MAT 221 ALGEBRA DISCUSSION

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Read the following instructions in order to complete this discussion, and review the example of how to complete the math required for this assignment: 

  1. Write your birth date or the birth date of someone in your family as mm/dd/yy. (Example: March 13, 1981 is written 3/13/81, and November 7, 1967 is written 11/7/67).   
    • Now let a = the one- or two-digit month number, 
    • b = the negative of the one- or two-digit day number, and 
    • c = the two-digit year number.

      (Our example:  a = 3, b = -13, and c = 81 or a = 11, b = -7, and c = 67) 

       

  2. Use the following algebraic expressions for parts 3-5 of the discussion: 
    •   
    •    
    •  

       

  3. Evaluate the three given expressions using the a, b, and c from your birth date.  Make sure that b is negative when you plug in the values. 

     

  4. After you have your math worked out on scratch paper, go back and verbally describe the steps you took to evaluate the expressions. Make sure to use each of the vocabulary words at least once in your writing.
    • Did you notice anything interesting about the results of  and ? Was this coincidence or do you think there is a reason for this?

       

  5. Incorporate the following five math vocabulary words into your discussion. Use bold font to emphasize the words in your writing (Do not write definitions for the words; use them appropriately in sentences describing your math work.): 
    • Exponent 
    • Integer 
    • Variable 
    • Lowest terms 
    • Divisor

Your initial post should be 150-250 words in length. Respond to at least two of your classmates’ posts by Day 7. Do you agree with how your classmates used the vocabulary? Did the student handle the negatives in the formulas accurately?

Finish times (to the nearest hour) for 10 dogsled teams are shown below: Make a frequency table showing class

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Finish times (to the nearest hour) for 10 dogsled teams are shown below:  Make a frequency table showing class limits, class boundaries, midpoints, frequency, relative frequencies, and cumulative frequencies.  Use three classes. (Round your answer for relative frequency to the nearest hundredth and for midpoint to the nearest tenth).

310   279   271   262   236   284   246   259    283    256

a). 

Class limits               Boundaries            Midpoint      Freq      Relative Freq       Cumulative Freq

236 – 260                  235.5 – 260.5          248              4           0.44

261 – 284                  260.5 – 284.5          273              5           0.59

285 – 308                  284.5 – 308.5          297              1           0.1                         10

 

 

b).

Class limits               Boundaries            Midpoint      Freq      Relative Freq       Cumulative Freq

236 – 260                  235.5 – 260.5          248              4           0.44

261 – 285                  260.5 – 285.5          273              5           0.59

286 – 310                  285.5 – 310.5          298              1           0.1                         10

 

 

c).

Class limits               Boundaries            Midpoint      Freq      Relative Freq       Cumulative Freq

236 – 260                  235.5 – 260.5          248              5           0.44

261 – 284                  260.5 – 285.5          273              7           0.59

285 – 310                  285.5 – 310.5          297              2           0.1                         10

 

 

d).

Class limits               Boundaries            Midpoint      Freq      Relative Freq       Cumulative Freq

236 – 260                  235.5 – 260.5          248              5           0.44

261 – 284                  260.5 – 284.5          273              7           0.5 9

285 – 308                  284.5 – 308.5          297              2           0.1                         10

 

 

e).  None of these choices

 

 

In a metal fabrication process, metal rods are produced to a specified target length of 15 feet

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In a metal fabrication process, metal rods are produced to a specified target length of 15 feet. Suppose that the lengths are normally distributed. A quality control specialist collects a random sample of 16 rods and finds the sample mean length to be 14.8 feet and a standard deviation of 0.65 feet. The standard error of the mean is 

A 0.65 ft.
B 0.1625 ft.
C 0.0098 ft.
D 0.346 ft.
E 1.625 ft.

EU (European Union) countries report that 46% of their labor force is female. The United Nations wants to determine if the percentage of females in the U.S. labor force is the same. Based on a sample of 500 employment records, representatives from the United States Department of Labor find that the 95% confidence interval for the proportion of females in the U.S. labor force is 0.357 to 0.443. Which of the following is the correct interpretation? 

A We are 95% confident that between 35.7% and 44.3% of the persons in the U.S. labor force is female
B The margin of error for the true percentage of females in the U.S. labor force is between 35.7% and 44.3%
C The percentage of females in the U.S. labor force is between 35.7% and 44.3%
D All samples of size 500 will yield a percentage of females in the U.S. labor force that falls within 35.7% and 44.3%
E None of these

In a metal fabrication process, metal rods are produced to a specified target length of 15 feet. Suppose that the lengths are normally distributed. A quality control specialist collects a random sample of 16 rods and finds the sample mean length to be 14.8 feet and a standard deviation of 0.65 feet. Which of the following statements is true? 

A The sampling distribution for the sample mean follows the t-distribution with 15 degrees of freedom
B The mean of the sampling distribution for the sample mean is 14.8 feet
C The standard error is 0.65 feet
D The sampling distribution for the sample mean follows the t-distribution with 16 degrees of freedom
E The sampling distribution for the sample mean is Normal with a mean of 14.8 feet and standard deviation of 0.65 feet

 

There is a great deal of debate in the medical community concerning how to treat

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There is a great deal of debate in the medical community concerning how to treat pregnancies when the mother is pregnant with twins.
Some doctors believe that when the mother shows certain symptoms before the due date (i.e., preterm), it is best to induce labor or perform a cesarean; other doctors think that preterm births should be allowed to occur without induced labor or a cesarean. Still other doctors believe as much as possible should be done to promote a full term pregnancy.
The head of the birthing center at a large city hospital examined the hospital’s twin birth records for several years. The data is summarized in the table below and is in this Excel file: Twin Births

1999 2004 2009
Preterm (induced or cesarean) 11 13 19
Preterm (without procedures) 13 14 18
Full term 27 26 32

Important Note: To arrange the data in the Excel file into a table in the worksheet, do the following:
1) click on any cell in the data;
2) click on the Insert tab;
3) click on PivotTable at the extreme left of the menu ribbon;
4) in the resulting window check that the Table/Range area shows Sheet1!$A$1:$C$10; in addition, click the radio button next to “Existing Worksheet” and specify the address of an empty cell in the “Location” area; then click OK;
5) in the PivotTable Field list, drag Duration to the Row Labels area;
6) in the PivotTable Field list, drag Year to the Column Labels area;
7) in the PivotTable Field list, drag Counts to the Values area;
8) you should now have a table showing the number of pregnancies with each duration/year combination.
Is there evidence that the way the hospital deals with pregnancies involving twins may have changed? Answer the following questions to perform the appropriate statistical test.

Question 1: What is the degrees of freedom for this test?

Question 2: What is the value of the chi-square test statistic? (use 4 decimal places) 

 

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?

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?