Describe the three properties (characteristics) of a sampling distribution of the sample mean which include properties on the shape, mean and standard error

This file includes the guidelines/instructions on the homework as well as Homework Problems

 

Students can use IS 242 Business Statistics or Basic Business Statistics by Berenson, et al 13th ed. The homework file shows page numbers of both books, IS 242 Business Statistics and the original text – Basic Business Statistics by Berenson, Levine, Szabat 13th edition. Where BB means Basic Business Statistics 13th ed by Berenson et al.

 

There are 4 sets of homework, one set for each block. Some problems in the homework require using the Minitab – version 19 program and data files.

 

  • Minitab program is installed on some PCs on the SCSU campus.
  • Off-campus, users can access the Minitab program through AppsAnywhere-2 or buy it from OnTheHub, https://stcloudstate.onthehub.com/WebStore/Welcome.aspx
  • All the data files for the Homework are posted on the D2L in Excel Format Data for Homework or Minitab Format Data for Homework Download the Excel/Minitab data files on your USB (or a PC with the Minitab program installed). Log on a Campus PC or your PC and open Minitab 19, or access Minitab 19 through AppsAnywhere-2. Refer to the “Accessing Minitab” file posted on D2L.
  • After opening Minitab-19, follow the steps explained in each problem. For additional information refer to the “Minitab Basics” file, and Lecture Notes (for a specific Minitab task) posted on D2L, or contact Technology Help at SCSU E-mail: huskytech@stcloudstate.edu; Phone: (320) 308-7000 for assistance on AppsAnywhere-2.

 

All your homework is due at the period in the schedule by submitting it in the submission box on D2L. On your syllabus, the due date for individual blocks is indicated as well as on the corresponding submission box. The submission box will be closed after the due date. Before submitting homework, check carefully to make sure your submission answers the questions by following the guides provided.

 

Late homework should be submitted by email. The submission box will be closed after the due date. The email includes the reason for the late submission to be considered, and any permission in advance if it exists as well as homework answers. All late homework will be subjected to a penalty rate of my discretion. The penalty rate depends on the reason for late submission, permission, and timing of submission. No homework without permission in advance or without a legitimate reason will be accepted after 5 days from the due date or the last day of class.

 

The homework should be submitted in an MS Word format file. Use copy and paste to include Minitab outputs and graphs in the Word file to submit. No screenshots and pictures of Minitab outputs and graphs and handwritten answers except for a few notations are accepted for submission. Homework should be independent work for each student. All the submitted contents will be evaluated and graded. Any violations of homework guides, unnecessary or inconsistent content will be subjected to a penalty rate of the instructor’s discretion.

 

Submit all questions in each block in one file/packet with clear identification information such as the name of the student, course number with section number, homework number, problem number, sub-number, etc. No pre-submission/submission draft for verification or confirmation, nor re-submission for re-grading is allowed. Submit an answer to each problem/question with your works if you want, but do not submit a copy of the problems or instructions.

 

 

Block 1 Homework Problems (100 pts):

 

  • Describe the differences with one example each between descriptive statistics and inferential statistics, limit the answer including the description of differences and examples with 50 or fewer words, a penalty would be applied against an answer with more than 50 words (20 pts).
  • Show 2 examples of statistics, and 2 examples of parameter; the examples should be in notation not in word (20 pts)

 

1.3 What is the level of measurement for each of the following variables (20 pts)?

  1. The temperature of patients in Fahrenheit
  2. Player numbers of Vikings football team
  3. Heights in inches

1.4 For the Retirement Funds file (20 pts).

 

  1. Show descriptive statistics of 1YrReturn%, and
  2. A Boxplots of 1YrReturn% in various types (growth and value) computed/created by Minitab 19.

 

To get the needed Minitab output for 1.4:

 

Descriptive statistics and Box plot of Retirement Funds

 

Log on to a PC with Minitab-19 installed, or access Minitab-19 through AppsAnywhere -2.

 

At File, click Open, look for Minitab data or Excel data file downloaded from D2L in USB, select Retirement Funds & click Open> if it says unsupported format file and the file does not open then Open Excel format Retirement Funds file (Open an Excel format file, Minitab 19 may not open a Minitab format file),

 

You are ready to Minitab analysis on the file you opened ie Retirement Funds data

 

Go to the top line menu,

click Stat à Basic statistics à Display Descriptive Statistics

 

In the dialog box, double-click the name of the column containing data (1YrReturn%) in the left column.

