Depression engulfs millions of Americans every day. A new federal study reported

Question

Depression engulfs millions of Americans every day. A new federal study reported that 10.9% of adults aged 18-24 identified with some level of depression versus 6.8% of adults aged 65 or older. Suppose 250 young adults (18-24 years old) and 200 older adults (65 years or older) responded to the study.

Let p1 represent the population proportion of young adults (18-24) with depression and p2 the population proportion of older adults (65+) with depression. Let young adults and older adults represent population 1 and population 2, respectively. a. Develop the appropriate null and alternative hypotheses to test whether the proportion of young adults suffering from depression is greater than the proportion of older adults suffering from depression. O Ho: P1 -P2 20; HA: P1-P20 0H0: p1-p2:0; HA: p1-p2#0 b. Calculate the value of the test statistic and the p-value. (Round intermediate calculations to 4 decimal places, "test statistic" value to 2 decimal places and "p-value" to 4 decimal places.) Test statistic p-value C. At the 5% significance level, what is the conclusion? Do the sample data suggest that young adults suffer more from depression than older adults? O Yes, since we reject Ho No, since we reject Ho No, since we do not reject Ho O Yes, since we do not reject Ho

Explanation / Answer

Null, Ho: p1 - p2 <= 0
Alternate, H1: p1 - p2 > 0
Test Statistic
Sample 1 : X1 =27.25, n1 =250, P1= X1/n1=0.109
Sample 2 : X2 =13.6, n2 =200, P2= X2/n2=0.068
Finding a P^ value For Proportion P^=(X1 + X2 ) / (n1+n2)
P^=0.091
Q^ Value For Proportion= 1-P^=0.909
we use Test Statistic (Z) = (P1-P2)/(P^Q^(1/n1+1/n2))
Zo =(0.109-0.068)/Sqrt((0.091*0.909(1/250+1/200))
Zo =1.504
| Zo | =1.504
Critical Value
The Value of |Z | at LOS 0.05% is 1.645
We got |Zo| =1.504 & | Z | =1.645
Make Decision
Hence Value of |Zo | < | Z | and Here we Do not Reject Ho
P-Value: Right Tail -Ha : ( P > 1.5043 ) = 0.06625
Hence Value of P0.05 < 0.06625,Here We Do not Reject Ho

[ANSWERS]
a. Ho: p1 - p2 <= 0, H1: p1 - p2 > 0
b. Zo =1.504, P = 0.06625
c. Do not Reject Ho

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