homework 5 statistics hypothesis testing regression and analysis of variance

Predict the percent weight gain when the metal is exposedfor 3.2 hours... The estimated regression line is Predict the percent weight gain when the metal is exposed for3.2 hours x=3.2Que

CITY INTERNATIONAL UNIVERSITY DEPARTMENT OF MATHEMATICS

HOMEWORK 5 – STATISTICS

HYPOTHESIS TESTING, REGRESSION

AND ANALYSIS OF VARIANCE

Ho Chi Minh City, Vietnam

December 2022

I CHAPTER 8 - HYPOTHESIS TESTING 3

Question 38 3

Question 40 3

Question 57 4

Question 58 5

II CHAPTER 9: REGRESSION 6

Question 1 6

Question 3 7

Question 4 9

Question 5 11

Question 6 12

Question 10 14

Question 11 14

Question 12 16

Question 20 18

III CHAPTER 10 – ANALYSIS OF VARIANCE 19

Question 1 19

Question 4 20

Question 5 21

Question 6 21

Question 7 22

Question 19 24

Question 20 25

I CHAPTER 8 - HYPOTHESIS TESTING

Question 38

To learn about the feeding habits of bats, 22 bats were taggedand tracked by radio Of these 22 bats, 12 were female and 10were male The distances flown (in meters) between feedingswere noted for each of the 22 bats, and the following summarystatistics were obtained

Test the hypothesis that the mean distance flown betweenfeedings is the same for the populations of both male and offemale bats Use the 5 percent level of significance

is the service’s claim believable at the

a 5 percent level of significance

b 1 percent level of significance

We have that p-value of this test is equal to

a 5 percent level of significance

Testing

We have :

Thus, we accept

II CHAPTER 9: REGRESSION

Question 1

The following data relate x, the moisture of a wet mix of acertain product, to Y, the density of the finished product

a Draw a scatter diagram

b Fit a linear curve to the data

Solution

a Draw a scatter diagram

b Fit a linear curve to the data

 Check by excel

The estimated regression line is

Question 3

The corrosion of a certain metallic substance has been studied

in dry oxygen at 500 degrees centigrade In this experiment,the gain in weight after various periods of exposure was used

as a measure of the amount of oxygen that had reacted withthe sample Here are the data:

a Plot a scatter diagram

b Fit a linear relation

c Predict the percent weight gain when the metal is exposedfor 3.2 hours

a Plot a scatter diagram

b Fit a linear relation

c

 Check by excel

The estimated regression line is

Predict the percent weight gain when the metal is exposed for3.2 hours (x=3.2)

Question 4

The following data indicate the relationship between x, thespecific gravity of a wood sample, and Y, its maximum crushingstrength in compression parallel to the grain

a Plot a scatter diagram Does a linear relationship seemreasonable?

b Estimate the regression coefficients

c Predict the maximum crushing strength of a wood samplewhose specific gravity is 43

Solution

a Plot a scatter diagram

A linear relationship seem unreasonable

b Estimate the regression coefficients

The estimated regression line is

c Predict the maximum crushing strength of a wood samplewhose specific gravity is 0.43 (x=0.43)

Question 5

The following data indicate the gain in reading speed versusthe number of weeks in the program of 10 students in a speed-reading program

a Plot a scatter diagram to see if a linear relationship isindicated

b Find the least squares estimates of the regressioncoefficients

c Estimate the expected gain of a student who plans to takethe program for 7 weeks

Solution

a Plot a scatter diagram

b Find the least squares estimates of the regressioncoefficients

 Check by python

The estimated regression line is

c Estimate the expected gain of a student who plans to takethe program for 7 weeks (x=7)

Question 6

Infrared spectroscopy is often used to determine the naturalrubber content of mixtures of natural and synthetic rubber Formixtures of known percentages, the infrared spectroscopy gavethe following readings:

The estimated regression line is

If a new mixture gives an infrared spectroscopy reading of 1.15,estimate its percentage of natural rubber (y = 1.15)

 Check by excel

Then, the line of linear regression is

Question 12

The following data set presents the heights of 12 male lawschool classmates whose law school examination scores wereroughly equal It also gives their first-year salaries Each ofthem went into corporate law The height is in inches and thesalary in units of $1000

a Do the above data establish the hypothesis that a lawyer’ssalary is related to his height? Use the 5 percent level ofsignificance

b What was the null hypothesis in part (a)?

