 # STA 250 Paper - Statistics 250 Solutions

Instructions: You MUST answer question 1 and choose 3 other questions for a total of Four.

1. Topics: Probability distribution, population moments, sampling distribution of sample mean, standard error of the sample mean, Confidence Interval (CI) for the mean, parts: a - j

(REQUIRED) Consider the finite population of size N = 3, measurements on the variable x: {1, 2, 3}:

a. derive the population probability distribution for the variable x.
b. compute the population mean, µx = E(x) = , and standard deviation,
sx = (Var(x))1/2 = .

c. Given the above population, sampling with replacement and considering order, how many samples of size n = 2 are possible?

d. derive all samples of n = 2 in the manner as described in c.

e. derive the sampling distribution of the mean as per your results obtained in b.

f. compute the expectation, E(m), or mean of the sampling distribution of the sample mean, m.

g. use the formula for the standard error of the mean, sm =

h. Using your result obtained in g, derive for each sample in your population of samples (cf., result obtained in d) a 90% confidence interval.

i. How many of your sample confidence interval cover the true population mean?

j. Does the nominal and actual confidence levels coincide or are they different?

2. Topics: Ratio level data, Correlation, Causality, Control, and Confounding, two parts: a and b

For the below datasets compute the respective simple (zero-order) correlation between the
independent variable x and the dependent variable y. Then use the first-order (partial correlation)
correlation to determine whether the relationship is between the variables x and y is causal or is due to
confounding by the variable z which precedes both x and y in time. Draw the path diagram appropriate
to the observed relationship for each dataset.

a. Data set 1

x y z
2 3 1
1 2 1
0 0 0
0 2 0
0 0 0
1 1 0
0 0 1

b. Data set 2

x y z
0 0 -0.03
1 0 0
2 1 0
0 0 -0.03
0 1 0
0 0 -0.03
0 0 -0.03

3. Topics: Mean, Median, Empirical Influence Function, Influence curve, breakdown point, two parts: a and b

Given the sample dataset of size n = 4: {1, -3, 1, x}. Note that x is a real number.

a. Obtain the empirical influence function for the mean and sketch its influence curve. What does this influence function tell you about the mean (hint: try x = -4 or x =
-100,000)? What is the breakdown point?

b. Now suppose that x < -3, compute the median. If x = -4, what is the median? If x =
-100,000, what is the median? What do you observe regarding the stability of the median for this dataset?

4. Topics: Ratio level data, Correlation, Statistical Significance, Effect size, Coefficient of
determination

A random sample of measurements on two quantitative variables x and y is taken from a population.
The data for the variables x and y are: {(x, y) | (2, 1), (3, 2), (4, 3)}. Assume that x is an independent
variable and y is a dependent variable. Calculate the Pearson correlation coefficient. What does the
correlation mean? Is the correlation statistically significant at the 5% level? What is the effect size?
What is the coefficient of determination?

5. Topics: statistical test, Confidence Interval (CI)

The average life span of American men is 75.6 years. A random sample of n = 25 male college lecturers
is found to have an average life span of 77.6 years, with a sample standard deviation 2.0 years. Would
you conclude that college lecturers live longer than average? Use the appropriate two-tailed test with
an a = 0.05. Also, construct a 95% confidence interval. Do the statistical test and the confidence interval
agree regarding whether college lecturers live longer? Write up your results.

You should be able to download solutions to the above questions in word and pdf formats! You can edit them as you wish.

Subjects: Statistics
Topic: Various topics in College Level Statistics
Level: College / University
Tags:

### Probability distribution, population moments, sampling distribution of sample mean, standard error of the sample mean, Confidence Interval

Price\$2.95

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Project Details
Subjects: Statistics
Topic: Various topics in College Level Statistics
Level: College / University
Tags:

## Probability distribution, population moments, sampling distribution of sample mean, standard error of the sample mean, Confidence Interval

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