Question

In a study examining the effect of caffeine on reaction time, Liguori and Robinson (2001) measured reaction time 30 minutes after participants consumed one 6-ounce cup of coffee and they used standardized driving simulation task for which the regular population averages μ = 415 msec. The distribution of reaction times is approximately normal with σ = 40. Assume that the researcher obtained a sample mean of M = 400 msec for the n = 36 participants in the study. Can the researcher conclude that caffeine has a significant effect on reaction time? Use a two-tailed test (i.e., non-directional) with α = .05.

a. Given your statistical decision, what type of decision error could you have made and what is the probability of making that error?

b. Compute Cohen’s d to measure the size of the effect. Interpret what this effect size really means in this context (don’t just say “large effect” or “small effect”)

c. In the above question, if you changed the alpha level from .05 to .01, would that have affected your statistical decision? If so, explain your answer

d. If the alpha level is changed from .05 to .01, what happens to the boundaries for the critical region?

e. If the alpha level is changed from .05 to .01, what happens to the probability of a Type I error?

Answer 1

a.
Given that,
population mean(u)=415
standard deviation, σ =40
sample mean, x =400
number (n)=36
null, Ho: μ=415
alternate, H1: μ!=415
level of significance, α = 0.05
from standard normal table, two tailed z α/2 =1.96
since our test is two-tailed
reject Ho, if zo < -1.96 OR if zo > 1.96
we use test statistic (z) = x-u/(s.d/sqrt(n))
zo = 400-415/(40/sqrt(36)
zo = -2.25
| zo | = 2.25
critical value
the value of |z α| at los 5% is 1.96
we got |zo| =2.25 & | z α | = 1.96
make decision
hence value of | zo | > | z α| and here we reject Ho
p-value : two tailed ( double the one tail ) - ha : ( p != -2.25 ) = 0.024
hence value of p0.05 > 0.024, here we reject Ho
ANSWERS
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i.
null, Ho: μ=415
alternate, H1: μ!=415
test statistic: -2.25
critical value: -1.96 , 1.96
decision: reject Ho
p-value: 0.024
we have enough evidence to support the claim that caffeine has a significant effect on reaction time for which the regular population averages μ = 415 msec.
Type 1 error is possible because it reject the null hypothesis.
ii.
Given that,
Standard deviation, σ =40
Sample Mean, X =400
Null, H0: μ=415
Alternate, H1: μ!=415
Level of significance, α = 0.05
From Standard normal table, Z α/2 =1.96
Since our test is two-tailed
Reject Ho, if Zo < -1.96 OR if Zo > 1.96
Reject Ho if (x-415)/40/√(n) < -1.96 OR if (x-415)/40/√(n) > 1.96
Reject Ho if x < 415-78.4/√(n) OR if x > 415-78.4/√(n)
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Suppose the size of the sample is n = 36 then the critical region
becomes,
Reject Ho if x < 415-78.4/√(36) OR if x > 415+78.4/√(36)
Reject Ho if x < 401.9333 OR if x > 428.0667
Suppose the true mean is 400
Probability of Type I error,
P(Type I error) = P(Reject Ho | Ho is true )
= P(401.9333 < x OR x >428.0667 | μ1 = 400)
= P(401.9333-400/40/√(36) < x - μ / σ/√n OR x - μ / σ/√n >428.0667-400/40/√(36)
= P(0.29 < Z OR Z >4.21 )
= P( Z <0.29) + P( Z > 4.21)
= 0.6141 + 0 [ Using Z Table ]
= 0.6141
b.
cohen'd size = mean difference / standard deviation
cohen'd size = (415-400)/40 =0.375
medium effect.
c.
if level of significance changes from 0.05 to 0.01
Given that,
population mean(u)=415
standard deviation, σ =40
sample mean, x =400
number (n)=36
null, Ho: μ=415
alternate, H1: μ!=415
level of significance, α = 0.01
from standard normal table, two tailed z α/2 =2.576
since our test is two-tailed
reject Ho, if zo < -2.576 OR if zo > 2.576
we use test statistic (z) = x-u/(s.d/sqrt(n))
zo = 400-415/(40/sqrt(36)
zo = -2.25
| zo | = 2.25
critical value
the value of |z α| at los 1% is 2.576
we got |zo| =2.25 & | z α | = 2.576
make decision
hence value of |zo | < | z α | and here we do not reject Ho
p-value : two tailed ( double the one tail ) - ha : ( p != -2.25 ) = 0.024
hence value of p0.01 < 0.024, here we do not reject Ho
ANSWERS
---------------
null, Ho: μ=415
alternate, H1: μ!=415
test statistic: -2.25
d.
critical value: -2.576 , 2.576
decision: do not reject Ho
p-value: 0.024
we do not have enough evidence to support the claim that caffeine has a significant effect on reaction time for which the regular population averages μ = 415 msec.
e.
Type 1 error is possible because it fails to reject the null hypothesis sothat,
Type 2 error is possible.