Applied quantitative methods

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5-1: Indicate the different ways an individual could forecast his or her weight 10 years from now.  Do these methods change based upon whether the individual is 5, 14, 24, or 45 years old?  If so why?  (15 points)

5-2:  Using the assumption of the past predicts the future write an equation for the weight forecast.  Do the same for the assumption of cause and effect.  How does the concept of error play into each?  (15 points)

5-3:  Provide examples from the field of health services management of phenomena that are probably best forecasted using genius forecasting.  Why? (10 points)

5-4:  Determine the number of weekdays and weekend days in this month?  Compare this with the equivalent numbers of next year and last year.  What phenomenon forecasted by the health services manager might be influenced by variation in the number and types of days in a month?  Be specific and cite examples.  (15 points)

5-5:  Calculate the expected number of infant needing neonatal intensive care in a hospital if the historic rate is 5 per 1000 births, and you expect 575 births this year.  (5 points)

Chapter 6 Extra Credit (12 points)

Using the Northern College Health Services visit volume in Appendix 6-1 on page 113, provide a forecast of the number of clinic visits for week XX using:

6-1:  Extrapolation based upon Average Change

6-2:  Extrapolation based upon a Confidence Interval

6-3:  Extrapolation based upon Average Percent Change

6-4:  Extrapolation based upon Moving Averages

6-5:  Extrapolation based upon Exponential Smoothing

6-6:  Of all methods used, which is best and why?

Math Problem

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  1. Direct Materials Purchases Budget

    Marino’s Frozen Pizza Inc. has determined from its production budget the following estimated production volumes for 12” and 16” frozen pizzas for June 2016:

      Units
      12″ Pizza 16″ Pizza
    Budgeted production volume 12,900   24,000  

    There are three direct materials used in producing the two types of pizza. The quantities of direct materials expected to be used for each pizza are as follows:

      12″ Pizza 16″ Pizza
    Direct materials:
      Dough 0.90  lb. per unit 1.50 lbs. per unit
      Tomato 0.60   1.00  
      Cheese 0.80   1.30  

    In addition, Marino’s has determined the following information about each material:

      Dough Tomato Cheese
    Estimated inventory, June 1, 2016 650 lbs. 180 lbs. 360 lbs.
    Desired inventory, June 30, 2016 680 lbs. 170 lbs. 390 lbs.
    Price per pound $1.1   $2.4   $3.3  

    Prepare June’s direct materials purchases budget for Marino’s Frozen Pizza Inc. When required, enter unit prices to the nearest cent. 

    Marino’s Frozen Pizza Inc.
    Direct Materials Purchases Budget
    For the Month Ending June 30, 2016
      Dough Tomato Cheese Total
    Units required for production:        
    12″ pizza [removed] [removed] [removed]  
    16″ pizza [removed] [removed] [removed]  
      [removed] [removed] [removed]  
    Total [removed] [removed] [removed]  
      [removed] [removed] [removed]  
    Total units to be purchased [removed] [removed] [removed]  
    Unit price x $[removed] x $[removed] x $[removed]  
    Total direct materials to be purchased $[removed] $[removed] $[removed] $[removed]

MAT 540 WEEK 1 TO WEEK 11( LATEST VERSION)

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Week 4

Chapter 15

 

1.      The manager of the Carpet City outlet needs to make an accurate forecast of the demand for Soft Shag carpet (its biggest seller). If the manager does not order enough carpet from the carpet mill, customer will buy their carpet from one of Carpet City’s many competitors. The manager has collected the following demand data for the past 8 months:

 

Month

Demand for Soft Shag

 

Carpet (1,000 yd.)

 

 

 

 

 

 

1

10

 

 

 

 

2

9

 

 

 

 

3

8

 

 

 

 

4

9

 

 

 

 

5

10

 

 

 

 

6

12

 

 

 

 

7

14

 

 

 

 

8

11

 

 

 

 

 

 

a.   Compute a 3-month moving average forecast for months 4 through 9.

