MATH HOMEWORK STATISTICS

MAT 222 (Spring 2016) HW Problems (for chapters 9, 10, and 11)

Chapter 9

From the Textbook:

9.25, 9.26, 9.27, 9.28, 9.31, 9.32, 9.33, 9.34, 9.40, 9.42, 9.55

Additional Problems:

1. (From MAT222 Spring 2006 Final Exam) A company’s workforce was cross-classified according to job type and

gender. That classification appears in the following contingency (two-way) table.

Men Women Total

Management/Sales 39 47 86

Research/Development 28 22 50

Labor 188 116 304

Total 255 185 440

Using a Chi-Square test and the 0.05 level of significance, determine if there is a significant association between

gender and the job type. Your test should include appropriate hypotheses, the computation of a test statistic,

mention of the number of degrees of freedom involved, the p-value, a decision (reject H0 or not) with

justification, and a conclusion.

2. A research study about student retention is interested in assessing whether there is an association between the

students’ initial major and the program they transferred to. The data collected by the study is as follows:

Initial Major

Program transferred to

Total Engineering Management Liberal Arts Other

Biology 13 25 158 202 398

Chemistry 16 15 19 64 114

Mathematics 3 11 20 38 72

Physics 9 5 14 33 61

Total 41 56 211 337 645

a. Complete the following table of expected counts:

Initial Major

Program transferred to

Engineering Management Liberal Arts Other

Biology 25.30 130.20 207.95

Chemistry 9.90 37.29 59.56

Mathematics 4.58 6.25 37.62

Physics 3.88 5.30 19.96

b. Use Chi-Square test with the significance level 1% to check whether there is a significant relationship

between the students’ initial major and the program they transferred to. Provide the hypotheses, the

test statistic, its degrees of freedom, the p-value, and the conclusion in context.

3. If both parents have hybrid genotypes Aa, a simplified genetic theory predicts that the child will have exactly

one of the genotypes AA, Aa, and aa with respective probabilities 0.25, 0.50, and 0.25. In a random sample of 24

offsprings each having both parents with hybrid genotype Aa, it was found that 10 had genotype AA, 10 had

genotype Aa, and 4 had genotype aa. By carrying out a Chi-Square goodness of fit test on this sample data, at

level of significance α= 0.10, investigate whether the simplified genetic theory is incorrect. Clearly indicate the

hypotheses, the test statistic, its degrees of freedom, the p-value, and the conclusion in context.

4. A forestry researcher is interested in assessing whether the longleaf pine trees are distributed randomly across a

certain tract. The tract was divided into four equal quadrants. A random sample of 100 longleaf pine trees was

taken and the number of trees in each quadrant was counted. Below are the data:

Quadrant Number of Trees

Q1 18

Q2 22

Q3 39

Q4 21

a. If the longleaf pine trees are randomly distributed, we expect to find equal number of trees in each

quadrant. Find the expected count for each quadrant.

b. Perform a goodness of fit test to determine if the longleaf pine trees are randomly distributed. Clearly

indicate the null hypothesis, the test statistic, its degrees of freedom, and the p-value.

c. Based on the answer to the previous part, is the evidence significant at level 10%? At level 5%? At level

1%?

Chapter 10

From the Textbook (problems marked with * are recommended to be done with MINITAB):

10.8, 10.9, 10.16*, 10.17*, 10.18*, 10.19*, 10.21*, 10.22*, 10.23*, 10.30*, 10.39*, 10.40*, 10.44, 10.45, 10.46, 10.47, 10.51*, 10.52*

Additional Problems:

1. A regression model was run to predict the SAT score based on the ACT score. Partial minitab output is below:

The regression equation is

sat = ________________________________________________

Predictor Coef SE Coef T P

Constant 253.19 62.67 4.04 0.000

act 31.206 2.895 _____ 0.000

S = ______________ R-Sq = __________% R-Sq(adj) = 66.1%

Analysis of Variance

Source DF SS MS F P

Regression __ 1276586 _______ _______ 0.000

Residual Error __ _______ 10989

Total 59 1913973

a. Fill in the missing parts in the output above. b. How many observations were included in this analysis? c. Provide the details of a t-test to assess whether ACT score is a significant predictor of SAT score. State

the hypotheses. Provide the t-statistic, its degrees of freedom, and the p-value. State the conclusion in context at α=5%.

