The multiple regressions serve to explain the behavior of one variable (dependent variable) though a set of some explanatory variables for which we can find a logical/theoretically founded relationship with the dependent variable.

Please discuss three business situations (either real or a business situation) with proposed set of 5 explanatory variable. Could you define the expected sign (positive or negative) of these selected explanatory variables?

As e have discussed the usage of the dummy variables propose at least in one of the three cases you discuss previously one or two dummy variables you think are good explanatory variables for the case you are discussing.

Another researcher is interested in how caffeine will affect the speed with which people read but decides to include a third condition, a placebo group (a group that gets a pill that looks like the caffeine group does, but it does not contain caffeine). The researcher randomly assigns 12 people into one of three groups: 50mg Caffeine (n1=4), No Caffeine (n2=4), and Placebo (n3=4). An hour after the treatment, the 12 participants in the study are asked to read from a book for 1 minute; the researcher counts the number of words each participant finished reading. The following are the data for each group:

50mg Caffeine (group 1)

450 400 500 450

No Caffeine (group 2)

400 410 430 440

Placebo (group 3)

400 410 430 440

Answer the following questions using the Analysis of Variance instead of the t-test

a. What is the research hypothesis?

b. What is the null hypothesis?

c. What is dfbetween and dfwithin? What is the total df for this problem?

d. What is SSbetween and SSwithin? What is the total SS for this problem?

e. What is MSbetween and MSwithin?

f. Calculate F.

Use an a-level of .05 to answer the questions below

g. Draw a picture of the F distribution for dfbetween and dfwithin above, and locate F on the x-axis.

h. What is the critical value of F, given dfbetween and dfwithin? Indicate the critical value of F (and its value) in your drawing. Also indicate what the area is in the tail beyond the critical value of F.

i. Can you reject the null hypothesis?

j. Can you accept the research hypothesis?

4. Two randomized-controlled trials of routine ultrasonography screening during pregnancy were carried out, to see whether routine ultrasound imaging influenced outcomes of pregnancy such as birthweight and mode of delivery. No significant differences were found. At ages 8 to 9 years, 2011 singleton children of the women who had taken part in these trials were followed up. Ultrasonography had actually been carried out on 92% of the ‘screened’ group and 5% of the control group. No significant differences were found in scores for reading, spelling, arithmetic or overall school performance. A subgroup of children underwent specific tests for dyslexia. The test results classified as dyslexic 21 of the 309 children in the screened group (7%, 95% confidence interval = 3-10%) and 26 of the 294 controls (9%, 95% CI = 4-12%]). (Lancet 1991; 339: 85-89.) a. What is meant by “randomized” and “controlled”? Why were these techniques used?

In a Randomized Complete Block Design one of the two factors in the analysis is an extraneous variable (we are not directly interested in it) that is called a block. Explain the goal of including the extraneous variable in the analysis.

Another paper, by Kristin Butcher and Anne Piehl (1998), compared the rates of institutionalization (in jail, prison, or mental hospitals) among immigrants and natives. In 1990, 7.54% of the institutionalized population (or 20,933 in the sample) were immigrants. The standard error of the fraction of institutionalized immigrants is 0.18. What is a 95% confidence interval for the fraction of the entire population who are immigrants? If you know that 10.63% of the general population at the time are immigrants, what conclusions can be made? Explain.

The mean of a population is 75 and the standard deviation is 12. The shape of the population is unknown. Determine the probability of each of the following occurring from this population. a. A random sample of size 35 yielding a sample mean of 78 or more b. A random sample of size 150 yielding a sample mean of between 73 and 76 c. A random sample of size 219 yielding a sample mean of less than 75.8

The population proportion is 0.28. What is the probability that a sample proportion will be within ±0.04 of the population proportion for each of the following sample sizes? (Round your answers to 4 decimal places.)

(a)n = 100

(b)n = 200

(c)n = 500

(d)n = 1,000

(R programming)

Generate 50 samples from a Poisson distribution with lambda to be 2 and define the log likelihood function

Use optimization to find the maximum likelihood estimator of lambda. Repeat for 100 times using forloop. You will need to save the results of the estimated values of lambda.

a)

b)

c)

In international Morse code, each letter in the alphabet is symbolized by a series of dots and dashes: the letter “a” for example is encoded as “×- ” while the most common letter “e” has the shortest code “×” (just a dot). What is the minimum number of dots and/or dashes needed to represent any letter in the English alphabet (26 letters)?

