1. Redo your midterm if you did any question wrong. You do not need to hand in your work for this part. Please verify your answer with the midterm answer key.
2. The following table shows that the economy next year has three possible states: Good , Average and Poor. It also shows the correponding probability of each states. The column of stock A shows the investment rate of return (%) for stock A; and the column of Stock B shows the invesment rate of return for stock B.
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|
Return (%)
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|
State
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Probability
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Stock A
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Stock B
|
|
Good
|
0.4
|
15
|
8
|
|
Average
|
0.5
|
9
|
10
|
|
Poor
|
0.1
|
6
|
12
|
a) Calculate the expected value of stock A and B's return
b) Calculate the variance of the return of Stock A and Stock B
c) Calculate the covariance and correlation of Stock A and Stock B's return
d) An investor invests in 40% of his money in stock A and 60% of his money in stock B, what is his portfolio's expected return? What is his portfolio's variance and standard deviation?
3.1 Evaluate the following statement. To answer this question please state the Central Limit Theorem and explain why central limit theorem is so important.
The samples mean of a random sample of n observations from a normal population with mean µ and variance σ2 is a sampling statistics. The sample mean is normally distributed with mean µ and variance σ2/n due to central limit theorem.
3.2. Find the sampling distribution of sample means if all possible samples of size 2 are drawn with replacement from the following population, please calculate the mean and variance of the sample means.
-
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X
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-2
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0
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2
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p(x)
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0.2
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0.6
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0.2
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3.3 Let the random variable X follow a normal distribution with a mean of μ and a standard deviation of σ. Let 1 be the mean of a sample of 16 observations randomly chosen from this population, and 2 be the mean of a sample of 25 observations randomly chosen from the same population.
Evaluate the statement P(μ - 0.2σ < 1 < μ + 0.2σ) < P(μ - 0.2σ < 2 < μ + 0.2σ) as to whether it is true or false.
4.1. Suppose that the amount of time teenagers spend on the internet is normally distributed with a standard deviation of 1.5 hours. A sample of 100 teenagers is selected at random, and the sample mean is computed as 6.5 hours. Construct the 95% confidence interval of the population mean and interpret what the 95% confidence interval estimate of the population mean tells you.
4.2. A furniture mover calculates the actual weight as a proportion of estimated weight for a sample of 31 recent jobs. The sample mean is 1.13 and the sample standard deviation is 0.16. Calculate a 90% confidence interval for the population mean.
4.3. Suppose that x1 and x2 are random samples of observations from a population with meanμ and variance σ2. Consider the following three point estimators, X, Y, and Z, of μ: X = (x1 +x2)/2, Y = (x1 + 3x2)/4, and Z = (x1 + 2x2)/3.
1) Show that all three estimators X, Y, and Z are unbiased.
2) Which of the estimators X, Y, and Z is the most efficient?
5. Redo the assignment two computer exercises. Generate 1000 series of data for Bernoulli distribution, Binomial Distribution, Uniform distribution and Normal distribution with the Random Number Generator from Excel as shown in Class. The number of data points for each series can be increased to 200. Then draw the histogram for the sampling mean and see whether this will assemble normal distribution better than the time you did last time, discuss the sampling distribution in terms of mean and variance. After you learn central limit theorem and law of large numbers, how these practices help you to understand these theorems?