d. Find the predicted SAT score for an ACT score of 20. e. Find the 95% confidence interval for the mean response of ACT score of 20 (i.e., for the average SAT

score among all students having ACT score x* = 20). Note 1: For degrees of freedom = 58 and 95% confidence level, t* is 2.002. Note 2: The following descriptive statistics provide the sample mean and the sample standard deviation of the predictor variable:

Descriptive Statistics: act Variable Mean StDev

act 21.133 4.714

2. A regression model was run to predict the “Assessed Value” of a house based on its “Sales Price”. Partial minitab

output is below:

Regression Analysis: Assessed Value versus Sales Price

The regression equation is

Assessed Value = _____________________________________

Predictor Coef SE Coef T P

Constant 49.87 28.53 1.75 0.091

Sales Price 0.7312 0.1386 ________ 0.000

S = __________ R-Sq = _______%

Analysis of Variance

Source DF SS MS F P

Regression __ 29462 29462 27.85 0.000

Residual Error __ _____ 1058

Total 29 59084

a) Fill in the missing parts in the output above.

b) How many observations were used in this analysis?

c) What is the correlation coefficient “r” between the variables “Assessed Value” and “Sales Price”

d) Is the variable “Sales Price”a significant predictor of “Assessed Value”? Answer this question by providing the details of a t-test. State the hypotheses to test, provide the t-statistic, and provide the p-value. State the conclusion of whether or not “Sales Price” is a significant predictor of “Assessed Value” at α=5%.

e) Provide a 95% confidence interval for β1. Clearly indicate the degrees of freedom and t*.

Chapter 11

From the Textbook (problems marked with * are recommended to be done with MINITAB):

11.7, 11.8, 11.9, 11.10, 11.11, 11.12, 11.17, 11.23*, 11.24*, 11.25*, 11.27*, 11.28*, 11.33*, 11.34*, 11.35*

Additional Problems:

1. A regression model was run to predict GPA based on HSM, HSE, and SATM. Partial minitab output is below:

Regression Analysis: GPA versus HSM, HSE, SATM

Predictor Coef SE Coef T P

Constant -1.0488 0.6024 -1.74 0.084

HSM 0.13765 0.04801 ____ 0.005

HSE 0.13555 0.05290 ____ 0.011

SATM 0.0024241 0.0008276 ____ 0.004

S = ________ R-Sq = 25.2% R-Sq(adj) = _____%

Analysis of Variance

Source DF SS MS F P

Regression ___ 25.0856 8.3619 _____ 0.000

Residual Error ___ 74.5893 0.5109

Total 149 99.6749

a) Fill in the missing parts in the output above.

b) Using the model given above, find the predicted GPA for a student with HSM=10, HSE=10, and SATM=640.

c) State the null and the alternate hypotheses for the F-test in the Analysis of Variance table.

d) Is the variable “HSE”a significant predictor of “GPA”? Answer this question by providing the details of a t-

test. State the hypotheses to test (in terms of one of β1, β2, or β3), provide the t-statistic, and provide the p-value. State the conclusion of whether or not “HSE” is a significant predictor of “GPA” at α=5% and at α=1%.

e) Provide a 95% confidence interval for β3, the slope of the predictor “SATM”. Clearly indicate the degrees

of freedom. Then circle the appropriate cell in the table below that can be used as t*. Then compute the confidence interval.

2. A study1 about the birth weight analyzed related variables. Study was based on a sample of all births occurring in

Philadelphia in 1990. The following regression model was run on two predictor variables (“YrsEduc” - Mother's

years of education and “GestWks” -Gestational age in weeks).

Regression Analysis: BirthWt versus YrsEduc, GestWks The regression equation is

BirthWt = ______________________________________

Predictor Coef SE Coef T P

Constant -3.4806 0.2070 -16.81 0.000

YrsEduc 0.022696 0.006429 3.53 0.000

GestWks 0.165333 0.005044 32.78 0.000

S = _________ R-Sq = 49.9% R-Sq(adj) = 49.8%

Analysis of Variance

Source DF SS MS F P

Regression __ 222.90 ______ _______ 0.000

Residual Error 1112 ______ ______

Total ____ 447.10

a) Find the missing parts in the output above. b) How many observations were used in this study? c) One of the data points had YrsEduc = 8, GestWks = 40, and BirthWt = 3.51. Find the predicted value and

the residual corresponding to this observation.

1 I.T. Elo, G. Rodríguez and H. Lee (2001). Racial and Neighborhood Disparities in Birthweight in Philadelphia. Paper presented at the Annual Meeting of the Population Association of America, Washington, DC 2001. Downloaded from: http://data.princeton.edu/wws509/datasets/