 

b.   Compute a weighted 3-month moving average forecast for months 4 through 9. Assign weights of 0.55, 0.35, and 0.10 to the months in sequence, starting with the most recent month.

 

c.  Compare the two forecasts by using MAD. Which forecast appears to be more accurate?

 

 

2.    The manager of the Petroco Service Station wants to forecast the demand for unleaded gasoline next month so that the proper number of gallons can be ordered from the distributor. The owner has accumulated the following data on demand for unleaded gasoline from sales during the past 10 months:

 

 

Month

Gasoline Demanded (gal.)

 

 

 

 

October

775

November

835

December

605

January

450

February

600

March

700

April

820

May

925

June

1500

July

1200

 

 

 

 

a.      Compute an exponential smoothed forecast, using an α value of 0.4

 

b.      Compute the MAD.

 

 

3.      Emily Andrews has invested in a science and technology mutual fund. Now she is considering liquidating and investing in another fund. She would like to forecast the price of the science and technology fund for the next month before making a decision. She has collected the following data on the average price of the fund during the past 20 months:

 

 

Month

Fund Price

 

 

1

$55 ¾
   

2

54 ¼
   

3

55 1/8
   

4

58 1/8
   

5

53 3/8
   

6

51 1/8
   

7

56 ¼
   

8

59 5/8
   

9

62 ¼
   

10

59 ¼
   

11

62 3/8
   

12

57 1/1

 

 

 

13

58 1/8
   

14

62 ¾
   

15

64 ¾
   

16

66 1/8
   

17

68 ¾
   

18

60.5
   

19

65.875
   

20

72.25

 

 

 

 

a.    Using a 3-month average, forecast the fund price for month 21.

 

b.    Using a 3-month weighted average with the most recent month weighted 0.5, the next most recent month weighted 0.30, and the third month weighted 0.20, forecast the fund price for month 21.

 

c.    Compute an exponentially smoothed forecast, using α=0.3, and forecast the fund price for month 21.

 

d.   Compare the forecasts in (a), (b), and (c), using MAD, and indicate the most accurate.

 

4.      Carpet City wants to develop a means to forecast its carpet sales. The store manager believes that the store’s sales are directly related to the number of new housing starts in town. The manager has gathered data from county records on monthly house construction permits and from store records on monthly sales. These data are as follows:

 

 

Monthly Carpet Sales

Monthly Construction

(1,000 yd.)

Permits

 

 

9

17

14

25

10

8

12

7

15

14

9

7

24

45

21

19

20

28

29

28

 

a.    Develop a linear regression model for these data and forecast carpet sales if 30 construction permits for new homes are filed.

 

b.    Determine the strength of the causal relationship between monthly sales and new home construction by using correlation.

 

5.     The manager of Gilley’s Ice Cream Parlor needs an accurate forecast of the demand for ice cream. The store orders ice cream from a distributor a week ahead; if the store orders too little, it loses business, and if it orders too much, the extra must be thrown away. The manager believes that a major determinant of ice cream sales is temperature (i.e. the hotter the weather, the more ice cream people buy). Using an almanac, the manager has determined the average day time temperature for 14 weeks, selected at random, and from store records he has determined the ice cream consumption for the same 14 weeks. These data are summarized as follows:

 

 

Week

Average Temperature

Ice Cream Sold

 

(Degrees)

(gal.)

 

 

 

 

 

 

 

1

68

80

 

 

 

 

 

2

70

115

 

 

 

 

 

3

73

91

 

 

 

 

 

4

79

87

 

 

 

 

 

5

77

110

 

 

 

 

 

6

82

128

 

 

 

 

 

7

85

164

 

 

 

 

 

8

90

178

 

 

 

 

 

9

85

144

 

 

 

 

 

10

92

179

 

 

 

 

 

11

90

144

 

 

 

 

 

12

95

197

 

 

 

 

 

13

80

144

 

 

 

 

 

14

75

123

 

 

 

 

 

 

 

 

a.      Develop a linear regression model for these data and forecast the ice cream consumption if the average weekly daytime temperature is expected to be 85 degrees.