 

In the dialogue box, click OK or Statistics to check the statistics you want

 

The Minitab pane shows output (results), Select (mark) results, copy and paste into Word file.

 

 

Boxplots:

 

With the Retirement funds file open,

Select Graph à boxplot

 

Choose one Y with groups > OK

 

Double click 1YrReturn% in the variable list to put 1YrReturn% in the Graph Variables box

 

Click Categorical variables box, and double-click Type in the variable list to add type in Categorical variables box

 

Click OK

 

Right-click on the Boxplot, copy graph and paste on the Word file to submit.

 

1.5 For these questions refer to The Normal Distribution and Other Continuous Distributions Chapter (Ch 6. Section 2). (20 pts).

 

When X has a normal distribution of mean = µ and standard deviation = σ, then z has the standard normal distribution, where z = (X – µ)/ σ and use the cumulative Standardized Normal Distribution table in Appendix or D2L Statistics Tables to determine probability on z and X.

 

For the standard normal distribution, determine the following probabilities:

 

  1. P(z < 1.05)
  2. P ( -1.96 < z < 1.96)

 

For x has a normal distribution with mean =100 and standard deviation = 10, determine the following probabilities:

 

  1. P( x   < 85)
  2. P( x < 80 or x > 125)

 

Block 2 Homework Problems (100 pts):

 

2.1   Describe the three properties (characteristics) of a sampling distribution of the sample mean which include properties on the shape, mean and standard error, limit the answer for 3 properties with 50 or fewer words, a penalty would be applied against an answer with more the word limit (20 pts). Refer to 7.2 of text and Lecture Notes posted on D2L.

 

  1. Property on the shape:
  2. Property on the mean:
  3. Property on the standard error:

 

2.2 The following problems are related to Section 2 in Sampling Distributions chapter and also similar to problem 1 or 2 on page 295 (Problem 7.1 page 261 in BB – Book by Berenson et al).

 

Note that when X has a normal distribution of mean = µ and standard deviation = σ, then z has the standard normal distribution, where z = ( ͞χ – µ)/ s/   and   ͞χ is mean of X. Use this and the standard normal distribution table to answer the questions.

 

Given a normal distribution of µ = 100 σ = 10 if you select sample of n=25, what is the probability that x̅ is (10 pts)

 

  1. Less than 95?
  2. between 95 and 97.5?
  3. above 90?
  4. there is a 65% chance that  x̅ is above what value?

 

.

2.3 (20 pts)

 

  1. If x̅ = 85,   σ = 8, and n = 64, constrict a 95% confidence interval estimate for the population mean, µ.
  2. If x̅ = 125,   σ = 24, and n = 36, constrict a 99% confidence interval estimate for the population mean, µ. (a and b are problems from Section 1 in Confidence Interval Chapter),

 

  1. If x̅ = 75,   s = 24, and n = 36, and assuming that the population is normally distributed, constrict a 95% confidence interval estimate for the population mean, µ. (Problems from Section 2 in Confidence Interval Chapter) and

 

 

  1. problems 34 on page 332 (Problem 8.34 on page 294 of BB)

 

  1. problems 35 on page 332 (Problem 8.35 on page 294 of BB) (d and e are problems from Section 4 in Confidence Interval Chapter)

 

 

Refer to the confidence interval is given as

 

σ is known, use     ͞χ     ±   zα/2  s/

For a 90% confidence interval use zα/2  = z 0.05 = 1.645

For a 95% confidence interval use zα/2  = z 0.25 = 1.96

For a 99% confidence interval use zα/2  = z 0.005 = 2.58

 

σ is unknown, use     ͞χ   ±     tα/2 , df  s/

Use tα/2 , df  from t-table with df = n-1 (df is degrees of freedom)

 

Sample size n = [ zα/2 σ/e]2  where zα/2 from a confidence level, e=size of error and σ population standard deviation; Specify the level of accuracy in terms of error (e) and confidence level and determine the required sample size.

