Chapter 8.2, Problem 8.1LM

The purpose of this exercise is to compare the variability of ${\bar{x}}_1$ and ${\bar{x}}_2$with the variability of $\left({\bar{x}}_1-{\bar{x}}_2\right)$.

1. a. Suppose the first sample is selected from a population with mean μ₁= 150 and variance ${\sigma }_1^2=900$. Within what range should the sample mean vary about 95% of the time in repeated samples of 100 measurements from this distribution? That is, construct an interval extending 2 standard deviations of ${\bar{x}}_1$on each side of μ₁
2. b. Suppose the second sample is selected independently of the first from a second population with mean μ₂ = 150 and variance ${\sigma }_2^2=900$. Within what range should the sample mean vary about 95% of the time in repeated samples of 100 measurements from this distribution? That is, construct an interval extending 2 standard deviations of ${\bar{x}}_2$on each side of μ₂
3. c. Now consider the difference between the two sample means $\left({\bar{x}}_1-{\bar{x}}_2\right)$. What are the mean and standard deviation of the sampling distribution of $\left({\bar{x}}_1-{\bar{x}}_2\right)$?
4. d. Within what range should the difference in sample means vary about 95% of the time in repeated independent samples of 100 measurements each from the two populations?
5. e. What, in general, can be said about the variability of the difference between independent sample means relative to the variability of the individual sample means?