 

b.   Determine the strength of the linear relationship between temperature and ice cream consumption by using correlation.

 

c.      What is the coefficient of determination? Explain its meaning.

6 POINT QUESTION

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One common type of calculation that is made frequently out there in the real world is a “fixed and variable cost problem” – what I call a Garfield problem because it’s a “lump and per” scenario. You pay a fixed cost (or a lump sum) to rent the car, or have phone service. Then, in addition, you have to pay a variable cost (so much per mile or per minute). People use math all the time (or should!) to decide which company has the best plan for their needs.

 

For the purposes of this Forum, we are going to discuss a subject near and dear to everyone’s heart these days – the price of operating your vehicle. To do this, we need to collect three types of data – our fixed costs, our variable costs and our usage. If you either don’t own a car just borrow a friend’s data or use a hypothetical example based on internet research. One student even did the analysis on his aircraft carrier!

 

For your initial post (worth 6 points), in 250 words or more, give a full and complete answer to the following questions. Then reply to 2 classmates on substantive posts of 50 words or more.

 

A – What are your fixed monthly car expenses? (1 point)

 

Although most Americans view their car expenses as an inevitable part of life and not part of their actual car calculations, we are going to figure out what our car expenses really are.

 

I want you to add up all the payments that you must make (car payment, garage rent, license tags, insurance, etc.)–regardless of whether or not old Betsy ever emerges from the garage. If you only pay insurance once a year then divide that number by 12 to add it to your monthly fixed expenses. You don’t have to share the specifics if you would prefer not to–you can just give the total.

 

 B – What are your variable monthly car expenses? (1 point)

 

Add up your variable expenses—the ones that depend on how many miles you drive the car. Don’t get into the nitty-gritty of your car’s miles per gallon—just give an approximation of what you spend each month on variable expenses (gas, oil, washer fluid, car washes, etc.).

 

C – What is your total cost per month? (1 point)

 

Add your fixed and variable costs together.

 

D – Approximately how many miles do you drive a month? (1 point)

 

All of the discussion in the media is about the price of gasoline per gallon, but to calculate your personal cost, a more relevant statistic for you is your cost per mile. That is one of the things that you should consider when you decide whether or not to drive home for lunch.

 

E – Divide C by D to figure your cost per mile. (1 point)

 

Pay attention here. If the money is more than the miles, the cost is over $1.00 per mile – if the miles are more than the money, the cost is under $1.00 per mile. What is your cost per mile?

Surprised? One student did the calculations wrong and got that she was spending $316.00 a mile. That would go way beyond surprised!

 

F – What changes might you make to save money on your total car expenses? (1 point)

 

Another student lived in a big city and only used the car on weekends. When he finished the calculations, he cut a deal with a local car rental company, sold his car, and wound up renting a very nice car every weekend for less than he was spending on a car of his own!

 

Best packers and movers in Noida Home

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While exchanging the undertaking including pressing notwithstanding moving is a testing notwithstanding muddled procedure. The particular circumstance compounds on the off chance that you are not an expert division handler. Truth be told taking things is normally themselves a major weight. It is truly greatly intense to assist load with up an assortment of items safely. Just pro packers can deal with this errand adequately and securely. In spite of the fact that having a modest bunch of cerebral vascular mischances of figuring out the basic ways it’s likewise conceivable to deal with the partition movement alongside brightness. Are this proposals which can be talked about with the gifted packers and in addition movers concerning quick a genuinely simple protected and sound supplying. 

 

System before you amass. Another pleasantly composed move will really succeed notwithstanding clean. Therefore, make a migration methodology before you begin taking and also exchanging elements. You must dispense time for every one of those schedules. It is better to make time span concerning different errands that you ought to perform. You must help make technique in this strategy that undertakings of a room ought to be expert together. Giving an area amid a period can without much of a stretch disentangle ones migration work; furthermore it helps individuals with nearing your own particular merchandise generally advantageous. 