Solution

b The null hypothesis in part (a)

Question 20

a Draw a scatter diagram relating cigarette use and deathrates from lung cancer

b Estimate the regression parameters α and β

c Test at the .05 level of significance the hypothesis thatcigarette consumption does not affect the death rate from lungcancer

d What is the p-value of the test in part (c)?

III CHAPTER 10 – ANALYSIS OF VARIANCE

Question 1

A purification process for a chemical involves passing it, insolution, through a resin on which impurities are adsorbed Achemical engineer wishing to test the efficiency of 3 differentresins took a chemical solution and broke it into 15 batches.She tested each resin 5 times and then measured theconcentration of impurities after passing through the resins.Her data were as follows:

Test the hypothesis that there is no difference in the efficiency

of the resins

Solution

Test The sample means are

The average of the three sample means is

The sample variance of the data is

 Check by python

Question 4

A machine shop contains 3 ovens that are used to heat metalspecimens Subject to random fluctuations, they are allsupposed to heat to the same temperature To test thishypothesis, temperatures were noted on 15 separate heatings.The following data resulted

Do the ovens appear to operate at the same temperature? Test

at the 5 percent level of significance What is the p-value?

Solution

Test The sample means are

The average of the three sample means is

The sample variance of the data is

Thus we reject the null hypothesis

Question 5

Four standard chemical procedures are used to determine themagnesium content in a certain chemical compound Eachprocedure is used four times on a given compound with thefollowing data resulting

Do the data indicate that the procedures yield equivalentresults?

Test The sample means are

The average of the three sample means is

The sample variance of the data is

Thus we reject the null hypothesis

Question 6

Twenty overweight individuals, each more than 40 poundsoverweight, were randomly assigned to one of two diets After

10 weeks, the total weight losses (in pounds) of the individuals

on each of the diets were as follows:

the two diets have equal effect

Solution

Test The sample means are

The average of the three sample means is

The sample variance of the data

Thus we “accept” the null hypothesis

 Check by python

Question 7

In a test of the ability of a certain polymer to remove toxicwastes from water, experiments were conducted at threedifferent temperatures The data below give the percentages ofthe impurities that were removed by the polymer in 21independent attempts

Test the hypothesis that the polymer performs equally well atall three temperatures Use the

(a) 5 percent level of significance

(b) 1 percent level of significance

Solution

Test

The average of the three sample means is

The sample variance of the data is

(a) 5 percent level of significance and

Thus, we can not reject the null hypothesis

 Check by python

(b) 1 percent level of significance

Thus, we can not reject the null hypothesis

 Check by python

Question 19

A study has been made on pyrethrum flowers to determine thecontent of pyrethrin, a chemical used in insecticides Fourmethods of extracting the chemical are used, and samples areobtained from flowers stored under three conditions: freshflowers, flowers stored for 1 year, and flower stored for 1 yearbut treated It is assumed that there is no interaction present.The data are in Table 10.4

Suggest a model for the preceding information, and use thedata to estimate its parameters

Solution

Estimateparameters

We have m = 3, n = 4

So

Question 20

The following data refer to the number of deaths per 10,000adults in a large Eastern city in the different seasons for theyears 1982 to 1986

a Assuming a two-factor model, estimate the parameters

b Test the hypothesis that death rates do not depend on theseason Use the 5 percent level of significance

c Test, at the 5 percent level of significance, the hypothesisthat there is no effect due to the year

Solution

a Estimate the parameters

So