
State the null and alternative hypothesis.
Find the degrees of freedom for the
Find the critical value of chi-square from Appendix E or from Excel’s
Calculate the chi-square test statistics at 0.10 level of significance and interpret the p-value.
Check whether the conclusion is sensitive to the level of significance chosen, identify the cells that contribute to the chi-square test statistic and check for the small expected frequencies.
Calculate the p-value using Excel’s function if necessary.
Perform a two-tailed, two-sample z test for

Answer to Problem 21CE
The null hypothesis is:
And the alternative hypothesis is:
The degrees of freedom for the contingency table is 1.
The critical-value using EXCEL is 2.706.
There is no enough evidence to conclude that grade and order handed in are independent.
The p-value for the hypothesis test is 0.382.
The conclusion is not sensitive to the level of significance chosen.
The cells (1, 1) and (1, 2) contribute the most to the chi-square test statistic.
There is no expected frequencies that are too small.
It is verified that
Explanation of Solution
Calculation:
The table summarizes the grade and order of papers handed in.
The claim is to test whether the data provide sufficient evidence to conclude that the grade and order handed in are independent. If the claim is rejected, then the grade and order handed in are not independent.
The test hypotheses are given below:
Null hypothesis:
Alternative hypothesis:
The degrees of freedom can be obtained as follows:
Substitute 2 for r and 2 for c.
Thus, the degrees of freedom for the contingency table is 1.
Procedure for critical-value using EXCEL:
Step-by-step software procedure to obtain critical-value using EXCEL software is as follows:
- Open an EXCEL file.
- In cell A1, enter the formula “=CHISQ.INV.RT(0.10,1)”
- Output using EXCEL software is given below:
Thus, the critical-value using EXCEL is 2.706.
Test statistic:
Software procedure:
Step by step procedure to obtain the chi-square test statistics and p-value using the MINITAB software:
- Choose Stat > Tables >Cross Tabulation and Chi-Square.
- Choose Row data (categorical variables).
- In Rows, choose Grade.
- In Columns, choose Order handed in.
- In Frequencies, choose Count.
- In Display, select Counts.
- In chi-square, select Chi-square test, Expected cell counts and Each cell’s contribution to chi-square.
- Click OK.
Output using the MINITAB software is given below:
Thus, the test statistic is 0.764 and the p-value for the hypothesis test is 0.382.
Rejection rule:
If the p-value is less than or equal to the significance level, then reject the null hypothesis
Conclusion:
Here, the p-value is greater than the level of significance.
That is,
Therefore, the null hypothesis is not rejected.
Thus, the data provide sufficient evidence to conclude that the grade and order handed in are independent.
Take
Here, the p-value is greater than the level of significance.
That is,
Therefore, the null hypothesis is not rejected.
Thus, the data provide sufficient evidence to conclude that the grade and order handed in are independent.
Thus, the conclusion is same for both the significance levels.
Hence, the conclusion is not sensitive to the level of significance chosen.
The cells (1, 1) and (1, 2) contribute most to the chi-square test statistic.
Since all
Two-tailed, two-sample z test:
The test hypotheses are given below:
Null hypothesis:
Alternative hypothesis:
The proportion of students who finish an exam first and get “B or better” on the exam is denoted as
Where
The proportion of students who finish later and get “B or better” on the exam is denoted as
Where
The pooled proportion is denoted as
Test statistic:
The z-test statistics can be obtained as follows:
Thus, the z-test statistic is 0.87.
The square of the z-test statistic is,
Thus the square of the z-test statistic is same as the chi-square statistics.
Procedure for p-value using EXCEL:
Step-by-step software procedure to obtain p-value using EXCEL software is as follows:
- Open an EXCEL file.
- In cell A1, enter the formula “=2*(1-NORM.S.DIST(0.8740,1))”
- Output using EXCEL software is given below:
Thus, the p-value using EXCEL is 0.382, which is also same as the p-value obtained in chi-square test.
Thus, it is verified that
Want to see more full solutions like this?
