Question

In New York City, a study was conducted to evaluate whether any information that is available at the time of birth can be used to identify children with special educational needs. In a random sample of 45 third-graders enrolled in the special education program of the public school system, 4 have mothers who have had more than 12 years of schooling.

A) Construct a 90% CI for the population proportion of children with special educational needs whose mothers have had more than 12 years of schooling.

B) In 1980, 22% of all third-graders enrolled in the NYC public school system had mothers who had more than 12 years of schooling. Suppose you wish to know whether this proportion is the same for children in the special education program. What are the null and alternative hypotheses of the appropriate test?

C) Conduct the test at the 0.05 level of significance

Should use exact confidence interval and exact test in parts a and c. Please include your R commands for these parts.

Answer 1

Solution-:

> # Let, X- mothers who had more than 12 years of scooling
> #(a)
> n=45;x=4
> prop.test(x,n,conf.level=0.90)

1-sample proportions test with continuity correction

data: x out of n, null probability 0.5
X-squared = 28.8, df = 1, p-value = 8.025e-08
alternative hypothesis: true p is not equal to 0.5
90 percent confidence interval:
0.03359594 0.19764422
sample estimates:
p
0.08888889

> #(b)
> po=0.22
> # Hypothesis: Ho: p=po=0.22 Vs H1:p not equal to 0.22
> #(c)here, alpha=0.05
> prop.test(x,n,p=0.22,conf.level=0.95)

1-sample proportions test with continuity correction

data: x out of n, null probability 0.22
X-squared = 3.7762, df = 1, p-value = 0.05199
alternative hypothesis: true p is not equal to 0.22
95 percent confidence interval:
0.02886221 0.22128500
sample estimates:
p
0.08888889

R-code for this example:

# Let, X- mothers who had more than 12 years of scooling
#(a)
n=45;x=4
prop.test(x,n,conf.level=0.90)
#(b)
po=0.22
# Hypothesis: Ho: p=po=0.22 Vs H1:p not equal to 0.22
#(c)here, alpha=0.05
prop.test(x,n,p=0.22,conf.level=0.95)