Search & Filtering
The following sample observations were randomly selected.
X 4 5 3 6 10
Y 4 6 5 7 7
Determine the regression equation. (6 marks)
Suppose X and Y are random variables with P(X = 1) = P(X = −1) = 1 2 ; P(Y = 1) = P(Y = −1) = 1 2 . Let c = P(X = 1 and Y = 1) (a) Determine the joint distribution of X and Y, Cov(X,Y), and r(X,Y). (b) For what value(s) of c are X and Y independent? For what value(s) of c are X and Y 100% correlated?
The following data gives the information on the ages (in years) and the number of breakdowns during the past month for a sample of 7 machines at a large industrial company.
Age (X): 12, 7, 2, 8, 13, 9, 4
Number of breakdowns (Y): 9, 5, 1, 4, 11, 7, 2
(a) Draw a scatter diagram to represent the above data.
(b) Calculate the value of Pearson’s correlation coefficient r. Interpret the value of r.
(c) Calculate coefficient of determination. Interpret your answer.
(d) Determine the equation of the regression line using least squares method.
(e) What is the expected breakdown for an eleven year old machine?
The following table refers to two properties: age and traffic violations of residents in Gauteng over 12 months.
- under 21: none=15, one or more=25
- [21,41): none=8, one or more=12
- 41 or more: none=30, one or more 10
(i) If a Gauteng resident is selected at random, what is the probability that
(a) he is under 21 and has one or more violations during the past 12 months?
(b) he is under 21?
(c) he has one or more violations during the past 12 months?
(ii) Are events “under 21” and “one or more violations during the past 12 months” independent?
(iii) Test at a 5% level of significance if age and traffic violations are independent of each other:
A population consists of three numbers (3, 6, 9). Consider all possible samples of size 2
which can be drawn without replacement from the population. Find the following:
a. Population mean
b. Population variance
c. Populations standard deviation
d. Mean of the sampling distribution of the means
e. Variance of the sampling distribution of the means
f. Standard deviation of the sampling distributions of the means
A population consists of three numbers (3, 6, 9). Consider all possible samples of size 2
which can be drawn without replacement from the population. Find the following:
a. Population mean
b. Population variance
c. Populations standard deviation
d. Mean of the sampling distribution of the means
e. Variance of the sampling distribution of the means
f. Standard deviation of the sampling distributions of the means
Find the M.D. & variance of the following distribution. x 2 3 4 5 6 7 8 9 10 f 1 1 2 4 4 3 7 5 3
2.3 of 60 new entrants in a are given university found to have a mean height of 68.60 inches and 50 seniors, a mean height of 69.51 inches. Is the evidence conclusive that the mean height of seniors is greater then that of the new entrants? Assume the s.d. of the height to be 2.48 inches?
A factory produces apple juice contained in a bottle of 1.5L. However, due to random fluctuations in the automatic bottling machine, the actual volume per bottle varies according to a normal distribution. It is observed that 10% of bottles are under 1.45L whereas 5% contain more than 1.55L. Calculate the mean and standard deviation of the volume distribution。
Consider the following subsets of 𝑆 = {1, 2, 3, … ,8,9}:
(𝑖) [{1,3,5},{2,4,6,8},{5,7,9}]
(𝑖𝑖) [{1,3,5},{2,4,6,8},{7,9}]
Does 𝑖 and 𝑖𝑖 form a partition on sample space 𝑆 [4]
