1. A research worker wishes to estimate the mean of a population using a sample large enough that the probability will be 0.95 that the sample mean will not differ from the population mean by more than 25 percent of the standard deviation. How large a sample should he take?

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Exercises. Show all necessary solutions.
1. A research worker wishes to estimate the mean of a population using a sample large enough that
the probability will be 0.95 that the sample mean will not differ from the population mean by
more than 25 percent of the standard deviation. How large a sample should he take?
2. Let X1, X2 be a random sample from N(0, 1). What is the distribution of each of the following?
а. (Х2 — X1)/V2
b. (X1+ X2)²/(X2 – X1)²
c. (X2 + X1)//(X1 – X2)²
d. X{/X?
3. Derive the variance of the F distribution.
4. If X has an F distribution with m and n degrees of freedom, argue that 1/X has an F distribution
with n and m degrees of freedom.
5. Derive the variance of the Student's t distribution.
6. If X is t-distributed, show that X? is F-distributed.
Transcribed Image Text:Exercises. Show all necessary solutions. 1. A research worker wishes to estimate the mean of a population using a sample large enough that the probability will be 0.95 that the sample mean will not differ from the population mean by more than 25 percent of the standard deviation. How large a sample should he take? 2. Let X1, X2 be a random sample from N(0, 1). What is the distribution of each of the following? а. (Х2 — X1)/V2 b. (X1+ X2)²/(X2 – X1)² c. (X2 + X1)//(X1 – X2)² d. X{/X? 3. Derive the variance of the F distribution. 4. If X has an F distribution with m and n degrees of freedom, argue that 1/X has an F distribution with n and m degrees of freedom. 5. Derive the variance of the Student's t distribution. 6. If X is t-distributed, show that X? is F-distributed.
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