4.2 Sampling Distributions Associated with Normal Populations

Page | 1

CHAPTER 4

Sampling Distributions

4.1 Introduction

4.2 Sampling distributions associated with Normal populations

4.3 Order Statistics

4.4 Large sample approximations

4.5 Chapter Summary

4.6 Computer examples

Projects for Chapter 4

Exercises 4.1

4.1.1.

(a) There are equally likely possible samples of size 3, so the probability for each is without replacement:

/ M / S
/ / / 1
/ / / 4
/ / 1 /
/ 0 / 0 / 1
/ / 1 /
/ 1 / 1 / 1

(iv)

(b) We can get =125 samples of size 3 with replacement

4.1.3. Population:{1,2,3}. , for in

(a) ,

(b)

/ 1 / / / 2 / / / 3
/ 1/27 / 1/9 / 2/9 / 7/27 / 2/9 / 1/9 / 1/27

(c), then

4.1.5. Since , we have

Assuming the sampling fron a population with mean and variance ,we have

4.1.7. Let X be the weight of sugar

Then is the mean weight, where .

By Corollary 4.2.2, , and . Then , and . Therefore, the probability requested is R Code:

4.1.9.Let X be the height. , and . Then,

4.1.11. Let X be the time. . Then . Therefore, , or 15.87% of measurement times will fall below 85 seconds.

4.1.13. According the information, and .

(a) If , we can assume , then 0.0007

(b) If , we can assume , then

(d) Increasing the sample size, decrement the probability

4.1.15. Let T be the temperature.

Since , we assume . Then . Therefore,

Exercises 4.2

4.2.1.We have that

(a) We can see, for example in a table, that . Then

(b) Choosing upper and lower tail area to 0.025, and since , and , then , then ,

(c)

4.2.3. If , then . In our case , then

(a) With , (b)

4.2.5. Since are i.i.d. , then

(a) Since , and , Z is Chi-square distributed with 4 degrees of freedom and Y is Chi-square distributed with 5 degrees of freedom

(b) Yes.

4.2.7. Since the random sample comes from a normal distribution, .

Setting the upper and lower tail area equal to 0.05, even this is not the only choices, and using a Chi-square table with degrees of freedom, we have , and . Then, with , , and

4.2.9. Since

(a) =0.99

(b)

4.2.11. According with the information, ,,, and , then . The degrees of freedom are , so the critic value is 1.729 at =0.05 -level. Then, the data tend to agree with the psychologist claims.

4.2.13. If , then, then

and

4.2.15. If is from then, by Theorem. 4.2.8, is from

by exercise 4.2.13, Since ,

Simplifying after multiplying by , we obtain

4.2.17. If X and Y are independent random variables from an exponential distribution with common parameter , then using 4.2.16 with n=1, and then

4.2.19. If

(a)

(b)

, where

Then and . Thus,

4.2.21. If the PDF is given by

Then

Let then and

and , then , which converges for .

For , where with the property .

Then

Similarly,

, which converges for

, .

Now, . Therefore,

and

4.2.23. If is a random sample from a normal population with mean and variance and if is a random sample from an independent normal population with mean and variance , then , ,, and .

Then and

then and

Then, since the samples are independent, we have by definition that

This after simplification becomes:

Q.E.D.

4.2.25. If with , then the pdf of is given by

Since and , then

, then Therefore,

4.2.27. where V is a chi-square random variable and n is the degrees of freedom associated with our random variable. Then since and . The F distributions relationship to chi-square statistics is in our case thus

Exercises 4.3

4.3.1. , then the cumulative distribution of is

Let Y represent the life length of the system, then and , then the pdf of Y is of the form and is given by

4.3.3.take values ; take values 1,2,3, and

Since the values of are less or equal to the values for , take values 0,1

Since the values of are greater than the values for , then

take values 1,2,3

Since , take values 0,1

4.3.5. Let be a random sample from exponential distribution with mean ,

then the common pdf is given by , if Using Theorem 4.3.2, the pdf of the k-th order statistic is given by ,where

then Then, the pdf of is , which is the pdf of an exponential distribution with mean , and the pdf of is

4.3.7. a random sample are i.i.d with pdf then if then Then, using theorem 4.3.3 the joint pdf of and is given by

,if and otherwise.

Now, let and , and consider the functions

then their inverses are Then the corresponding Jacobian of the one-to-one transformation isthe the join pdf of R and Z is

and otherwise.

Then, the pdf of the range is , if and , otherwise.

4.3.9.a random sample from ,

The CDF of is , where is the cdf of evaluated in y

Then, and

Then , Then

Therefore

4.3.11. is a random sample from

The joint pdf of and , according theorem 4.3.3, is given by

, if

Since for , the pdf is,

and, the DF is

In our case,, if

and , if

Then, the joint pdf , using the 4.3.3, is given by

, if , and , otherwise

Exercises 4.4

4.4.1., where , , then and

By theorem 4.4.1:

then

4.4.3. Let be the time spent by a customer coming to certain gas station to fill up gas

Suppose are independent random variables, with minutes, minutes, and

Then is the total time spent by the customers

Since, hours = 180 minutes,

and by Theorem 4.4.1, For practical purposes is large enough

Therefore, , where

There is approximately 0% chance that the total time spent by the customers is less than 3 hours.

4.4.5. 1250 students took it, and ,, and students (is large enough). Then, and

Then=1

There is almost 100% chance that the average score is less than 75.08%.

4.4.7. for , then

for , then

Therefore,this assumes independence or that the covariance is zero.

4.4.9. Binomial

Using normal approximation:

4.4.11. of person making reservations will not show up each day

Rental company reserves for 215 persons. 200 automobiles anvilable.

is the probability of the person making reservation will show up each day

Letbe the number of the persons making reservation will show up.

Then

The probability requested is

4.4.13. SIDS occurs between the ages 28 days and one year Rate of death due to SIDS is 0.0013 per year. Randon sample of 5000 infants between the ages 28 day and are your Let be the number of SIDS related deaths , Then The probability requested is