 

Lessen untouched stock before beginning taking to get a go. There ought to be various useless things in your home. Wipe out every such thing. This may spare awesome arrangement of your vitality. You may be protected from giving the inadequate products. On the off chance that the things are ordinarily in doing the employment issue you’ll have the capacity to hand over these to your buddies or even other people who live close-by, or even and afterward just annihilate them. 

 

Gather vital supplying items. On the off chance that you have settled on a choice to adapt to the appearance undertaking from your one of a kind, next numerous you have to do ought to be to gather the important appearance supplies. Remember to make a decent measure of additional cushioning and also bolster segments. You will require the vast majority of these parts despite the fact that taking sensitive notwithstanding weak things. 

 

Wrap up things appropriately. Satisfied partition totally relies on upon the specific appearance on the merchandise. Any harshness in precisely the same result in this annihilation of this adorable and fine things. The brilliant tip of danger free supplying will be given down beneath. Maintain the genuine ways sequentially and you will securely and safely load up different types including items. packers and movers in noida

Simple Linear Regression

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Simple Linear Regression

·          

·         https://assets.pinterest.com/images/PinExt.png

In your response to your peer’s initial post comment on other variables that might potentially confound the relationship between the two displayed variables.

Peer response

Simple linear regression analyses provide a broader scope of information than correlations do. In conducting a simple linear regression, researchers are provided with several values including the slope, y-intercept, r-squared and p-value.1 In a correlation, researchers are interested in the r and p-values and these are used to determine if two variables are associated with one another, or not. Simple linear regressions assist researchers in analyzing the actual line drawn through a correlation scatterplot which can be used to predict outcomes of additional participants.

For this assignment, I am investigating the relationship between life expectancy (dependent variable) and per capita income (independent variable) in 2015. The following values were gleaned from a simple linear regression analysis conducted in Stata:

Slope = .00013

Y-Intercept = 72.42

R-squared = 0.415

P= 0.0000

The slope value tells us that for every change observed in per capita income, we can expect to see a change of .00013 in life expectancy. The y-intercept provides that with a per capita income of zero, we can expect life expectancy to be 72.42 years. The r-squared value tells us that 41.5% of the variability observed in life expectancy is due to per capita income level. Finally, the p-value tells us that we can reject the null hypothesis.1 Hypotheses in this case are as follows:

H0: Slope value = 0

H1: Slope value does not equal 0

In summary, we can use these values to predict life expectancy, given we have data for income per capita. For example, if we know a country’s per capita income in 2015 was $20,000 we can use the following equation to determine life expectancy:

Life Expectancy = 72.42 + .00013(20000)

Life Expectancy = 75.02 years

 

matrix

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1 point) On the augmented matrix AA below, perform all three row operations in the order given, ((a) followed by (b) followed by (c)) and then write the resulting augmented matrix.

A=⎡⎣⎢⎢13−3−1−2−136135−4⎤⎦⎥⎥A=[1−1333−265−3−11−4]

