Question

1. Suppose we have the following paired observations of variables X and Y:

X         Y

18        40

14        30

20        20

22        20

            19        10

            27        0

Calculate the values of the sample covariance and sample correlation between X and Y. Using this information, how would you characterize the relationship between X and Y?

            (12 points)

2. Suppose X follows a normal distribution with mean µ = 50 and standard deviation σ = 5.

(10 points)

1. What is the probability that X > 58?
2. What is the probability that 45 < X < 62?
3. What is the probability that X < 59?
4. What is the probability that X = 50?

3. Suppose we know that income follows a normal distribution with a population average µ = $37,000 and a standard deviation σ = $9,000. The federal government want to provide families with free college tuition if they fall below a certain income. In order to save money, government officials want to set the income cutoff so only 1.7 % (or 0.0017) of the population qualify for free tuition. What should this income cutoff be set at for this policy to work? (10 points)

4. Suppose we know that a random variable X has a population mean µ = 100 with a standard deviation σ = 30. What are the following probabilities? (9 points)

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1. The probability that X > 102 when n = 1296.

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2. The probability that X > 102 when n = 900.

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3. The probability that X > 102 when n = 36.

5. Suppose we know that the population standard deviation for SAT scores (σ) is 200. Provide each of the following using this information. (11 points)

1. A random sample of n = 100 results in a sample mean of 1050. Provide a 95% confidence interval estimate of µ. What is the value for the margin of error? Interpret your results.
2. A random sample of n = 1600 results in a sample mean of 1050. Provide a 95% confidence interval estimate of µ.
3. A random sample of n = 100 results in a sample mean of 1050. Provide a 99% confidence interval estimate of µ.

6. Suppose we collect a random sample of n = 16 and find an average income of $45,000 with a sample standard deviation s = $12,000. Provide each of the following using this information. (11 points)

1. A 95% confidence interval estimate of the population mean µ. What is the value for the margin of error? Interpret your results.
2. A 90% confidence interval estimate of the population mean µ.
3. A 99% confidence interval estimate of the population mean µ.

7. A random sample of n = 1000 voters is taken and 650 of those surveyed indicate that they like pizza. What is the value of the sample proportion of those who like pizza? Provide a 95% confidence interval estimate of the proportion of individuals in the population who like pizza. Interpret your results. (7 points)

Answer 1

Following table shows the calculations:

	X	Y	(x-xmean)^2	(y-ymean)^2	(x-xmean)*(y-ymean)
	18	40	4	400	-40
	14	30	36	100	-60
	20	20	0	0	0
	22	20	4	0	0
	19	10	1	100	10
	27	0	49	400	-140
Total	120	120	94	1000	-230

Mean:

The co-variance of the data:

The variance is:

The correlation coefficient is

It shows a strong, linear and negative relationship between the variables.