# Mathematics

HWK #5

1) For the companying data set,

X 2 6 6 7 9

Y 3 2 6 9 5

(a) I Have done it

(b) By hand, compute the correction coefficient. The correlation coefficient is r= (……). (Rounds to three decimal places as needed)

(c) Determine whether there is a linear relation between x and y. Explain.

(d) Find the critical value correlation for the sample size n= 5. Used the table of critical value for correlation coefficient. Explain.

2) An engineer wanted to determine how the weight of a car affects gas mileages. The following data represent the weight of various cars and their gas mileage.

Car weight pound miles per gallon

A 3590 18

B 2690 26

C 3180 23

D 3565 20

E 3680 19

(a) I have done it

(b) I’ve done it

(c) Compute the linear correlation coefficient between the weight of a car and its miles per gallon. r = (…….) (Round to three decimal places as needed)

(d) Comment on the type of relation that appears to exist between the weight of a car and its miles per gallon based on the scatter diagram and the linear correlation coefficient.

3) Researcher wondered whether the size of a person’s brain was to related to the individual’s mental capacity. The selected 3 females and 3 males measured their MRI counts IQ scores. The data is below.

Females Males

MRI IQ MRI IQ

857,781 135 924,060 138

991,305 138 965,355 132

856,473 141 1,038,438 138

Treat the MRI as the explanatory variable. Compute the linear correlation coefficient between MRI count and IQ for both the males and the females. Do you believe that MRI count and IQ are linearly related?

A-the linear correlation coefficient for females is (…….)

B- the linear correlation coefficient for males is (…….)

(Rounds both to three decimal places as needed)

-Do you believe that MRI count and IQ are linearly related?

4) For the set below,

(a) Determine the list-squares regression line

(b) Graph the least-squares regression line on the scatter diagram.

X 3 4 5 6 8

Y 4 5 6 9 13

(a) Determine the list-squares regression line. Ŷ= (……) X + (…….)(Round to four decimal places as needed.

(b) Graph the least-squares regression line on the scatter diagram.

5)A pediatrician want to determine the relation that exists between a child’s height, x, and head circumference, y. She randomly selects 11 children from her practice, measures their heights and head circumferences and obtains the accompanying data. Complete parts (a) through (g).

Data table. In inches

Height, x Head Circumference, y Height, x Head Circumference, y

27.75 17.7 27 17.3

24.7 17.1 27.5 17.5

25.5 17.1 26.75 17.3

25.5 17.3 26.75 17.5

25 16.9 27.5 17.5

27.75 17.6

(a) Find the least -squares regression line treating height as the explanatory variable and head circumference as the response variable.

The least squares regression line is Ŷ= (……) X + (……)(Round to four decimal places as needed)

(b) Interpret the slope and y-intercept, if appropriate.

(c) Use the regression equation to predict the head circumference of the child who is 25 inches tall.

(d) Compute the residual based on the observed head circumference of the 25-inch-tall child in the table. Is the head circumference of child above average or below average?

(Residual=observed y – predicted y = y- Ŷ

(e) Draw the least squares regression line on the scatter diagram of the data and label the residual from part (d).

(f) Notice that two children are 27 inches tall. One has a head circumference 17.7 inch and other has a head circumference 17.3 inch. How can this be?

(g) Would it be reasonable to use the least-squares regression line to predict the head circumference of a child who was 34 inches tall?

6) An engineer wants to determine how the weight of a car, x, affects gas mileages, y. The following data represent the weight of various cars and their miles per gallon.

Car A B C D E

Weight pounds, x 2660 2910 3290 3860 4015

Miles per Gallon,y 22.2 22.9 21.8 15.9 14.1

(a) Find the least-squares regression line treating weight as the explanatory variable and miles per gallon as the response variable. Write equation for the least-squares regression line.

Ŷ= (…..) x + (…….)(Round to four decimal places as needed).

(b) Interpret the slop and intercept, if appropriate.

(c) Predict the miles per gallon of car B and compute the residual. Is the miles per gallon of this car above average or below average for cars of this weights?

(d) Draw the least squares regression line on the scatter diagram of the data and label the residual.

7) One of the biggest factors in determining the value of a home is the square footage. The accompanying data represent the square footage and asking prices (in thousand of dollars) for a random sample of home for sales. Complete parts (a) through (h).

Square Footage, x Asking Prince ($000s), y

1,148 154

1,096 159.9

1,151 169

1,288 169.9

1,322 170

1,466 179.9

1344 180

1,544 189

1,494 189.9

(a) And (b) have done.

(c) Determine the linear correlation coefficient between square footage and asking price.

The linear correlation coefficient between square footage and asking price is r = (……..)(Round to three decimal places as needed)

(d) Is there a linear relation between the square footage and asking price?

(e) Find the least-square regression line treating square footage as the explanatory variable.

(f) Interpret the slop

(g) Is it reasonable to interpret the y-intercept?

(h) One home that is1, 094 square feet is listed as 189,900. Is this home’s price above or below average for home of this size?

8) The following data represent the heights and weights of various baseball players. (in inches/ pounds)

Player Height, x Weight, y Player Height, x Weight, y

Nate 72 197 Greg 70 195

Derek 71 201 Randy 77 240

Mark 73 192 Josh 71 193

Jon 74 209 Pat 74 203

(a) Compute the least-square regression line. Ŷ=(…..)x- (……)(Do not round until the final answer, then round to three decimal places as needed)

(b) Remove the value corresponding to Randy and compute the least square regression line the correlation coefficient.