(a) R2=−3r1+r2R2=−3r1+r2
(b) R3=3r1+r3R3=3r1+r3
(c) R3=4r2+r3

Your struggle is the area between the two thrones

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 ‘ And it must be a disk of our expectations and understand how MMOs. Sometimes there are important ideas that come out of these jokes, which can actually make these titles better.So Let’s go through favorite MMO April Fools jokes of all time, and indexed by Yours! I have to tip my hat to the crew who went above and beyond to not only the Islamic idea to one of the questions most famous game in the form of music, but recording video, which seems to said musical, complete with songs, puppets, costumes and crazy happy years. Monkey punching and rsfunny two weeks and approaching milling carbon spring event! Resulting from a poll of society, and led the team RS players epic contest between Marimbo cabbage Prime.To attending the mini-games, go to the old camp Saradominist war where you will encounter Marimbo and Cabbage Head-to-face off along with the atheist Holstein, who looked uncomfortable. They try to RS 3 Gold find out who is the best. If you are a partner of one of them, fighting in the field below the ground at the beginning of names.To, and choose between the side Marimbo, cabbage or Holstein. Do not forget you have to listen to the voices of the Department of Defense Michele and Raven Liono on volumes. Your struggle is the area between the two thrones, where all players will wait. When assembling enough players, the cabbage Facepunch Bonanza start automatically.The simple rules should be in a limited time, players run over one side to the hallway and dribbling threats simultaneously. The game will be over a number of rounds. Points are given to those who are able to run on the next side, and coal ore mines and monkeys punch. Will the players who failed to reach the next goal in time become a gorilla and placed the center of the battlefield. The only purpose Gorillas is to kill human players that earns them points too. The game ends either when they turn all players to gorillas or when you turn ten players around the successfully.You points can be earned in the game to use the heads of the beauty and the capacity of the positive and negative, which aim Add the cabbage has used rid Facepunch Bonanza.In you will find some interesting articles stunts have been picked up by a recent poll, for example Seedicide things that seeds are killed in the fighting and the weakness of Agriculture XP if planted. Bombs that the patch can instantly scan patch of Agriculture for the full amount of the items and XP. Killer VIP codes you more than an alternative offer if you want a new killer task.

Which statement is NOT true regarding the slope of a regression equation?

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  1. Which statement is NOT true regarding the slope of a regression equation?
  a. The slope quantifies the steepness of the line.
  b. If the slope is negative, Y tends to decrease as X tends to decrease.
  c. If the slope is positive, Y tends to increase as X tends to increase.
  d. None of the above, all are true.

Question 2

  1. The scale of the correlation coefficient is:
  a. nominal
  b. interval
  c. ordinal
  d. ratio

Question 3

  1. A fellow researcher tells you that she got a p calculated of .04 from an inferential test she conducted. She asks you if her results are statistically significant. The most informed response would be:
  a. Yes, because the probability of obtaining your sample statistic was less than .05.
  b. Yes, because your alpha level was greater than your obtained p calculated.
  c. No,because the probability of obtaining your sample statistic was greater than .05.
  d. No, because your alpha level was greater than your obtained p calculated.
  e. It depends, what did you set your probability of Type I error to be?

Question 4

  1. Use the following scenario to answer the next 2 questions.

Working a full-time job and taking classes at night can be very stressful at times. Recently a study was conducted to determine whether the amount of sleep taken the night before a test differed for UNT graduate students with full-time jobs as compared to UNT graduate students in general, and you expect that graduate students with full-time jobs get less sleep. The data collected were as follows. Assume that we know the general population of UNT graduate students sleep 7 hours the night before a test and that this distribution is normal. The standard deviation around this population mean is 2.5 hours. The study’s sample of 25 UNT graduate students with full-time jobs slept 6 hours before a test with a standard deviation of 5. Assume alpha = .05. ROUND ALL CALCULATIONS TO TWO DECIMAL PLACES.

What is the calculated Z-test value?

  a. -.2
  b. -.4
  c. -1.0
  d. -2.0
  e. none of the above

Question 5

  1. What is the p calculated (probability) for your sample results?
  a. .0114
  b. .3446
  c. .0228
  d. .1554
  e. .1723

 

Question 6

  1. Statistical significance testing (null hypothesis tests) depends on the assumption that:
  a. you have a large sample size.
  b. the null hypothesis is true in the population.
  c. you have a low risk if Type I error.
  d. the alternative hypothesis cannot be proven true.
  e. none of the above

Question 7

  1. A statistically significant result means that, given the assumptions of the test, you have a(n):
  a. replicable result.
  b. important result
  c. unlikely result
  d. unimportant result

Question 8

  1. In the process of testing a null hypothesis, which of the following should precede the others.
  a. Computing the test statistic.
  b. Deciding whether to reject Ho.
  c. Determining your conclusion regarding group differences.
  d. Establishing the criterion for rejecting.