 

2.4 Do problem 13 on page 363 (Problem 9.13 page 321 in BB) (Problems for Section 1 in Fundamentals of hypothesis testing Chapter), limit the answer for 2.3 with 50 or fewer words (10 pts)

 

2.5 Confidence intervals; use MINITAB for the FORCE data file which is in the Minitab Data for Homework, construct confidence intervals of the population mean force (90%, 95%, and 99%) using t-distribution (10 pts).

To get the Minitab output:

To open Force file on a PC with Minitab 19:

 

Select.File à Open, select the data file (Force) à click open and OK( if Minitab format file does not open, open Excel format file)

 

Go to the top-line menu, and click Stat à Basic statistics à 1 Sample t

 

In the dialog box, choose One or more Samples, each in a column from the top-right pulldown menu box, click the second-right box then Force appears in the left box, and double-click Force in the left box, to copy it in the second-right box.

 

Click Options in the dialogue box (1-Sample t – Options) enter 90.0 in the Confidence level Box.

 

Click Ok and Ok.

 

Go to the top-line menu and repeat for confidence levels 95.0 and 99.0

 

The session window shows output (results), Select (mark) results, copy and paste on Word file to submit.

 

 

  1. 6. Hypothesis tests –Use Minitab, answer problem 30, part a) on page 368 (Problem 9.30 page 326 in BB) (applying the concepts in Fundamentals of hypothesis testing Chapter), use both t (T-value) and p-value for DRINK file use α = 0.05. (30 pts)

 

  1. Submit your own Minitab 19 output showing both t and p-value.

 

  1. Test the hypothesis using the t with following the 5 steps (combine steps 3 and 4 together, then the six-step The Critical Value Approach in pages 364-5 (page 322-324 in BB) becomes a five-step approach), and

 

  1. Repeat the test hypothesis using p-Value with the following 5 steps as on page 366 in the P-value approach.

 

To get the Minitab output:

 

To 0pen Drink file in a PC with Minitab 19:

Select.File à Open, select the data file (Drink) à click open and OK

 

Go to topline menu – click Stat à Basic statistics –> 1 Sample t

 

In the dialog box, choose One or more Samples, each in a column from the top-right pulldown menu box, click the second-right box then Amount appears in the left box, and double-click Amount in the left box to enter the name of the column Amount to the second-right box (or simply type in Amount)

 

Check to Perform hypothesis test, type 2 in the Hypothesized mean box

Click Ok

 

 

You will get the following output (T-Value and P-value are deleted intentionally)

 

 

 

Descriptive Statistics

N Mean StDev SE Mean 90% CI for μ
50 2.00072 0.04456 0.00630 (1.99015, 2.01129)

μ: mean of Amount

Test

Null hypothesis H₀: μ = 2
Alternative hypothesis H₁: μ ≠ 2
T-Value P-Value  
———– ———  

 

 

 

 

 

For the using t-value and p-value for the hypothesis test by following 5-step, refer to the Lecture Notes Block 2.

Both methods using t or p-value for the same set of data, should come up with the same conclusion.

 

 

Block 3 Homework Problems (100 pts):

 

3.1 χ 2 – Test of Independence (30 pts) Refer 12.3 in the textbook

 

Perform Hypothesis test with:

 

H0: Hotels and the reason for not returning are independent -not related vs.

H1: Hotels and reasons are not independent–related

 

To get the Minitab output (refer to Figure 10 page 411 (or Figure 12.10 page 465 in BB) in Chapter 12 Chi-Square and Nonparametric Tests),

 

Log on to a PC with Minitab 19 or access Minitab 19 through AppsAnywhere-2.

 

To get the Minitab output:

 

Click File > Open and select open PalmHotels data file ( if Minitab format file does not open, open the excel format file) to the worksheet; select Stat > Tables > χ 2 Chi-SquareTest for Association,. In the χ 2 – Test dialog box, choose Summarized data in a two-way table in the top-right pulldown menu, click Columns containing the table box, then variable names appear in the left box, enter the names of a column by double-clicking each variable name in the left box, and click Ok. Minitab generates an output on the output pane

 

Refer to the Minitab output of version 16 on the right side of Figure 10 page 411 (Figure 12.10 page 465 in BB) in the textbook Chi-square and Nonparametric Tests chapter, the output of the version19 is in the Lecture Notes file

 

From the Minitab output, use the χ 2 (Chi-square, Pearson) and p-Value to perform 5- step hypothesis tests using α = 0.05, and submit:

 

  1. Minitab 19 output
  2. 5-step Hypothesis tests using X2,
  3. 5-step Hypothesis tests using p-value.

 

Please refer to “5-step for hypothesis test using chi-sq and p-value” in the Lecture Notes for Block 3-4 file in the Lecture Notes Module posted on D2L.

