1
1 A manufacturer claims that a particular automobile model will
get 50 miles per gallon on the highway. The researchers at a
consumer-oriented magazine believe that this claim is high and
plan a test with a simple random sample of 30 cars. The
standard deviation between individual cars is calculated =2.3
miles per gallon, what should the researchers conclude if the
sample mean is 49 miles per gallon?
1.1 What is the outcome variable?
Quantity of miles attained by the car on the highway per gallon of gas consumed
1.2 What is the level of measurement for outcome variable?
Interval/ratio scale
1.3 Is the population parameter stated?
Yes. It is the mean miles per gallon on the highway attained by a particular automobile
model, which is equal to 50 miles/gallon of gas consumed.
1.4 State the statistical hypotheses?
Ho: μ = 50
Ha: μ ≠ 50
1.5 What test statistic would you use to test the null hypothesis?
Why?
t-distribution. The question is about a single mean and the population variance is
unknown
1.6 What is the best alpha level of significance for this question?
The usual alpha level of 0.05 would be sufficient for this analysis, since no serious
consequences are expected for failing to detect a difference between the sample and
company claim.
1.7 What is the formula for calculating the standard error?
√
√
SE=0.420
n=30
S=2.3 miles/hour
Statistic =
= 49 miles / hour
Hypothesized parameter = μ = 50 miles / hour
2
Decision rule =
t
1-α/2, df=n-1
t
0.975, df=30-1=29
2.05
Test statistic = - μ)/SE = 2.38095
|Test statistic| ≥ |Decision rule|
Reject the H
o
in favor of H
A
1.8 Select the most appropriate conclusion
A) There is no sufficient evidence to reject the manufacturer’s claim; 49 miles per
gallon is too close to the claimed 50 miles per gallon.
B) The manufacturer’s claim should not be rejected because the P-value is less than
0.001.
C) The manufacturer’s claim should be rejected because the sample mean is less than
the claimed mean.
(D) The P-value of >0.1 is sufficient evidence to reject the manufacturer’s claim.
(E) The P-value of <0.05 is sufficient evidence to prove that the manufacturer’s claim
is false.
3
2 The death rate from a particular form of cancer is 23% during the
first year. When treated with an experimental drug, only 15 out of
84 patients die during the initial year. Is this strong evidence to
claim that the new medication reduces the mortality rate?
2.1 What is the outcome variable?
Occurrence of death during the first year after cancer (frequency or count data)
2.2 What is the level of measurement for outcome variable?
Nominal scale (occurrence of death during the first year after cancer, yes or no)
2.3 Is the population parameter stated?
Yes. (P) It is death rate from a particular form of cancer is 23% during the first year.
2.4 State the statistical hypotheses?
H
o
: P = 0.23
H
A
: P ≠ 0.23
2.5 What test statistic would you use to test the null hypothesis?
Why?
Z-distribution. The question is about a single proportion
2.6 What is the best alpha level of significance for this question?
The usual alpha level of 0.05 would be sufficient for this analysis, since no serious
consequences are expected for failing to detect a difference.
2.7 What is the formula for calculating the standard error?
√
n=84
Statistic = P = (15/84) = 0.179
Hypothesized parameter = P = 0.23
SE= 0.042
2.8 Select the most appropriate conclusion
Decision rule =
Z
1-α/2
Z
0.975
=1.96
Test statistic = (P (1- P ))/SE= (0.179-0.23)/0.042=-1.214
|Test statistic| < |Decision rule|
Accept the H
o
4
2.9 Does this sample provide enough evidence to claim that the
new medication reduces the mortality rate?
(A) Yes, because the P-value is <0.05.
(B) Yes, because the P-value is <0.001.
(C) No, because the P-value is <0.05.
(D) No, because the P-value is above 0.05.
5
3 A historian believes that the average height of soldiers in World
War II was greater than that of soldiers in World War I. She
examines a random sample of records of 100 men in World War I
and 200 men in World War II and calculated standard deviations
of 2.5 and 2.3 inches in World War I and World War II,
respectively. If the average height from the sample of World War
II soldiers is 1 inch greater than from the sample of World War I
soldiers, what conclusion is justified from a two-sample
hypothesis test
3.1 What is the outcome variable?
Body height (inches)
3.2 What is the level of measurement for outcome variable?
Interval/ratio scale
3.3 What is the grouping variable?
World War-II Vs World War-II
3.4 What is the level of measurement for the grouping variable?
Nominal scale
3.5 Is the population parameter stated?
No population parameter is stated in the question.
