Question One
Consider the following two independent random samples selected from two normal populations of common unknown variance
|
Sample one
|
18
|
37
|
27
|
10
|
20
|
24
|
26
|
2
|
12
|
31
|
|
Sample two
|
23
|
28
|
25
|
14
|
26
|
21
|
28
|
17
|
24
|
20
|
23
|
53
|
a. Compute the mean, variance, standard deviation, median, Q1 and Q3 for each of the two samples.
b. Plot boxplots of both samples on the same graph.
c. Construct 95% confidence intervals for each of the two populations' means.
d. Construct 95% confidence interval for the difference between the two populations' means.
e. Test whether the mean of the first population is larger 18.
f. Test whether the mean of the second population is smaller than 30.
g. Test whether the means of the two populations are not equal.
Question Two
The table below displays summarized data of daily sales of two small stores. Assuming the sales are normally distributed, answer a-d
Variable Sample Size Mean SD
|
Variable
|
Sample Size
|
Mean
|
SD
|
|
Store I
|
12
|
19.6
|
8.7
|
|
Store II
|
10
|
24.7
|
16.2
|
a. Construct a 90% confidence interval for the variance of daily sales of the first store.
b. Test that the standard deviation of daily sales of the second store is larger than 12.
c. Construct a 90% confidence interval for the ratio of the variances of the daily sales of the two stores.
d. Test that the standard deviation of daily sales of the second store is larger than that of the first.
Question Three
A study is conducted to compare the relative effectiveness of two kinds of cough medicines in increasing sleep. Six people with colds are given medicine A the first night, and medicine B on the second. Their hours of sleep each night are recorded below.
|
Medicine A
|
5.2
|
4.3
|
6.1
|
5.8
|
5.5
|
7.5
|
|
Medicine B
|
4.3
|
4.5
|
5.2
|
5.8
|
5.8
|
7.2
|
a. Construct a 95 % Confidence interval for the difference of the two means.
b. Is one medicine more effective than the other?
Question Four
a. Let X ~ Poisson(4) , find P(X=3)
b. Let X ~ bin(17, 0.3), find P(X>3)
c. Let X ~ Hypergeometric(N, M,;n), N = 20, M =12 and sample size n = 7, find P(X < 5).
d. Let X ~ t(10), find the 90th percentile of X, (or t0.1(10) ).
e. Let X ~F(18, 10), find the 90th percentile of X, (or F0.1(18, 10) ).
f. Let X ~ χ2(15), find the 10th percentile of X, (or χ20.9(15) )
g. Let Let X ~ N(500, 900), find the 15th percentile of X.
h. Let Let Z ~ N(0.1), find the 95th percentile of Z, (or z0.5).
Question Five
Water samples were taken at four different locations in a river to determine whether the quantity of dissolved oxygen, a measure of water pollution, varied from one location to another. Locations 1 and 2 were selected above an industrial plant, one near the shore and the other in midstream; location 3 was adjacent to the industrial water discharge for the plant; and location 4 was slightly downriver in midstream. Five water specimens were randomly selected at each location, but one specimen, corresponding to location 4, was lost in the laboratory. The data are provided here (the greater the pollution, the lower the dissolved oxygen readings).
|
Location
|
Mean Dissolved Oxygen Content
|
|
1
|
6.0
|
6.2
|
6.3
|
6.2
|
6.0
|
|
2
|
6.3
|
6.6
|
6.5
|
6.5
|
6.6
|
|
3
|
4.8
|
4.4
|
5.0
|
4.8
|
5.1
|
|
4
|
6.1
|
6.3
|
6.1
|
5.8
|
|
|
a. Construct the ANOVA table needed to test whether the data provide sufficient evidence to indicate a difference in the mean dissolved oxygen contents for the four locations?
b. Use Tukey's test to compare water pollution in the four locations.