Chapter 15 Solutions
Applied Statistics in Business and Economics
- Question 1. Your manager asks you to explain why the Black-Scholes model may be inappro- priate for pricing options in practice. Give one reason that would substantiate this claim? Question 2. We consider stock #1 and stock #2 in the model of Problem 2. Your manager asks you to pick only one of them to invest in based on the model provided. Which one do you choose and why ? Question 3. Let (St) to be an asset modeled by the Black-Scholes SDE. Let Ft be the price at time t of a European put with maturity T and strike price K. Then, the discounted option price process (ert Ft) t20 is a martingale. True or False? (Explain your answer.) Question 4. You are considering pricing an American put option using a Black-Scholes model for the underlying stock. An explicit formula for the price doesn't exist. In just a few words (no more than 2 sentences), explain how you would proceed to price it. Question 5. We model a short rate with a Ho-Lee model drt = ln(1+t) dt +2dWt. Then the interest rate…arrow_forwardIn this problem, we consider a Brownian motion (W+) t≥0. We consider a stock model (St)t>0 given (under the measure P) by d.St 0.03 St dt + 0.2 St dwt, with So 2. We assume that the interest rate is r = 0.06. The purpose of this problem is to price an option on this stock (which we name cubic put). This option is European-type, with maturity 3 months (i.e. T = 0.25 years), and payoff given by F = (8-5)+ (a) Write the Stochastic Differential Equation satisfied by (St) under the risk-neutral measure Q. (You don't need to prove it, simply give the answer.) (b) Give the price of a regular European put on (St) with maturity 3 months and strike K = 2. (c) Let X = S. Find the Stochastic Differential Equation satisfied by the process (Xt) under the measure Q. (d) Find an explicit expression for X₁ = S3 under measure Q. (e) Using the results above, find the price of the cubic put option mentioned above. (f) Is the price in (e) the same as in question (b)? (Explain why.)arrow_forwardThe managing director of a consulting group has the accompanying monthly data on total overhead costs and professional labor hours to bill to clients. Complete parts a through c. Question content area bottom Part 1 a. Develop a simple linear regression model between billable hours and overhead costs. Overhead Costsequals=212495.2212495.2plus+left parenthesis 42.4857 right parenthesis42.485742.4857times×Billable Hours (Round the constant to one decimal place as needed. Round the coefficient to four decimal places as needed. Do not include the $ symbol in your answers.) Part 2 b. Interpret the coefficients of your regression model. Specifically, what does the fixed component of the model mean to the consulting firm? Interpret the fixed term, b 0b0, if appropriate. Choose the correct answer below. A. The value of b 0b0 is the predicted billable hours for an overhead cost of 0 dollars. B. It is not appropriate to interpret b 0b0, because its value…arrow_forward
- Using the accompanying Home Market Value data and associated regression line, Market ValueMarket Valueequals=$28,416+$37.066×Square Feet, compute the errors associated with each observation using the formula e Subscript ieiequals=Upper Y Subscript iYiminus−ModifyingAbove Upper Y with caret Subscript iYi and construct a frequency distribution and histogram. LOADING... Click the icon to view the Home Market Value data. Question content area bottom Part 1 Construct a frequency distribution of the errors, e Subscript iei. (Type whole numbers.) Error Frequency minus−15 comma 00015,000less than< e Subscript iei less than or equals≤minus−10 comma 00010,000 0 minus−10 comma 00010,000less than< e Subscript iei less than or equals≤minus−50005000 5 minus−50005000less than< e Subscript iei less than or equals≤0 21 0less than< e Subscript iei less than or equals≤50005000 9…arrow_forwardThe managing director of a consulting group has the accompanying monthly data on total overhead costs and professional labor hours to bill to clients. Complete parts a through c Overhead Costs Billable Hours345000 3000385000 4000410000 5000462000 6000530000 7000545000 8000arrow_forwardUsing the accompanying Home Market Value data and associated regression line, Market ValueMarket Valueequals=$28,416plus+$37.066×Square Feet, compute the errors associated with each observation using the formula e Subscript ieiequals=Upper Y Subscript iYiminus−ModifyingAbove Upper Y with caret Subscript iYi and construct a frequency distribution and histogram. Square Feet Market Value1813 911001916 1043001842 934001814 909001836 1020002030 1085001731 877001852 960001793 893001665 884001852 1009001619 967001690 876002370 1139002373 1131001666 875002122 1161001619 946001729 863001667 871001522 833001484 798001589 814001600 871001484 825001483 787001522 877001703 942001485 820001468 881001519 882001518 885001483 765001522 844001668 909001587 810001782 912001483 812001519 1007001522 872001684 966001581 86200arrow_forward
- For a binary asymmetric channel with Py|X(0|1) = 0.1 and Py|X(1|0) = 0.2; PX(0) = 0.4 isthe probability of a bit of “0” being transmitted. X is the transmitted digit, and Y is the received digit.a. Find the values of Py(0) and Py(1).b. What is the probability that only 0s will be received for a sequence of 10 digits transmitted?c. What is the probability that 8 1s and 2 0s will be received for the same sequence of 10 digits?d. What is the probability that at least 5 0s will be received for the same sequence of 10 digits?arrow_forwardV2 360 Step down + I₁ = I2 10KVA 120V 10KVA 1₂ = 360-120 or 2nd Ratio's V₂ m 120 Ratio= 360 √2 H I2 I, + I2 120arrow_forwardQ2. [20 points] An amplitude X of a Gaussian signal x(t) has a mean value of 2 and an RMS value of √(10), i.e. square root of 10. Determine the PDF of x(t).arrow_forward
- Big Ideas Math A Bridge To Success Algebra 1: Stu...AlgebraISBN:9781680331141Author:HOUGHTON MIFFLIN HARCOURTPublisher:Houghton Mifflin HarcourtGlencoe Algebra 1, Student Edition, 9780079039897...AlgebraISBN:9780079039897Author:CarterPublisher:McGraw HillHolt Mcdougal Larson Pre-algebra: Student Edition...AlgebraISBN:9780547587776Author:HOLT MCDOUGALPublisher:HOLT MCDOUGAL


