6) Suppose a multinomial regression model has two continuous explanatory variables ?1 and ?2 ,and they are represented in the model by their linear and interaction terms.
a) For a ? unit increase in ?1, derive the corresponding odds ratio that compares a category ? response to a category 1 response. Show the form of the variance that would be used in a Wald confidence interval.
b) Repeat this problem for a proportional odds regression model.

I. Proof of an assertion regarding a proportion:

1. The municipal government of a city uses two methods to register properties. The first requires the owner to go in person. The second allows registration by mail. A sample of 50 of the method I was taken, and 5 errors were found. In a sample of 75 of method II, 10 errors were found. Test at a significance level of .15 that the personal method produces fewer errors than the mail method.

2. A pharmaceutical firm is testing two components to regulate the pressure. The components were administered to two groups. In group I, 71 of 100 patients managed to control their pressure. In group II, 58 of 90 patients achieved the same. The company wants to prove at a level of significance of .05 that there is no difference in the effectiveness of the two drugs.

Playbill magazine reported that the mean annual household income of its readers is $120,255. (Playbill, January 2006). Assume this estimate of the mean annual household income is based on a sample of 80 households, and based on past studies, the population standard deviation is known to be σ = $33,225.

a. Develop a 90% confidence interval estimate of the population mean.

b. Develop a 95% confidence interval estimate of the population mean.

c. Develop a 99% confidence interval estimate of the population mean.

d. Discuss what happens to the width of the confidence interval as the confidence level is increase. Does this result seem reasonable? Explain.

Suppose the life of a particular brand of calculator battery is approximately normally distributed with a mean of 75 hours and a standard deviation of 9 hours.

What is the probability that 9 randomly sampled batteries from the population will have a sample mean life of between 70 and 80 ​hours?

According to a recent study annual per capita consumption of milk in the United States is 22.6 gallons. Being from the Midwest, you believe milk consumption is higher there and wish to test your hypothesis. A sample of 14 individuals from the Midwestern town of Webster City was selected and then each person's milk consumption was entered into the Microsoft Excel Online file below. Use the data to set up your spreadsheet and test your hypothesis.

Open spreadsheet

1. Develop a hypothesis test that can be used to determine whether the mean annual consumption in Webster City is higher than the national mean.

   H₀: ? _________> 22.6≥ 22.6= 22.6≤ 22.6< 22.6≠ 22.6

   H_a: ? _________> 22.6≥ 22.6= 22.6≤ 22.6< 22.6≠ 22.6

2. What is a point estimate of the difference between mean annual consumption in Webster City and the national mean?

   (2 decimals) ______

3. At ? = 0.01, test for a significant difference by completing the following.

   Calculate the value of the test statistic (2 decimals).

   _______

   The p-value is  (4 decimals)

4. Reject the null hypothesis?

   _____NoYes

   What is your conclusion?

   _________There is insufficient evidence to conclude that the population mean consumption of milk in Webster City is greater than the hypothesized mean.Conclude the population mean consumption of milk in Webster City is greater than the hypothesized mean.

Let X and Y be independent Exponential random variables with common mean 1.

Their joint pdf is f(x,y) = exp (-x-y) for x > 0 and y > 0 , f(x, y ) = 0 otherwise. (See "Independence" on page 349)

Let U = min(X, Y) and V = max (X, Y).

The joint pdf of U and V is f(u, v) = 2 exp (-u-v) for 0 < u < v < infinity, f(u, v ) = 0 otherwise. WORDS: f(u, v ) is twice f(x, y) above the diagonal in the first quadrant, otherwise f(u, v ) is zero.

(a). Use the "Marginals" formula on page 349 to get the marginal pdf f(u) of U from joint pdf f(u, v) HINT: You should know the answer before you plug into the formula.

(b) Use the "Marginals" formula on page 349 to get the marginal pdf f(v) of V from joint pdf f(u, v) HINT: You found f(v) in a previous HW by finding the CDF of V. You can also figure out the answer by thinking about two independent light bulbs and adding the probabilities of the two ways that V can fall into a tiny interval dv.

(c) Find the conditional pdf of V, given that U = 2. (See page 411). HINT: You can figure out what the answer has to be by thinking about two independent light bulbs and remembering the memoryless property.

(d) Find P( V > 3 | U= 2 ). (See bottom of page 411. Do the appropriate integral, but you should know what the answer will be.)

(e) Find the conditional pdf of U, given that V = 1. (See page 411).

(f) Find P ( U < 0.5 | V = 1).

HINT: You should know ahead of time whether the answer is > or < or = 1/2.

Using the

2 Superscript k Baseline greater than or equals n

​rule, determine the number of classes needed for the following data set sizes.

50

400

1250

2500

​a) The number of classes needed when

nequals

50is

nothing

.

​b) The number of classes needed when

nequals

400is

nothing

.

​c) The number of classes needed when

nequals

1250is

nothing

.

​d) The number of classes needed when

nequals

2500is

nothing

.

