EXERCISES Section 5.3 (37-45)

37

A particular brand of dishwasher soap is sold in three sizes: $25\mathrm{oz},40\mathrm{oz}$ , and $65\mathrm{oz}$ . Twenty percent of all purchasers select a 25-oz box, 50% select a 40-oz box, and the remaining $30\%$ choose a 65-oz box. Let $X_1$ and $X_2$ denote the package sizes selected by two independently selected purchasers.

a. Determine the sampling distribution of $\bar{X}$ , calculate $E\left(\bar{X}\right)$ , and compare to $\mu$ .

b. Determine the sampling distribution of the sample variance $S^2$ , calculate $E\left(S^2\right)$ , and compare to ${\sigma }^2$ .

38

There are two traffic lights on a commuter’s route to and from work. Let

• $X_1$ be the number of lights at which the commuter must stop on his way to work,
• $X_2$ be the number of lights at which he must stop when returning from work.

Suppose these two variables are independent, each with pmf given in the accompanying table (so $X_1,X_2$ is a random sample of size $n=2$ ).

$x_1$	0	1	2
$p(x_1)$	0.2	0.5	0.3

$$\mu =1.1,{\sigma }^2=.49$$

a. Determine the pmf of $T_o=X_1+X_2$ .

b. Calculate ${\mu }_{T_o}$ . How does it relate to $\mu$ , the population mean?

c. Calculate ${\sigma }_{T_o}^2$ . How does it relate to ${\sigma }^2$ , the population variance?

d. Let $X_3$ and $X_4$ be the number of lights at which a stop is required when driving to and from work on a second day assumed independent of the first day. With $T_o=$ the sum of all four $X_i$ ’s, what now are the values of $E\left(T_o\right)$ and $V\left(T_o\right)$ ?

e. Referring back to (d), what are the values of $P\left(T_o=8\right)$ and $P\left(T_o\ge 7\right)$ [Hint: Don’t even think of listing all possible outcomes!]

39

It is known that $80\%$ of all brand A external hard drives work in a satisfactory manner throughout the warranty period (are “successes”). Suppose that $n=15$ drives are randomly selected. Let $X=$ the number of successes in the sample. The statistic $X/n$ is the sample proportion (fraction) of successes. Obtain the sampling distribution of this statistic. [Hint: One possible value of $X/n$ is .2, corresponding to $X=3$ . What is the probability of this value (what kind of rv is $X)$ ?]

40

A box contains ten sealed envelopes numbered $1,\dots ,10$ . The first five contain no money, the next three each contains \ 5$,andthereisa$$ {10}$billineachofthelasttwo.Asampleofsize3isselectedwithreplacement(sowehavearandomsample),andyougetthelargestamountinanyoftheenvelopesselected.If${X}{1},{X}{2}$,and${X}{3}$denotetheamountsintheselectedenvelopes,thestatisticofinterestis$M =$themaximumof${X}{1},{X}{2}$,and${X}{3}$ .

a. Obtain the probability distribution of this statistic.

b. Describe how you would carry out a simulation experiment to compare the distributions of $M$ for various sample sizes. How would you guess the distribution would change as $n$ increases?

41

Let $X$ be the number of packages being mailed by a randomly selected customer at a certain shipping facility. Suppose the distribution of $X$ is as follows:

$x$	1	2	3	4
$p(x)$	0.4	0.3	0.2	0.1

a. Consider a random sample of size $n=2$ (two customers), and let $\bar{X}$ be the sample mean number of packages shipped. Obtain the probability distribution of $\bar{X}$ .

b. Refer to part (a) and calculate $P\left(\bar{X}\le 2.5\right)$ .

c. Again consider a random sample of size $n=2$ , but now focus on the statistic $R=$ the sample range (difference between the largest and smallest values in the sample). Obtain the distribution of $R$ . [Hint: Calculate the value of $R$ for each outcome and use the probabilities from part (a).]

d. If a random sample of size $n=4$ is selected, what is $P\left(\bar{X}\le 1.5\right)$ ? [Hint: You should not have to list all possible outcomes, only those for which $\bar{x}\le 1.5$ .]

42

A company maintains three offices in a certain region, each staffed by two employees. Information concerning yearly salaries (1000s of dollars) is as follows:

Office	1	1	2	2	3	3
Employee	1	2	3	4	5	6
Salary	29.7	33.6	30.2	33.6	25.8	29.7

a. Suppose two of these employees are randomly selected from among the six (without replacement). Determine the sampling distribution of the sample mean salary $\bar{X}$ .

b. Suppose one of the three offices is randomly selected. Let $X_1$ and $X_2$ denote the salaries of the two employees. Determine the sampling distribution of $\bar{X}$ .

c. How does $E\left(\bar{X}\right)$ from parts (a) and (b) compare to the population mean salary $\mu$ ?

43

Suppose the amount of liquid dispensed by a certain machine is uniformly distributed with lower limit $A=8\mathrm{oz}$ and upper limit $B=10$ oz. Describe how you would carry out simulation experiments to compare the sampling distribution of the (sample) fourth spread for sample sizes $n=5,10,20$ , and 30 .

44

Carry out a simulation experiment using a statistical computer package or other software to study the sampling distribution of $\bar{X}$ when the population distribution is Weibull with $\alpha =2$ and $\beta =5$ , as in Example 5.20. Consider the four sample sizes $n=5,10,20$ , and 30, and in each case use 1000 replications. For which of these sample sizes does the $\bar{X}$ sampling distribution appear to be approximately normal?

45

Carry out a simulation experiment using a statistical computer package or other software to study the sampling distribution of $\bar{X}$ when the population distribution is lognormal with $E\left(\ln \left(X\right)\right)=3$ and $V\left(\ln \left(X\right)\right)=1$ . Consider the four sample sizes $n=10,20,30$ , and 50, and in each case use 1000 replications. For which of these sample sizes does the $\bar{X}$ sampling distribution appear to be approximately normal?