Question 1

1. What does it mean for two events A and B to be statistically independent?

2. What is the difference between the standard deviation for Continuous data and the standard deviation for Discrete data (you cannot state that the formula is different)?

Question 2

Digital data are transmitted as a sequence of signals that represent 0s or 1s. Suppose that such data are being transmitted to a satellite and then relayed to a distant site. Suppose that due to electrical interference in the atmosphere, there is a 1-in-1000 chance that the transmitted 0 will be reversed between the sender and the satellite (i.e. distorted to the extent that the satellites receiver interprets the 0 as a 1) and a 2-in-1000 chance that a transmitted 1 will be reversed. Suppose that 40% of the transmitted digits are 0s.

1. What is the probability that a transmitted digit is correctly received by the satellite?

2. Assuming independence, what is the probability that all the digits are received correctly
   1. if 1000 digits are transmitted

2. if 10,000 digits are transmitted.

3. Suppose that between the satellite and the receiver, the chances of reversal are twice as large as they were between the sender and the satellite. Assuming independence, what is the probability that a digit reaches the receiver as originally sent?

Question 3

Of 200 adults, 176 own one TV set, 22 own two TV sets, and 2 own three TV sets. A person is chosen at random.

1. What is the probability function of X?

2. What is the expected value for the probability distribution?

3. What are the variance and standard deviations for the probability distribution?

Dual-energy X-ray absorptiometry (DXA) is a technique for measuring bone health. One of the most common measures is total body bone mineral content (TBBMC). A highly skilled operator is required to take the measurements. Recently, a new DXA machine was purchased by a research lab, and two operators were trained to take the measurements. TBBMC for eight subjects was measured by both operators. The units are grams (g). A comparison of the means for the two operators provides a check on the training they received and allows us to determine if one of the operators is producing measurements that are consistently higher than the other. Here are the data.

(a)

Take the difference between the TBBMC recorded for Operator 1 and the TBBMC for Operator 2. (Use Operator 1 minus Operator 2. Round your answers to four decimal places.)

x = _______ (x bar equals)

s = _______

Describe the distribution of these differences using words.

The sample is too small to make judgments about skewness or symmetry.

The distribution is Normal.     

The distribution is right skewed.

The distribution is left skewed.

The distribution is uniform.

(b)

Use a significance test to examine the null hypothesis that the two operators have the same mean. Give the test statistic. (Round your answer to three decimal places.)

t = _______

Give the degrees of freedom. _______

Give the P-value. (Round your answer to four decimal places.) _______

Give your conclusion. (Use the significance level of 5%.)

We cannot reject H₀ based on this sample.

We can reject H₀ based on this sample.     

(c)

The sample here is rather small, so we may not have much power to detect differences of interest. Use a 95% confidence interval to provide a range of differences that are compatible with these data. (Round your answers to four decimal places.)

(d)

The eight subjects used for this comparison were not a random sample. In fact, they were friends of the researchers whose ages and weights were similar to the types of people who would be measured with this DXA machine. Comment on the appropriateness of this procedure for selecting a sample, and discuss any consequences regarding the interpretation of the significance-testing and confidence interval results.

The subjects from this sample may be representative of future subjects, but the test results and confidence interval are suspect because this is not a random sample.

The subjects from this sample, test results, and confidence interval are representative of future subjects.     

Distribution A: xi Distribution A: P(X=xi) Distribution B: xi Distribution B: P(X=xi)

                        0 0.03                             0 0.49

                        1 0.08                             1 0.23

                        2 0.17                            2 0.17

                       3    0.23                            3 0.08

                       4 0.49 4 0.03

c. What is the probability that x will be at least 3 in Distribution A and Distribution​ B?

d. Compare the results of distributions A and B

The previous answers to A & B were:

a. What is the expected value for distribution​ A?

mu

equals3.07

​(Type an integer or decimal rounded to three decimal places as​ needed.)

What is the expected value for distribution​ B?

mu

equals0.93

​(Type an integer or decimal rounded to three decimal places as​ needed.)

b. What is the standard deviation for distribution​ A?

sigma

equals1.116

​(Type an integer or decimal rounded to three decimal places as​ needed.)

What is the standard deviation for distribution​ B?

sigma

equals1.116

A data set includes 106 body temperatures of healthy adult humans having a mean of 98.9degreesF and a standard deviation of 0.62degreesF. Construct a 99​% confidence interval estimate of the mean body temperature of all healthy humans. What does the sample suggest about the use of 98.6degreesF as the mean body​ temperature? Click here to view a t distribution table. LOADING... Click here to view page 1 of the standard normal distribution table. LOADING... Click here to view page 2 of the standard normal distribution table. LOADING... What is the confidence interval estimate of the population mean mu​? nothingdegreesFless thanmuless than nothingdegreesF

BAYES THEOREM

Policy Pollsters [PP] is a market research firm specializing in political polls. Records indicate that in past elections, when a candidate was elected, Policy Pollsters had accurately predicted this 85% of the time and was wrong only 15% of the time. Records also show that, for losing candidates, Policy Pollsters accurately predicted that they lose 75% of the time and was wrong 25% of the time. Before the poll is conducted, there is a 50% chance of winning the election.

a. If a poll conducted by PP predicts a candidate will lose the election, what is the probability that the candidate will win the election? Show your work.        [2 points]

b. What is the probability that a poll conducted by PP predicts a candidate will win the election and the candidate loses the election? Show your work.                  [2 points]

c. Are the PP poll predicting a loss and the candidate losing the election independent events? Please show how you arrived at your answer.                                 [1 point]

Consumer price indices in 2018 in RK were:

Feb. 722.7

Mar. 726.5

Apr. 729.3

May 730.8

Jun. 732.2

Jul. 733.1

Aug. 734.2

Sep. 736.9

Oct. 739.9

Nov. 746.4

Dec. 751.5

a)      Produce the simple linear regression analysis. Present the fitted line analytically. Is there a significant linear dependence of СPI on time? Explain.

b)      Construct the 95% confidence predicted interval for the expected in the mean CPI for Jan. 2019 .

c)      How much is r squared and what is its meaning?

d)      What assumptions do you need for that prediction?

e)      Draw the corresponding graph and show on it your predicted interval.