 

3.2 Regression outputs (30 pts).

 

Use MINITAB, get regression output of Table 1 on page 444 (Table 13.1 page 494 in BB) in Simple Linear Regression Chapter (use SiteSelection data file). Annual Sales is y (Response), and Profiled Customers (both in millions) is x (Predictors). Also, predict Annual Sales when the Profiled Customers is 4.

 

To get Minitab output:

 

Log on a PC with Minitab 19 or access Minitab 19 through AppsAnywhere-2, and open SiteSelection data (if Minitab format file does not open, open Excel format file)

 

select Statà Regression à Regression à Fit Regression Model In the dialogue box, double-click Annual Sales to copy it to top-right box, move the cursor to Continuous predictors box and double-click Profiled Customers, and OK, copy and paste regression output into a Word file and continue for prediction below

 

For prediction select Statà Regression àRegression àPredict In the dialogue box, enter 4 in the Profiled Customers, and Ok

 

Copy and paste prediction output into Word file to submit.   Navigator Pane (left pane) shows the list of all Minitab outputs, to see an output, click it in the list

 

Your submission includes

 

  1. complete Minitab output of regression
  2. regression equation, r2, predicted sales for Profiled Customers is 4
  3. 5-step Hypothesis tests on slope β1 using t with α = 0.05, and
  4. 5-step Hypothesis tests on slope β1 using p with α = 0.05

 

 

Please refer to the steps for hypothesis test using t and p-Value in the Lecture Notes for Block 3-4 file in Lecture Notes Module posted on D2L.

 

 

3.3 Table 1 on page 498 (page 544 in BB) (Developing multiple regression Model in Introduction to Multiple regression Chapter), use OMNIPower data file and Minitab, get regression output. Predict sales at price =79 and promotion = 400 (30 pts).

 

To get Minitab output:

 

To get the Minitab output of OMNIPower, follow similar procedures with SiteSelection data

 

After opening Minitab, File -> Open by selecting OMNIPower, clicking open and OK (the PC should have Minitab installed) ( or copy and Paste cells for an Excel file),

 

 

select Statà Regression à Regression -> Fit regression Model. In the dialogue box, double-click Sales, and move the cursor to the continuous predictors box, double-click Price, promotion (sales is in the response box, and price, promotion are in the Continuous Predictors). OK, copy and paste regression output into a Word file and continue for prediction below

 

For prediction select Statà Regression àRegression àPredict In the dialogue box, enter 79 400 in the price, Promotion. Ok

 

Copy and paste prediction output into Word file to submit. Navigator Pane (left pane) shows the list of all Minitab outputs, to see an output, click it in the list.

 

Submit:

 

  1. Minitab output including predicted value,
  2. Perform a Hypothesis test for the significance of the overall model by F use α = 0.05, and
  3. Perform a Hypothesis test for the significance of the overall model by p-value, use α = 0.05.

 

Refer to the Steps for hypothesis test using F and p-Value in the Lecture Notes for Block 3-4 file in Lecture Notes Module posted on D2L in Multiple Regression Chapter

 

 

3.4 Indicator variables or Dummy Variables (10 pts) Refer 14.6 in textbook

 

To get regression outputs of SilverSpring data file:

 

After opening Minitab 19, File -> Open by selecting SilverSpring, clicking open

select Statà Regression à Regression -> Fit Regression Model  In the dialogue box, double-click Asking Price, and move cursor to continuous predictors box, double-click Size, Fireplace Coded in continuous Predictors. Click OK.

 

copy and paste the output into a Word file to submit.

.

 

Block 4 Homework Problems (100 pts):

 

4.1 Use MINITAB, construct X-bar chart and R chart for the data in Hotel2 file (20 pts). Interpret the results.

 

To get the Minitab output:

 

After opening Minitab 19, File -> Open, select Hotel2, click open and OK. (The PC should have Minitab installed or downloaded by AppsAnywhere-2).

