21. Research has shown that IO scores have been increas-ng for years (Flynn, 1984, 1999). The phenomenonis called the Flynn effect and the data indicate that thencrease appears to average about 7 points per decade.To examine this effect, a researcher obtains an IQ testwith instructions for scoring from 10 years ago significant increase in IQ over the past 10 years.stingand plans to administers the test to a sample of25 of today's high school students. Ten yearsago, the scores on this IQ test produced a standard-ized distribution with a mean of un =100 and astandard deviation o =15. If there actually has beena 7-point increase in the average IQ during the past10 years, then find the power of the hypothesis testfor each of the following.a. The researcher uses a two-tailed hypothesis testwith a = .05 to determine if the data indicate asignificant change in IQ over the past 10 years.b. The researcher uses a one-tailed hypothesiswith a =test.05 to determine if the data indicate a22, Briefluinfluences

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21. Research has shown that IO scores have been increas-
ng for years (Flynn, 1984, 1999). The phenomenon
is called the Flynn effect and the data indicate that the
ncrease appears to average about 7 points per decade.
To examine this effect, a researcher obtains an IQ test
with instructions for scoring from 10 years ago

significant increase in IQ over the past 10 years.
sting
and plans to administers the test to a sample of
25 of today's high school students. Ten years
ago, the scores on this IQ test produced a standard-
ized distribution with a mean of u
n =
100 and a
standard deviation o =
15. If there actually has been
a 7-point increase in the average IQ during the past
10 years, then find the power of the hypothesis test
for each of the following.
a. The researcher uses a two-tailed hypothesis test
with a = .05 to determine if the data indicate a
significant change in IQ over the past 10 years.
b. The researcher uses a one-tailed hypothesis
with a =
test
.05 to determine if the data indicate a
22, Brieflu
influences

Expert Solution

Step 1

Given
a)
sample size $n = 25$
Ten years ago
The mean IQ score $μ = 100$
The standard deviation $σ = 15$
level of significance $α = 0.05$

Null hypothesis : $H_{0} : μ = 100$
Alternate hypothesis : $H_{a} : μ \neq 100$

Type II error occurs when a false null hypothesis is failed to reject.
The power of a hypothesis test is not committing type II error $P o w e r = 1 - β$

The Z critical value for two tailed test with 5% level of significance is $Z = \pm 1.96$

As it is given that IQ score increasing by 7 points in a decade, let us consider right side critical value 1.96.
We reject the null hypothesis if test statistic $Z > 1.96$
$Z > 1.96 \to \frac{X - μ}{\frac{σ}{\sqrt{n}}} > 1.96 X > 1.96 * \frac{15}{\sqrt{25}} + 100 X > 105.88$

So we reject the null hypothesis if IQ score is greater than 105.88

The power of test is probability of rejecting null hypothesis given that alternate hypothesis is correct.
If alternate hypothesis is correct then the IQ score increase by 7 points, so mean IQ score is 107.

Power of test is calculated as shown below
$P (X > 105.88 | μ = 107) = P (\frac{X - μ}{\frac{σ}{\sqrt{n}}} > \frac{105.88 - μ}{\frac{σ}{\sqrt{n}}} | μ = 107) = P (Z > \frac{105.88 - 107}{\frac{15}{\sqrt{25}}}) = P (Z > - 0.37)$

With respect to Z table we get the following
$P (Z < - 0.37) = 0.3557 P (Z > - 0.37) = 1 - P (Z < - 0.37) = 1 - 0.3557 = 0.6443$

Thus the power of the test is 64.43%

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