question archive A scientist wants to determine whether or not the height of cacti, in feet, in Africa is significantly higher than the height of Mexican cacti
A scientist wants to determine whether or not the height of cacti, in feet, in Africa is significantly higher than the height of Mexican cacti
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A scientist wants to determine whether or not the height of cacti, in feet, in Africa is significantly higher than the height of Mexican cacti. He selects random samples from both regions and obtains the following data.
Africa:
Mean = 12.1
Sample size = 201
Mexico:
Mean = 11.2
Sample size = 238
(a) Which of the following would be the correct hypothesis test procedure to determine if the height of cacti, in feet, in Africa is significantly higher than the height of Mexican cacti?
a. Paired t-test
b. Two-sample t-test
c. Two-sample test for proportions
(b) What is the value of the sample statistic to test those hypotheses? (2 decimal places)
(c) If the T test statistic is 2.169, and df = 202, find the p-value. (3 decimal places)
(d) Select the correct conclusion at alpha = 0.05.
a. The null hypothesis is rejected. There is sufficient evidence that African cacti are taller on average.
b. The null hypothesis is rejected. There is insufficient evidence that African cacti are taller on average.
c. The null hypothesis is not rejected. There is insufficient evidence that African cacti are taller on average.
d. The null hypothesis is not rejected. There is sufficient evidence that African cacti are taller on average.
(e) Explain the type of error, in context, that might have been made.
a. Type I error, which means the scientist concluded there is a significant difference between average height of cacti in Africa and cacti in Mexico, when in reality there is no difference.
b. Type I error, which means the scientist concluded there is not a significant difference between average height of cacti in Africa and cacti in Mexico, when in reality there is a difference.
c. Type II error, which means the scientist concluded there is a significant difference between average height of cacti in Africa and cacti in Mexico, when in reality there is no difference.
d. Type II error, which means the scientist concluded there is not a significant difference between average height of cacti in Africa and cacti in Mexico, when in reality there is a difference.
(f) What would the p-value have been if we had done a two-tailed test? (3 decimal places)
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a) b.
b) 0.90
c) 0.016
d) a.
e) a.
f) 0.031
Step-by-step explanation
The data is as:
Africa;
X1= 12.1
n1=201
Mexico;
X1=11.2
n2=238
Determining whether African cacti height is substantially higher than that of Mexico.
Let µ1 be the true mean height of African cacti and µ2 for Mexican cacti
The hypothesis is;
Ho: µ1- µ2=0
Ha: µ1- µ2>0
a) The two sample t-test is the correct hypothesis test procedure because this is a test for difference in two independent population means.
b) Sample statistic value;
X1-X2 = 12.1-11.2
=0.90
c) t=2.169, df=202
P-value= P(t>2.169)
= 1- P(t≤2.169)
= 1-0.9844
= 0.0156 ~ 0.016
d) significance level α= 0.05
The null hypothesis will be rejected if the level of significance is greater than the p-value. 0.016<0.05, hence, the null hypothesis is rejected.
The answer is (a).
e) The correct answer is (a). Type I error rejects the null hypothesis when it is true and it concludes that the African cacti are significantly taller on average but in actual sense the Mexican and the African cacti are averagely of the same height. Type II error fail to reject the null hypothesis when they are actually false. It concludes that the both the Mexican and the African cacti are of the same height on average while the African cacti is actually taller
f) t=2.169
df=202
p-value= P(t>2.169)+P(t<-2.169)
= 2P(T<-2.169)
= 2(0.0156)
= 0.0312 ~ 0.031