Question 9

  1. Which of the following is not true about Pearson r and Spearman rho?
  a. r ranges from -1 to +1.
  b. When Spearman rho = 1.00, then Pearson r will always be 1.00.
  c. When Pearson r = 1.00, then Spearman rho will always be 1.00.
  d. r and rho cannot be compared without more information.

Question 10

  1. Which alpha level below corresponds to the most risk of Type I error?
  a. alpha = .10
  b. alpha = .05
  c. alpha = .01
  d. alpha = .001

Question 11

  1. Which of the following is an example of an alternative hypothesis?
  a. Exercising 30 minutes a day for 2 weeks does not decrease heart rate by 10%.
  b. Taking 500mg of ibuprofen within 20 minutes of onset will decrease headache pain.
  c. Studying more than 5 hours for a statistics final does not impact your grade.
  d. There is no difference between the mean self-esteem scores of mean and women athletes at UNT.

Question 12

  1. Graphic r-Q1 of 2
  a. .8
  b. -.8
  c. -1.0
  d. .7
  e. .6

Question 13

  1. If we now add the two plotted points marked with (g), which of the following is not true?
  a. The line of best fit will come closer to catching all points.
  b. r will increase in magnitude.
  c. r will approach its numerical limit.
  d. None of the above; all are true.

Question 14

  1. The primary goal of inferential statistics is to:
  a. describe characteristics of samples based on population parameters.
  b. suggest characteristics of populations based on sample statistics.
  c. find exact values for population parameters with our sample statistics.
  d. find exact values for sample statistics given estimated population parameters.
  e. drive graduate students “batty”.

Question 15

  1. Use the following data for the next four questions.
  X Y
Linda 3 -2
Jane 4 0
Dean 5 2
Lesley 6 8
  1. Which statement is true?
  a. Pearson r is positive.
  b. Pearson r is positively skewed.
  c. Pearson r = 1.
  d. Pearson r = Spearman rho.
  e. None of the above, all are false.

Question 16

  1. If we change Lesley’s X score to 4, which statement is true?
  a. Pearson r becomes larger.
  b. Spearman rho becomes smaller.
  c. Spearman rho = Pearson r.
  d. Pearson r becomes negatively skewed.
  e. None of the above.

Question 17

  1. Now that we have changed Lesley’s X score to 4, if we change Linda’s X score to -20 which statement is true?
  a. r is negative.
  b. r becomes smaller.
  c. rho becomes smaller.
  d. rho = r.
  e. None of the above.

Question 18

  1. On the original data set, if we change Lesley’s Y score (not X) to 4, which of the following is true?
  a. r = rho
  b. r is less than rho
  c. r is greater than rho
  d. Not enough information to determine.

Question 19

  1. Standard error is the standard deviation of a sampling distribution. The larger the standard error, the greater risk of _____________ error.
  a. sampling
  b. Type I
  c. Type II
  d. Type III
  e. none of the above

Question 20

  1. Which of the following is not true of Pearson r and COV?
  a. r and COV always have the same sign.
  b. r is the COV with the impact of SD of X and SD of Y removed.
  c. r has minimum and maximum limits while COV does not.
  d. r = COV only when the SDs of both variables are 1.
  e. None of the above are false, all are true.

Question 21

  1. Given r = .8 and a p calculated of .6, when we test the null hypothesis that r = 0, we can say that:
  a. there is an 80% probability of obtaining at least r = .8 from any random sample, if the r should have been 0.
  b. there is a 60% probability of obtaining at least r = .8 from any random sample, if the r should have been 0.
  c. there is a 60% probability of not obtaining at least r = .8 from any random sample, if the r should have been 0.
  d. there is an 80% probability of not obtaining at least r = .8 from any random sample, if the r should have been 0.

Question 22

  1. Which of the following is true regarding weak correlation coefficients?
  a. They indicate a slight causation between variables.
  b. They are found only with Pearson r coefficients.
  c. They may be the result of restricted range for one of the variables.
  d. They are rarely found in social science research.
  e. They account for large amounts of dependent variable variability.

