Question

. In s study done by the Ohio Department of Job and Family Services, it was determined 38 (out of 62) poor children who attended preschool needed social services later in life compared to 49 (out of 61) poor children who did not attend preschool. A. Does this study provide significant evidence that preschool reduces the need for social services later in life? Do a complete and appropriate hypothesis test using α = .01. B. Construct a 98% confidence interval estimate of the difference in the proportion of poor children who attended preschool and poor children who did not attend preschool who needed social services later in life. Interpret the practical meaning of the resulting interval estimate, in the context of the problem, in plain English.

Answer 1

Given that,
sample one, x1 =38, n1 =62, p1= x1/n1=0.613
sample two, x2 =49, n2 =61, p2= x2/n2=0.803
finding a p^ value for proportion p^=(x1 + x2 ) / (n1+n2)
p^=0.707
q^ Value For Proportion= 1-p^=0.293
null, Ho: p1 = p2
alternate, H1: p1 < p2
level of significance, α = 0.01
from standard normal table,left tailed z α/2 =2.326
since our test is left-tailed
reject Ho, if zo < -2.326
we use test statistic (z) = (p1-p2)/√(p^q^(1/n1+1/n2))
zo =(0.613-0.803)/sqrt((0.707*0.293(1/62+1/61))
zo =-2.32
| zo | =2.32
critical value
the value of |z α| at los 0.01% is 2.326
we got |zo| =2.32 & | z α | =2.326
make decision
hence value of |zo | < | z α | and here we do not reject Ho
p-value: left tail - Ha : ( p < -2.3201 ) = 0.01017
hence value of p0.01 < 0.01017,here we do not reject Ho
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null, Ho: p1 = p2
alternate, H1: p1 < p2
test statistic: -2.32
critical value: -2.326
decision: do not reject Ho
p-value: 0.01017
no evidence that preschool reduces the need for social services later in life
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CI = (p1-p2) ± sqrt( p1 * (1-p1)/n1 + p2 * (1-p2)/n2 )
where,
p1, p2 = proportion of both sample observation
n1,n2 = size of both group
a = 1 - (confidence Level/100)
Za/2 = Z-table value
CI = confidence interval
CI = [ (0.6129-0.8033) ± 2.33 * 0.0801]
= [ -0.377 , -0.0037 ]
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interpretations:
1) we are 98% sure that the interval [ -0.377 , -0.0037] contains the difference between
true population proportion P1-P2
2) if a large number of samples are collected, and a confidence interval is created
for each sample, 98% of these intervals will contains the difference between
true population mean P1-P2
we conclude that difference in the proportion of poor children who attended preschool and poor children who did not attend preschool who needed social services later in life