(c) Do you thing that Randy is an influential observation.

9) The time it takes for a planet to complete its orbit around a particular star is called the planet’s sidereal year.The sidereal year of the planet is related to the distance the planet is from the star. Accompanying data show the distance of the planets from a particular star and their sidereal years. Complete part a through e. x million of miles/ y sidereal year

Planet Distance from the star x Sidereal Year, y

Planet1 36 0.22

Planet2 67 0.64

Planet3 93 1.00

Planet4 142 1.88

Planet5 483 11.9

Planet6 887 29.3

Planet7 1,785 84.0

Planet8 2,797 165.0

Planet9 3.675 248.0

(a) Has been done.

(b) Determine the correlation between distance and sidereal year. The correlation between distance and sidereal year is (……)(Round to three decimal places as needed).

(c) Compute the least square regression line.

(d) Plot the residuals against the distance from the star

(e) Do you think the least squares regression line is good model?

1) For the data set shown below, do they following. (a) Compute the standard error the point estimate for σ. (b) Assuming the residuals are normally distributed, determine Sb1. (c) Assuming the residuals are normally distributed, test H0;β1=0 versus H1; β1≠0 at the a=0.05 level of significance. The null hypothesis is that x and y are not linearly related.

X 3 4 5 7 8

Y 5 6 8 13 15

a) Determine the point estimate for σ. Se ≈ (…….)(Do not round until the final answer. Then round to four decimal places as needed)

b) Recall that b1 is the estimate of slope of the regression line β1. The sample standard error of b1 is given by the following formula.

c) In order to conduct the hypothesis test, first calculate the test statistic t0.(Explain the result; rejected or do not reject; do the relation exists between x and y?).

2) For the data set shown below, do they following. (a) Compute the standard error the point estimate for σ. (b) Assuming the residuals are normally distributed, determine Sb1. (c) Assuming the residuals are normally distributed, test H0;β1=0 versus H1; β1≠0 at the a=0.05 level of significance.

X 20 30 40 50 60

Y 98 95 91 81 68

a) Determine the point estimate for σ. Se = (…….)(Do not round until the final answer. Then round to four decimal places as needed)

b) Assuming the residuals are normally distributed, determine Sb1.

c) In order to conduct the hypothesis test, first calculate the test statistic t0.(Explain the result; rejected or do not reject; do the relation exists between x and y?).

3) A concrete cures it gains strength. The following data represent 7 days and 28 day strength in pounds per square inch (psi) of a certain type of concrete.

7 day strength psi, x 3330 2620 3380 2300 2480

28 day strength psi, y 4850 4190 5020 4070 4120

a) Compute the standard error of the estimate, Se. Se≈(……) (Do not round until the final answer, Then round to four decimal places as needed)

b) Determine whether the residuals are normally distributes

c) If the residuals are normally distributed, determine Sb1

d) If the residuals are normally distributed, test whether a linear relation exists between 7 day strength and 28 day strength a= 0.05 level of significance.

e) If the residuals are normally distributed construct a 95% confidence interval about the slop of the true least squares regression line.

f) What is the estimate mean 28 day strength of this concrete if the 7 day is 3000 psi?

4) Supposed a multiple regression model is given by ŷ= -3.98×1 – 9.03×2 – 37.73. What would an interception of the coefficient of x1 be? Fill in the blank below

An interception of the coefficient of x1 would be “if x1 increases by 1 unit, then the response variable will decrease by (……) units, on average, while holdingx2 constant.”

5) The regression equation ŷ = 56,777.848 + 4401.106x – 46.38 describe the median income, y, at different ages, x, for residents of a certain country. Complete parts (a) through (c)

(a) And (b) have been done

(c)How much does median income change from 40 to 50 years of age?

The median income (Increases / decreases) by $(……..) from 40 to 50 of age (Round to the nearest cent as needed)

6) Researchers develop a model to predict the age, y, of the individual based on the gender of the individual ,x1 (0=female, 1= male), the height of the second premolar, x2, the number of teeth with root development, x3, and sum of the normalized heights of seven teeth on the left side of the month, x4. The normalized height of the seven teeth was found by dividing the distance between teeth by the height of the tooth. Their model is shown below. Complete parts (a) through (d).

Ŷ= 9.139 + 0.392×1 + 1.256×2 + 0.678×3 – 0.90×4 – 0.174x3x4

(a) Based on this model, what is the expected age of a female with x2 = 27mm, x3=9, and x4= 16mm? (………) years.( Round to one decimal place as needed)

(b) Based on this model, what is the expected of a male with x2=27, x3=9 and x4=16mm

(c) What is the interaction term? What variable interact?

(d) The Coefficient of determination for this model is 85.5%. Explain what this means.

7) Determine if there is a linear relation among air temperature x1, wind speed x2, and wind child y. The following data show the measurement value for various days.

X1 -10 30 0 -30 20 0 30 30 -20 -30 -30 -20 10 -10

X2 50 30 10 100 90 40 20 50 30 50 70 40 80 20

Y -62 0 -22 -104 -32 -44 6 -8 -64 – 88 -98 -70 -44 -44

(a) Find the least squares regression equation Ŷ= b0 + b1x1 + b2x2, where x1 is air temperature and x2 is wind speed, and y is the respond variable, wind chill.

Ŷ= (……) + (…….)x1 + (…….)x2 (Round to three decimal places as needed).

(b) Draw residual plots to assess the adequacy of the model. (Create a residual plot for air temperature and for wind speed).

What might you conclude based on the plot of residuals against wind speet.