Question 23

  1. A Pearson r of .70 is how much stronger than a Pearson r of .00 in terms of shared variance between the two variables?
  a. 30%
  b. 40%
  c. 49%
  d. 70%
  e. None of the above

Question 24

  1. In hypothesis testing, if we found a statistically significant difference between the means of two groups, then we:
  a. rejected the null.
  b. failed to reject the null.
  c. committed Type I error.
  d. committed Type II error.
  e. none of the above

Question 25

  1. When choosing a non-directional hypothesis, we:
  a. are choosing a one-tailed test.
  b. decrease the likelihood of Type II error.
  c. increase the likelihood of Type I error.
  d. decrease the likelihood of statistical significance.
  e. none of the above

Question 26

  1. As a special education researcher, you know that easy distractibility and diminished attention at school have been strongly correlated with low scores on certain tests of sensory integration. A child is found to have trouble paying attention in his classes. The school counselor asks for sensory integration therapy to remediate the attention problem. Should the therapist provide therapy right away?
  a. Yes, because the strong correlation proves that therapy which improves sensory integration will definitely improve attention.
  b. No, because therapy caseloads in public schools are high and this child s problems are relatively minor.
  c. No, because a strong correlation doesn t mean that one condition necessarily causes the other; more investigation is needed to determine the possible causes of the child s distractibility.
  d. Yes, because the school administrators are in favor of the therapy.

Question 27

  1. Dr. Marcela conducted a study on SAT and GRE scores and calculated the Pearson r correlation between these scores for a sample of graduate students. Which correlation coefficient is least likely to reflect the relationship between these variables?
  a. -1.0
  b. -0.1
  c. 1.4
  d. 1.0
  e. None of the above, all are equally plausible

Question 28

  1. The bivariate reference point that is analogous to the univariate mean is called the:
  a. covariance.
  b. correlation.
  c. coordinate.
  d. centroid.

Question 29

  1. In a regression linear equation for one predictor and a dependent variable, in which of the following scenarios will the a weight always be 0?
  a. When the SDs of both variables are the same.
  b. When the means of both variables are the same.
  c. When the SDs of both variables are 0.
  d. When the variables are in Z score form.
  e. None of the above

Question 30

  1. I am more informed of basic statistical practices now than when I began this class, and I am thusly truly grateful.
  a. True
  b. False
  c. I don’t know because I think this is somehow a trick question
  d. Maybe
  e. It depends

 

PROBABILITY AND STATISTICS Final Exam

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PROBABILITY AND STATISTICS

Final Exam

 

This is an open-book take-home exam. Good luck!

 

1.  Let Xi be the life length of an item. Consider X1, X2,…Xn to be independently and identically distributed, each with normal distribution N(m,s2). Assume that s2=16, but that m is unknown. Suppose 100 tests yield an average life of =501.2 hours.

 

a) Construct a 95% confidence interval for the reliability of the item for a service time of t hours given by

            R(t; m)=P(X>t).

 

b) Compute numerical values for a) if t = 500 hours.

 

2.  For a random sample of size n from f(x|θ)= θLθx-(1+θ) for x>L, where L is known and θ>0,

a) Find the maximum likelihood estimator of θ and express it as a function of g=, the geometric mean of the observations.

b) Find the set of admissible rejection regions in terms of g for a likelihood ratio test of H0: θ=5 versus H1: θ=2.

 

3.  For a normal data-generating process with m and s not known but the coefficient of variation c=s/m known, find the maximum likelihood estimates of m and s2 if c=0.25 and the data are: 16, 27, 24, 21, 23, 12, 21, 18, 17, 23. Compare these estimates with estimates that would be obtained if no information were available concerning c.