 

Select Statà Control Chartsà Variable chart for Subgroups à R Chart, In the dialogue box, top of right-side pulldown menu, select Observations for a subgroup are in one row of column, Click the second-right box below, double-click C2 C3 C4 C5 C6 in the left box, next click R-Options, in the R Chart-Options dialog box, click Tests tab and choose to Perform all tests for special causes from the drop-down list; click Limits tab and enter 1 2 3, there is a space between numbers 1 2 and 3, OK, and OK

 

To enter variables by double-clicking each variable or c2-c6, if you don’t see the variable list, click the second-right box. Limits command subdivides the control chart into Zone A, B, and C.

After R-Chart, do X-bar chart, always do R-Chart first (replace R by X-bar, and follow the above steps).

 

Copy control chart and session window, and paste into a Word file to submit.

 

Submit:

 

  1. A copy of control chart and session window (output pane)
  2. Interpretation of results

 

4.2 Construct p-chart for the data in ERRORSPC file (20 pts). Interpret the results

 

To get Minitab output:

 

Open ERRORSPC file. Select Stat à Control Charts à Attributes Charts à P.

 

In the P Chart dialog box, enter Nonconforming Items in the variable box by double-click Nonconforming Items in the variable list, enter 200 in the Subgroup size box, and click P Chart Options,

 

In P Chart-Options, click Tests tab and choose to Perform all tests for special causes from the drop-down list, and click Limits tab, enter 1 2 3, OK, and OK.

 

Copy control chart and Test Results for P Chart for Nonconforming Items, and paste into a Word file to submit.

 

Submit:

 

  1. A copy of control chart and session window (output pane)
  2. Interpretation of results

 

4.3 Construct c-chart for the data in COMPLAINTS file (20 pts). Interpret the results

 

To get Minitab output:

 

Open COMPLAINTS file and select Stat à Control Charts à Attributes Charts à C.

in the C Chart dialog box, enter Complaints in the left variable box by double-click Complaints in the variable list in the left box. Click C Chart Options, in C Chart-Options box, click Tests tab and choose to Perform all tests for special causes from the drop-down list and click Limits tab enter 1 2 3, OK and OK

 

Copy control chart and Test results — and paste into a Word file to submit.

 

Only X-bar chart applies 8 tests and all others (P, C, and R-charts) apply the first 4 tests.

Refer to the list of tests in the References Module in D2L content

 

Submit:

 

  1. A copy of control chart and session window (output pane)
  2. Interpretation of results

 

 

4.4

 

Note on Converting the payoff table to opportunity loss (OL) table

 

Refer to Table 3 or Table 4 in pages 670-1 of the textbook (Tables 20.3 -4 page 20-4/5 in BB) -in Decision making chapter

 

For the Payoff table (Table 3);

 

  A1: Gradual A2 Concentrated
E1: low demand 1 -5
E2: High demand 4 10

 

You may think this way to get opportunity loss table:

 

  • if in Low demand, E1, you will choose gradual alternative and profit is 1, OL of gradual action is 0, but OL of the Concentrated alternative is 6, because you lose 5, in fact, you could make 1, so the amount of regret(OL) is 6

 

OL = largest payoff in the line (event) – the payoff of the cell

 

  • If in High demand, E2, you will choose Concentrated, OL is 0, for Gradual OL= 10 – 4 = 6

 

Key is to think in one event i.e. low demand at a time and move to the next event i.e. high demand

 

The Opportunity Loss table is:

 

  A1: Gradual A2 Concentrated
E1: low demand 0 6 ß(1- (-5))
E2: High demand 6 ß (10 – 4) 0

 

 

 

Question:  For the following Payoff table, answer questions a) – e) (40 Pts)

 

Events Probability A1 A2
E1 .3 1 -5
E2 .5 4 1
E3 .2 6 16

 

 

  1. Convert to the Opportunity Loss table
  2. Find the optimal act under the maximin criterion
  3. Find the EMV (A1) and EMV(A2) and optimal act under the EMV criterion
  4. Find the EOL (A1) and EOL(A2) and optimal act under the EOL criterion
  5. Find the EVPI