Question 18

A large hospital uses a certain intravenous solution that it maintains in inventory. Assume the hospital uses reorder point method to control the inventory of this item. Pertinent data about this item are as follows:

------------------------------------------------------------

Forecast of demand^a = 1,000 units per week

Forecast error^a, std. dev. =100 units per week

Lead time = 4 weeks

Carrying cost = 25 % per year

Purchase price, delivered = $52 per unit

Replenishment order cost = $20 per order

Stockout cost = $10 per unit

In-stock Probability during the lead time =90%

^a Normally distributed

------------------------------------------------------------

Due to possible rounding effect, please pick the closest number in the following options.

Question 19

If the hospital orders 400 units each time, what’s the total annual costs (holding cost + ordering cost + stock-out cost) excluding purchasing costs?

Question 19 options:

Use the following information to answer questions 17-20.

A large hospital uses a certain intravenous solution that it maintains in inventory. Assume the hospital uses reorder point method to control the inventory of this item. Pertinent data about this item are as follows:

------------------------------------------------------------

Forecast of demand^a = 1,000 units per week

Forecast error^a, std. dev. =100 units per week

Lead time = 4 weeks

Carrying cost = 25 % per year

Purchase price, delivered = $52 per unit

Replenishment order cost = $20 per order

Stockout cost = $10 per unit

In-stock Probability during the lead time =90%

^a Normally distributed

------------------------------------------------------------

Due to possible rounding effect, please pick the closest number in the following options.

Question 20

If the lead time is normally distributed with a mean of 4 weeks and a standard deviation of 0.5 weeks, what’s the reorder point?

Question 20 options:

In a hypothesis test, the significance level α (alpha) represents the threshold for how unlikely sample data has to be, assuming the null hypothesis, in order for us to reject the null hypothesis. Often alpha is taken to be 0.05, or 1/20, which means that even if the null hypothesis is true, there is a 1/20 chance that we will reject it because our sample data happens to have a P-value less than 0.05.

We want to study the heights of students. Our null hypothesis H_0 is that the average height is 67 inches, and the alternate hypothesis is that the average height is not 67 inches. We use significance level 0.05.

A) Suppose we run a hypothesis test. In the test, the data had a P-value less than 0.05. What should the conclusion of the test be?

B) Suppose we run 20 hypothesis tests on this, each with a different sample. In nineteen of these tests, our sample data had a P-value greater than 0.05. In one test, the data had a P-value less than 0.05. Does this evidence, as a whole, support the hypothesis that the average height is 67 inches? Explain your thinking.

Five students visiting the student health center for a free dental examination during National Dental Hygiene Month were asked how many months had passed since their last visit to a dentist. Their responses were as follows. 6 19 10 24 27 Assuming that these five students can be considered a random sample of all students participating in the free checkup program, construct a 95% confidence interval for the mean number of months elapsed since the last visit to a dentist for the population of students participating in the program. (Give the answer to two decimal places.) ( , )

In a study of academic procrastination, the authors of a paper reported that for a sample of 411 undergraduate students at a midsize public university preparing for a final exam in an introductory psychology course, the mean time spent studying for the exam was 7.34 hours and the standard deviation of study times was 3.90 hours. For purposes of this exercise, assume that it is reasonable to regard this sample as representative of students taking introductory psychology at this university.

(a) Construct a 95% confidence interval to estimate μ, the mean time spent studying for the final exam for students taking introductory psychology at this university. (Round your answers to three decimal places.)
(  ,  )

(b) The paper also gave the following sample statistics for the percentage of study time that occurred in the 24 hours prior to the exam.

n = 411      x = 43.98      s = 21.96

Construct a 90% confidence interval for the mean percentage of study time that occurs in the 24 hours prior to the exam. (Round your answers to three decimal places.)

Please show formulas, I just want to verify that I was doing the work correctly and please show one example of the case rate and ratio of cases, Thank you

1. The population of a state is 2,100,000. In a surveillance project lasting three months, investigators found communicable disease reports documenting one case of anthrax, three cases of mumps, 132 cases of pertussis, 27 cases of tuberculosis, 10 cases of primary and secondary syphilis, and 374 cases of gonorrhea.

Calculate the ratio of cases to population for each disease. Then, calculate the case rate (per 100,000 population per year) for each disease.

Calculate the ratio of cases to population for each disease.

Anthrax- 1 case for 3 months ratio- R=X/Y thus 1/2,586,000

Mumps-    …………………………… ratio- R=X/Y thus 3/2,586,000 reduced to 1/862,000

Calculate the case rate(per 100,000 population per year) for each disease

Anthrax- 1 case per 3 months thus 4 per year ……. [4/2,100,000] * 100,000 = 0.19

Mumps- 3 cases per 3 months thus 12 per year …..[12/2,100,000]* 100,000= 0.57

The table below gives the number of hours spent unsupervised each day as well as the overall grade averages for seven randomly selected middle school students. Using this data, consider the equation of the regression line, yˆ=b0+b1xy^=b0+b1x, for predicting the overall grade average for a middle school student based on the number of hours spent unsupervised each day. Keep in mind, the correlation coefficient may or may not be statistically significant for the data given. Remember, in practice, it would not be appropriate to use the regression line to make a prediction if the correlation coefficient is not statistically significant.