 

4.  In a survey, some of the questions concern sensitive issues (e.g., income, drug use, sexual experiences).  As a result, some respondents do not answer the questions truthfully.  Denote the proportion of the members of a particular population that had incomes over $100,000 last year by p.  A random sample of n members of this population is taken, and each person in the sample is asked “Was your income over $100,000 last year?”  If a person really had an income over $100,000, the probability that she will give a truthful answer to this question is 1-l1.  If a person’s income was not over $100,000, the probability that she will give a truthful answer is 1-l2.  From past experience, l1 and l2 are known, with 0<l1<0.5, 0<l2<0.5.

a) For a sample of size one, find the likelihood function if the answer is “yes” and find the likelihood function if the answer is “no.”

b) For a random sample of size n, find the likelihood function and sufficient statistics.

c) Find the maximum likelihood estimator for p.

d) Assume that l1=0.1, l2=0, and there is one “yes” answer in a random sample of size 10. What is your best estimate of p and why?

e) Consider the same scenario as in (d), but assume that l1 is unknown (0<l1<1). In this case, what would be your best estimate of p and why?

 

5.  Let X1, X2,…Xn be the times in months until failure of n similar pieces of equipment. If the equipment is subject to wear, a model often used is the one where X1, X2,…Xn (i.i.d) is a sample from a Weibull distribution with density

, xi>0.

Here c is a known positive constant and l>0 is the (scale) parameter of interest.

a) Show that  is an optimal test statistic for testing H0: 1/l<1/l0 versus H1: 1/l>1/l0, i.e., show that for a UMP test, the rejection and acceptance regions are defined in terms of the statistic .

b) If random variable X has a Weibull distribution specified above, find the distribution of the random variable .

 

6.  A journal editor says: “If we only publish papers with results that are statistically significant at the a=0.05 level, at most 5% of our papers will have erroneous results.” Denote by p the proportion of researchers with true H0 and false H1. Suppose that each researcher performs one test, sends the paper to the journal, and the paper is accepted if the results of the test are significant at the a=0.05 level.

a) If in a given year the journal publishes n papers, find the distribution of the papers with erroneous results that are published in this year. Assume that all the tests in all papers have the same b, probability of type II error.

b) What is this distribution if p=1, i.e., if all researchers, submitting the papers this year, had true H0 and false H1?

c) Overall, comment on the above statement of a journal editor.

 

7.  Suppose that a single observation X is to be drawn from an unknown distribution P, and that the following simple hypotheses are to be tested:

H0: P is a uniform distribution on the interval [0,1],

H1: P is a standard normal distribution.

Determine the most powerful test of size 0.01, and calculate the power of the test when H1 is true.

 

8.  An unethical experimenter desires to test the following hypotheses:

H0: q=q0,

H1: q¹q0.

She draws a random sample X1, X2,…Xn from a distribution with the pdf f(x|q) and carries out a test of size a. If this test does not reject H0, she discards the sample, draws a new independent random sample of n observations, and repeats the test based on the new sample. She continues drawing new independent samples in this way until she obtains a sample for which H0 is rejected.

a)     What is the overall size of this testing procedure?

b)     If H0 is true, what is the distribution of the number of samples that the experimenter will have to draw until she rejects H0? In particular, what is the expected number of samples for a=0.05?

 

9.  Consider the following situation. There are N job applicants, and, with probability pi, ni of them (i=1,2,…M; 0<ni <N) are invited for an interview. All pi and ni are known to all job applicants, and if ni applicants are invited, then each of N applicants has the same chance ni/N to be invited.

a) Given that a job applicant is invited for an interview, what are her expectations about the total number of applicants invited for an interview? 1) Find the corresponding probability distribution – i.e., the posterior distribution (conditional on an applicant being invited for an interview) for the number of applicants invited for an interview. 2) For this distribution, find the expected number of invited applicants.

b) Assume that if ni applicants are invited, each of them has equal (1/ni) chance of getting a job. Before the applicant is invited, what are her chances of getting a job? After the applicant is invited, what are her chances of getting a job? What are the chances to be invited? Do these three numbers agree with each other?

 

c) Repeat questions a) and b) for a special case M=2, p1=p2=0.5, n1 = 1, n2 = 100, N=1000 – i.e., out 1000 applicants, either 1 or 100 are invited for an interview. Do the answers make sense?