1. What is the mean of the sample values 2 cm, 2 cm, 3 cm, 5 cm, and 8 cm?

2. What is the median of the sample values listed in Exercise 1?

3. What is the mode of the sample values listed in Exercise 1?

4. If the standard deviation of a data set is 5.0 ft, what is the variance?

5. If a data set has a mean of 10.0 seconds and a standard deviation of 2.0 seconds, what is the z score corresponding to the time of 4.0 seconds?

6. Fill in the blank: The range, standard deviation, and variance are all measures of _____.

7. What is the symbol used to denote the standard deviation of a sample, and what is the symbol used to denote the standard deviation of a population?

8. What is the symbol used to denote the mean of a sample, and what is the symbol used to denote the mean of a population?

9. Fill in the blank: Approximately _____ percent of the values in a sample are greater than or equal to the 25th percentile.

10. True or false: For any data set, the median is always equal to the 50th percentile.

A consumer is trying to decide between two long-distance calling plans. The first one charges a flat rate of 10 cents per minute. The second charges a flat rate of 99 cents for calls up to 20 minutes in duration and then 10 cents for each additional minute exceeding 20. (Assume that calls lasting a non-integer number of minutes are charged proportionately to a whole-minutes charge). If the duration of a randomly selected call of this consumer is exponentially distributed and its expected value is 15 minutes, compute the expected value of the corresponding charge by each plan.

Hint: Denote by X the duration of a random call, by Y1 the charge of the first plan, and by Y2 the charge of the second plan. Then, express Y1 and Y2 as functions of X, i.e., Y1 = h1(X) and Y2 = h2(X)

The National Sleep Foundation used a survey to determine whether hours of sleeping per night are independent of age (Newsweek, January 19, 2004). The following show the hours of sleep on weeknights for a sample of individuals age 49 and younger and for a sample of individuals age 50 and older. Hours of Sleep Age Fewer than 6 6 to 6.9 7 to 7.9 8 or more Total 49 or younger 37 58 71 74 240 50 or older 34 60 79 87 260

a.Conduct a test of independence to determine whether the hours of sleep on weeknights are independent of age. Use = .05.

Compute the value of the 2 test statistic (to 2 decimals).

b.Using the total sample of 500, estimate the percentage of people who sleep less than 6, 6 to 6.9, 7 to 7.9, and 8 or more hours on weeknights (to 1 decimal).

Less than 6 hours %

6 to 6.9 hours %

7 to 7.9 hours %

8 or more hours %

A 0.01 significance level is used for a hypothesis test of the claim that when parents use a particular method of gender​ selection, the proportion of baby girls is different from 0.5. Assume that sample data consists of 66 girls in 144 births, so the sample statistic of 11/24 results in a z score that is 1 standard deviation below 0.

a. Identify the null hypothesis and the alternative hypothesis. Choose the correct answer below.

A.H0​:p=0.5

H1​:p>0.5

B. H0​: p≠0.5

   H1​: p=0.5

C.H0​: p=0.5

H1​: p<0.5

D.H0​: p=0.5

H1​: p≠0.5

b. What is the value of α​?

α=________

​(Type an integer or a​ decimal.)

c. What is the sampling distribution of the sample​ statistic?

Normal distribution

χ2

Student​ (t) distribution

d. Is the test​ two-tailed, left-tailed, or​ right-tailed?

___________

e. What is the value of the test​ statistic?

The test statistic is ____________

f. What is the​ P-value?

The​ P-value is _____________

g. What are the critical​ value(s)?

The critical​ value(s) is/are ________

h. What is the area of the critical​ region?

The area is ________

For a newsvendor product the probability distribution of demand X (in units) is as follows:

xi 0 1 2 3 4 5 6
pi 0.05 0.1 0.2 0.3 0.2 0.1 0.05

The newsvendor orders Q = 4 units.

a) Derive the probability distributions and the cumulative distribution functions of lost sales as well as leftover inventory.

b) Knowing that the expected total cost function is convex in the order quantity Q, demonstrate that Q = 4 gives the minimal expected total cost.