3.6 State the statistical hypotheses?
H
o
: μ
1
- μ
2
= 0
H
A
: μ
1
- μ
2
≠
0
3.7 What test statistic would you use to test the null hypothesis?
Why?
t-distribution. The question is about the difference between 2 means and the
population variances are unknown
3.8 What is the best alpha level of significance for this question?
The usual alpha level of 0.05 would be sufficient for this analysis, since no serious
consequences are expected for failing to detect a difference between the sample and
company claim.
3.9 What is the formula for calculating the standard error?
We need on further assumption about the unknown population variances
If we are willing to assume equal population variances (since they are unknown)
6
√
√
(
)
n
1
=100
S
1
=2.5 inch
n
2
=200
S
2
=2.3 inch
(
2
-
1
)= 1 inch
S
2
p
=
5.609
SE
equal V
= 0.290
Otherwise we assume unequal population variances (since they are unknown)
√
n
1
=100
S
1
=2.5 inch
n
2
=200
S
2
=2.3 inch
statistic = (
2
-
1
) = 1 inch
Hypothesized parameter (
μ
1
- μ
2
)= 0
SE
unequal V
=
0.298
3.10 What is your conclusion?
Decision rule (tabulate t)
t
1-α/2, df=n1+n2-2
t
0.975, df=100+200-2=298
1.97
Test statistic=(1-0)/SE
Test statistic under the assumption of equal variances (
Test statistics
equal V=
3.448)
Test statistic under the assumption of equal variances (
Test statistics
unequal V=
3.356)
|Test statistic| ≥ |Decision rule| under both assumptions)
Reject the H
o
in favor of H
A
(A) The observed difference in average height is significant.
(B) The observed difference in average height is not significant.
(C) A conclusion is not possible without knowing the mean height in each sample.
(D) A conclusion is not possible without knowing both the sample means and the two original
population sizes.
7
4 A psychiatric social worker believes that in both City-A and City-
B, the proportion of adolescents suffering from PTSD (Post-
traumatic stress disorders) is 20%. In a sample of 150
adolescents from City-A, 15 had PTSD, while in another sample
of 100 from City-B, the number was 18. Does the current data
provide enough evidence to reject the claim that the prevalence
(relative frequency) of adolescents with PTSD is not different in
the two cities.
4.1 What is the outcome variable?
Relative frequency of PTSD among adolescents (Frequency or counts data)
4.2 What is the level of measurement for outcome variable?
Nominal scale
4.3 What is the grouping variable?
City of residence (City-A Vs City-B)
4.4 What is the level of measurement for the grouping variable?
Nominal scale (dichotomous variable)
4.5 Is the population parameter stated?
Although the population parameter is stated in the question (prevalence of PTSD of
20% in each city). We just need to know that the hypothesized parameter, which is the
difference between 2 population proportion (P
1
-P
2
) is equal to zero.
4.6 State the statistical hypotheses?
H
o
:
P
1
-P
2
= 0
H
A
:
P
1
-P
2
≠
0
4.7 What test statistic would you use to test the null hypothesis?
Why?
Z-distribution. The question is about the difference between 2 proportions
4.8 What is the best alpha level of significance for this question?
The usual alpha level of 0.05 would be sufficient for this analysis, since no serious
consequences are expected for failing to detect a difference between the populations.
4.9 What is the formula for calculating the standard error?
√
n
1
=150
P
1
=15/150=0.10
n
2
=100
P
2
=18/100=0.18
SE=0.046
8
4.10 What is your conclusion?
Decision rule (tabulate t)
Z
1-α/2=1-(0.05/2)=0.975
=1.96
Statistic (
P
1
-
P
2
)= (0.1-0.18)=-0.08
Hypothesized parameter (
P
1
- P
2
)= 0
Test statistic=(-0.08-0)/SE= -1.739
|Test statistic| < |Decision rule|
Accept the H
o
The difference observed in relative frequency of PTSD between the 2 cities samples, is
too small to reflect a real difference in the two populations. i.e. the difference
observed in proportion between the 2 samples is not significant statistically
(P>0.05) and does not provide enough evidence to reject the
claim that the
prevalence (relative frequency) of adolescents with PTSD is not different in the two
cities.
9
5 A sample of 100 patients on clozapine had their blood pressure
recorded soon after one dose of clozapine was given. The mean
systolic BP was 150 mmHg. If the standard deviation of this
observation was 10 mmHg, calculate the standard error of the
mean blood pressure recorded in this sample.
a) 1.2
b) 1.5
c) 15
d) 1
√
√