Question

Ten sampled students of 18-21 years of age received special training. They are given an IQ test that is N (100, 102) in the general population. Let μ be the mean IQ of these students who received special training. The observed IQ scores: 121, 98, 95, 94,102, 106, 112, 120, 108, 109. Test if the special training improves the IQ score using significance level α = 0.05.

a.What is the rejection region?

b.Calculate the p-value and state your conclusion.

c.What if the variance is unknown?

Use R studio to solve this problem. What codes you have to put in ?

Answer 1

#Loading library containing z.test

> library(TeachingDemos)
> IQ=c(121,98,95,94,102,106,112,120,108,109)
(a)> qnorm(0.05,lower.tail=F)
[1] 1.644854
REJECTION REGION : Z > Z_0.05 = 1.645
(b)> z.test(IQ,mu=100,stdev=sqrt(102),) #To test H0:mu=100 ag. H1:mu>100, where population SD is known

One Sample z-test

data: IQ
z = 2.0352, n = 10.0000, Std. Dev. = 10.1000, Std. Dev. of the sample
mean = 3.1937, p-value = 0.02091
alternative hypothesis: true mean is greater than 100
95 percent confidence interval:
101.2468 Inf
sample estimates:
mean of IQ
106.5
Since p-value < 0.05, we reject H0 and conclude that the IQ of the students have improved significantly after the special training.

(c)> t.test(IQ,mu=100,) # To test H0:mu=100 ag. H1:mu>100, where population SD is unknown

One Sample t-test

data: IQ
t = 2.1633, df = 9, p-value = 0.02937
alternative hypothesis: true mean is greater than 100
95 percent confidence interval:
100.9922 Inf
sample estimates:
mean of x
106.5

Since p-value < 0.05, we reject H0 and conclude that the IQ of the students have improved significantly after the special training.