2 =750, P2= X2/n2=0.515

b)

Critical Value
The Value of |Z ?| at LOS 0.1% is 1.645

c)
Finding a P^ value For Proportion P^=(X1 + X2 ) / (n1+n2)
P^=0.498
Q^ Value For Proportion= 1-P^=0.502
we use Test Statistic (Z) =   (P1-P2)/?(P^Q^(1/n1+1/n2))
Zo =(0.474-0.515)/Sqrt((0.498*0.502(1/500+1/750))
Zo =-1.409
| Zo | =1.409

We got |Zo| =1.409 & | Z ? | =1.645
Make Decision
Hence Value of |Zo | < | Z ? | and Here we Do not Reject Ho
P-Value: Two Tailed ( double the one tail ) -Ha : ( P != -1.4087 ) = 0.1589
Hence Value of P0.1 < 0.1589,Here We Do not Reject Ho