Compare and contrast the Poisson and Negative binomial model. What happens if the Poisson distribution is the true data-generated process and a negative binomial regression model is estimate. What happens if the negative binomial distribution is the true data-generating process and a Poisson regression model is estimated?

1. Problem 16-15 (Algorithmic)

   A large corporation collected data on the reasons both middle managers and senior managers leave the company. Some managers eventually retire, but others leave the company prior to retirement for personal reasons, including more attractive positions with other firms. Assume that the following matrix of one-year transition probabilities applies with the four states of the Markov process being retirement, leaves prior to retirement for personal reasons, stays as a middle manager, and stays as a senior manager.

   		Leaves-	Middle	Senior
   	Retirement	Personal	Manager	Manager
   Retirement	1.00	0.00	0.00	0.00
   Leaves-Personal	0.00	1.00	0.00	0.00
   Middle Manager	0.02	0.07	0.79	0.12
   Senior Manager	0.08	0.01	0.04	0.87

   1. What states are considered absorbing states?
      
        
      
      Why?
      
      The input in the box below will not be graded, but may be reviewed and considered by your instructor.
      
      
      
   2. Interpret the transition probabilities for the middle managers.
      
      Probability of retirement =  
      
      Probability of leaving (personal) =  
      
      Probability of staying middle manager =  
      
      Probability of promotion to senior manager =
      
   3. Interpret the transition probabilities for the senior managers.
      
      Probability of retirement =  
      
      Probability of leaving (personal) =  
      
      Probability of staying middle manager =  
      
      Probability of promotion to senior manager =
      
   4. What percentage of the current middle managers will eventually retire from the company? What percentage will leave the company for personal reasons? If required, round your answers to one decimal place.
      
      Retire from the company =  %
      
      Leave the company for personal reasons =  %
      
   5. The company currently has 940 managers: 670 middle managers and 270 senior managers. How many of these managers will eventually retire from the company? How many will leave the company for personal reasons? If required, round your answers to one decimal place.
      
      Retire from the company =  %
      
      Leave the company for personal reasons =  %

Why do you think it is important for an engineer to measure variability? Give examples un your engineering specialty.

Birth rates (births per 1000 population) for 30 randomly selected countries are given below: 44.6 39.1 46.5 20.6 15.5 12.1 41.5 12.1 36.2 30.6 12.9 47.3 33.8 47.3 39.8 42.2 14.5 21.4 19.4 16.9 27.7 17.2 14.1 11.2 33.5 13.5 34.7 37.4 36.7 45.5 a) Construct a relative frequency histogram for the data (use k = 5 intervals !!!), b) Construct a cumulative relative frequency histogram (an empirical distribution function), c) Using grouped data estimate the unknown population mean µ and the population variance. d) What is the estimate of probability that a random variable X that has generated those data, will take the value of 12.1? Explain.

A process produces components with mean tensile strength of 85 MPa and standard deviation of 13 MPa. If the lowest acceptable strength is 40 MPa, what ppm of samples would be defective?

Using 50 observations, the following regression output is obtained from estimating y = β₀ + β₁x + β₂d₁ + β₃d₂ + ε.

a. Compute yˆy^ for x = 232, d₁ = 1, and d₂ = 0; compute yˆy^ for x = 232, d₁ = 0, and d₂ = 1. (Round your answers to 2 decimal places.)

b-1. Interpret d₁ and d₂. (You may select more than one answer. Single click the box with the question mark to produce a check mark for a correct answer and double click the box with the question mark to empty the box for a wrong answer. Any boxes left with a question mark will be automatically graded as incorrect.)

• When d₁ = 1, ŷ is 13.30 units greater than when d₁ = 0, holding everything else constant.unanswered
• When d₂ = 1, ŷ is 5.45 units greater than when d₂ = 0, holding everything else constant.unanswered
• When d₁ = 1, ŷ is 13.30 units less than when d₁ = 0, holding everything else constant.unanswered
• When d₂ = 1, ŷ is 5.45 units less when d₂ = 0, holding everything else constant.unanswered

b-2. Are both dummy variables individually significant at the 5% level?

• Yes, both dummy variables are individually significant at the 5% level.

• No, only the dummy variable d₂ is significant at 5% level.

• No, none of the dummy variables are individually significant at 5% level.

• No, only the dummy variable d₁ is significant at 5% level.

COMBINATIONS

Do the following problems using combinations.

1) How many different 5-player teams can be chosen from eight players?

2) How many 13-card bridge hands can be chosen from a deck of cards?

COMBINATIONS INVOLVING SEVERAL SETS

Following problems involve combinations from several different sets.

1) A club has 4 men, 5 women, 8 boys and 10 girls as members. In how many ways can a group of 2 men, 3 women, 4 boys and 4 girls be chosen?

2) How many 4-letter word sequences consisting of two vowels and two consonants can be made from the letters of the word PHOENIX if no letter is repeated?

From a set of all eight-digit natural numbers, where only the digits from the set {0,1, 3, 5, 7, 9} are present in decimal notation. we draw one. Calculate the probability of the event that the sum of digits of the drawn number is equal to 3.