Question on Probability

[Qurratulain]

Question 1
The probability of rain tomorrow is .30, and the probability of all members of ECN 215 being present in class is .6 (let us say). What is the probability of both these events occurring?
Question 2
For each of the following pairs, state whether you think the events are statistically independent or not. Explain briefly.
1. Disposable income and consumption in the US.
2. Consumption in the US and GNP in Britain.
3. Rainfall and the rate of inflation.
4. Rainfall and the price of corn.
5. An individual's shoe size and her height.
6. An individual's shoe size and her income.
Question 3
Suppose that Wake Forest students have a mean height of 68 inches, with a standard deviation of 3 inches. If heights are normally distributed, what is the probability that a randomly-selected Wake Student is between 63 and 73 inches tall?
Question 4
(continuation of question 3) Rachel, who is not aware of the above information, intends to draw a random sample of 50 Wake students in order to estimate the population mean height. What is the probability that the sample mean height she obtains lies between 63 and 73 inches?
Question 5
(continuation of question 3) Can you think of any reason why the heights of Wake Forest students might not follow the normal distribution?

[Qurratulain]

Question 6

You are trying to estimate the average houshold income in Forsyth County. You randomly sample 200 households, and come up with the following sample statistics: mean = $28000, standard deviation = $5000. Draw up the 95 per cent confidence interval for your estimate of the population mean income.

[Qurratulain]

Answer to question 6


We have and s = 5000, for a sample with n = 200. The figure of 28000 is our point estimate of the population mean, . What is our 95 per cent confidence interval? Well, we figure that with probability .95 our sample mean was drawn from the central 95 per cent of its sampling distribution, which is centered on the unknown population mean , and has standard error . If this sampling distribution is normal (and it will tend to be, for large n, even if the parent population is non-normal), then the central 95 per cent is given by (approximately) plus or minus two standard errors. Therefore, we can be 95 per cent confident that our sample mean of 28000 is no more than two standard errors away from (or in other words the maximum error for this degree of confidence is two standard errors). Now in this case, strictly speaking, the standard error is unknown since the population standard deviation, , is unknown. But we can take s as an estimate of , which than gives us an estimated standard error of

Our interval is then:

(Note: in this sort of case, where is unknown, and is replaced by its estimator, s, we should strictly speaking move from the assumption of a normal sampling distribution for to the t-distribution, with degrees of freedom . On the other hand, when the sample size is `large', i.e. greater than 30, the difference in results is negligible.)