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?
What is the outcome variable?
Quantity of miles attained by the car on the highway per gallon of gas consumed
What is the level of measurement for outcome variable?
Interval/ratio scale
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.
State the statistical hypotheses?
Ho: μ = 50
Ha: μ ≠ 50
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
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.
What is the formula for calculating the standard error?
SE=S2n=2S2n
SE=0.420
n=30
S=2.3 miles/hour
Statistic = x̅ = 49 miles / hour
Hypothesized parameter = μ = 50 miles / hour
Decision rule = t1-α/2, df=n-1 t0.975, df=30-1=29 2.05
Test Statistic= Statistic-ParameterSE
Test statistic = ( ͞x - μ)/SE = 2.38095
|Test statistic| ≥ |Decision rule|Reject the Ho in favor of HA
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.
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?
What is the outcome variable?
Occurrence of death during the first year after cancer (frequency or count data)
What is the level of measurement for outcome variable?
Nominal scale (occurrence of death during the first year after cancer, yes or no)
Is the population parameter stated?
Yes. (P) It is death rate from a particular form of cancer is 23% during the first year.
State the statistical hypotheses?
Ho: P = 0.23
HA: P ≠ 0.23
What test statistic would you use to test the null hypothesis? Why?
Z-distribution. The question is about a single proportion
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.
What is the formula for calculating the standard error?
SE=2P̅(1-P̅)n
n=84
Statistic = P̅ = (15/84) = 0.179
Hypothesized parameter = P = 0.23
SE= 0.042
Select the most appropriate conclusion
Decision rule = Z1-α/2 Z0.975=1.96
Test Statistic= Statistic-ParameterSE
Test statistic = (P̅(1- P̅))/SE= (0.179-0.23)/0.042=-1.214
|Test statistic| < |Decision rule|
Accept the Ho
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.
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
What is the outcome variable?
Body height (inches)
What is the level of measurement for outcome variable?
Interval/ratio scale
What is the grouping variable?
World War-II Vs World War-II
What is the level of measurement for the grouping variable?
Nominal scale
Is the population parameter stated?
No population parameter is stated in the question.
State the statistical hypotheses?
Ho: μ1 - μ2 = 0
HA: μ1 - μ2 ≠ 0
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
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.
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)
SE=Sp2n1+Sp2n2 or Sp x1n1+1n2
Sp2=n1-1S12+(n2-1S22)n1-1+n2-1
n1=100 S1=2.5 inch
n2=200 S2=2.3 inch
(x̅2 - x̅1)= 1 inch
S2p=5.609
SE equal V= 0.290
Otherwise we assume unequal population variances (since they are unknown)
SE=S12n1+S22n2n1=100 S1=2.5 inch
n2=200 S2=2.3 inch
statistic = (x̅2 - x̅1) = 1 inch
Hypothesized parameter (μ1 - μ2)= 0
SE unequal V= 0.298
What is your conclusion?
Decision rule (tabulate t) t1-α/2, df=n1+n2-2 t0.975, df=100+200-2=298 1.97
Test Statistic= Statistic-ParameterSE
Test statistic=(1-0)/SE
Test statistic under the assumption of equal variances (Test statisticsequal V=3.448)
Test statistic under the assumption of equal variances (Test statisticsunequal V= 3.356)
|Test statistic| ≥ |Decision rule| (under both assumptions)
Reject the Ho in favor of HA
(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.
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.
What is the outcome variable?
Relative frequency of PTSD among adolescents (Frequency or counts data)
What is the level of measurement for outcome variable?
Nominal scale
What is the grouping variable?
City of residence (City-A Vs City-B)
What is the level of measurement for the grouping variable?
Nominal scale (dichotomous variable)
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 (P1-P2) is equal to zero.
State the statistical hypotheses?
Ho: P1-P2 = 0
HA: P1-P2 ≠ 0
What test statistic would you use to test the null hypothesis? Why?
Z-distribution. The question is about the difference between 2 proportions
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.
What is the formula for calculating the standard error?
SE=P̅1(1-P̅1)n1+P̅2(1-P̅2)n2
n1=150 P̅1=15/150=0.10
n2=100 P̅2=18/100=0.18 SE=0.046
What is your conclusion?
Decision rule (tabulate t) Z1-α/2=1-(0.05/2)=0.975=1.96
Test Statistic= Statistic-ParameterSE
Statistic (P̅1 - P̅2)= (0.1-0.18)=-0.08
Hypothesized parameter (P1 - P2)= 0
Test statistic=(-0.08-0)/SE= -1.739
|Test statistic| < |Decision rule|
Accept the Ho
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.
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