a. If ? ~ ?(?,?), find the distribution (name and parameters) of ?=?X+?.

b. If ?~?(0,?), find the density of ?=|?|.

c. If ? ~ ?amma(?,?), find the distribution (name and parameters) of ?=?X.

d. If ? ~ Uniform(0,1), find the density function of ?=√?.

e. If Θ ~ Uniform(−?/2,?/2), find cdf and the density function of ?=tan(Θ).

The weight of male babies less than 2 months old in the United States is normally distributed with mean 12.3 pounds and standard deviation 3.8 pounds.

(a) Find the 84 th percentile of the baby weights.

(b) Find the 11 th percentile of the baby weights.

(c) Find the third quartile of the baby weights.

Use the TI-84 Plus calculator and round the answers to at least two decimal places.

A random sample of size n = 71 is taken from a population of size N = 639 with a population proportion p = 0.73.

a-1. Is it necessary to apply the finite population correction factor?

a-2. Calculate the expected value and the standard error of the sample proportion. (Round "expected value" to 2 decimal places and "standard error" to 4 decimal places.)

b. What is the probability that the sample proportion is less than 0.66? (Round “z” value to 2 decimal places, and final answer to 4 decimal places.)

In cohort studies of the roles of a suspected factor in the etiology of a disease, it is essential that: There be equal numbers of persons in both study groups At the beginning of the study, those with the disease and those without the disease have equal risks of having the factor The study group with the factor and the study group without the factor be representative of the general population The exposed and nonexposed groups under study be as similar as possible with regard to possible confounding factors Both b and c

The observations are listed : 5.32, 9.87, 11.25, 10.94, 5.58, 6.29, 7.47, 10.75, 6.22, 8.00. Apply backward empirical rule, IQR/S, and normal probability plot to check the normality assumption.(detail)

A researcher performed a retrospective study of the records of patients receiving care for hypercholesterolemia. The data gives measurements of total cholesterol for patients before and 6 weeks after taking a statin drug. Is there sufficient evidence at the α=.05 level significance to conclude that on an average the drug would result in reduction in total cholesterol in hypercholesterolemia patients. Assume total cholesterol in hypercholesterolemia patients is normally distributed

Using a Paired Sample t-test, what is the critical value of the test statistic?

Forecasting: Measuring Forecast Accuracy

Tires for You, Inc. (TFY), founded in 1987, is an automotive repair shop specializing in replacement tires. Located in Altoona, Pennsylvania, TFY has grown successfully over the past few years because of the addition of a new general manager, Ian Overbaugh. Since tire replacement is a major portion of TFY’s business (it also performs oil changes, small mechanical repairs, etc.), Ian was surprised at the lack of forecasts for tire consumption for the company. His senior mechanic, Skip Grenoble, told him that they usually stocked for this year what they sold last year. He readily admitted that several times throughout the season, stockouts occurred and customers had to go elsewhere for tires. Although many tire replacements were for defective of destroyed tires, most tires were installed on cars whose original tires had worn out. Most often, four tires were installed at the same time. Ian was determined to get a better idea of how many tires to hold in stock during the various months of the year. Listed below is a summary of individual tire sales by month.

Period 2010 October November December 2011 January February March April May June July August September October November December Tires Used 9,800 11,000 11,000 9,700 8,800 9,300 10,700 9,300 8,700 10,200 10,800 9,700 10,200 11,600 11,100

Ian has hired you to determine the best technique for forecasting TFY demand based on given data.

1. Calculate forecasts for August, 2011 through December 2011 using a simple four-month moving average. Round your answers to the whole number.

Period 2011 August September October November December Forecasts

2. Calculate forecasts for August, 2011 through December 2011 using the exponential smoothing method with α = 0.4. Assume the forecast for August, 2011 is 11,000. Round your answers to the whole number.

Period 2011 August September October November December Forecasts 11,000

3. Calculate the forecast errors, the MAD, the MSE, and the MAPE for the forecasts you made in Question 1. Use the actual sales data for 2011. Round your answers to the whole number.

Period August September October Actual Sales 10,800 9,700 10,200 Forecasts Errors Absolute Error Error2 Absolute % Error November 11,600 December 11,100 MAD= MSE= MAPE= 2

4. Calculate the forecast errors, MAD, MSE, and MAPE for the forecasts you made in Question 2. Use the actual sales data for 2011 given below. Round your answers to the whole number.

Period August September October Actual Sales Forecasts 10,800 11,000 9,700 Errors Absolute Error Error2 Absolute % Error 10,200 November 11,600 December 11,100 MAD= MSE= MAPE=

5. Based on the two methods used to calculate forecasts for TFY, which method produced the best forecast? The moving average method or the exponential smoothing method?

1. The regression model below examines factors that impact the DV home sale price among recently sold homes in a suburban community. The IV is total rooms within the home, and the EVs include 1) total bedrooms, 2) total bathrooms, 3) whether the home has a basement or not, and 4) the total number of days the home was on the market prior to its sale. Please answer the following regarding the regression output below:

   A) Identify the variables that are statistically significant predictors (at the .05 percentlevel) of a home’s sale price. [3]

   B) Of the variables you identified in Part A, what statistics did you examine todetermine each variable’s statistical significance? Explain the thresholds used to determine statistical significance. [3]

   C) Explain what the coefficient for the EV “Total Bathrooms” means. [3]D) Explain what the coefficient for the EV “Days on Market” means. [3]E) What does the model’s R-square of .33 mean? [3]

Kindly show how to do in Excel: Suppose that the weight of a typical American female follows a normal distribution with μ = 140 lb. and σ = 20 lb. Also, suppose 80.15% of all American females weigh more than I weigh. What fractions of American females weigh more than 165 pounds? How much do I weigh? If I weighed 20 pounds more than I do, what percentile would I be in? A production process manufactures items with weights that are normally distributed with mean 10 pounds and standard deviation 0.1. An item is considered to be defective if its weight is less than 9.8 pounds or greater than 10.2 pounds. Suppose that these items are currently produced in batches of 1000 units. Find the probability that at most 5% of the items in a given batch will be defective Find the probability that at least 85% of these items in a given batch will be acceptable.

1. Toyota has been manufacturing small automobiles that have averaged 50 miles per gallon of gasoline in highway driving. The company has developed a more efficient engine for its small cars and now advertises that its new small cars average more than 50 miles per gallon in highway driving. An independent testing service road-tested 36 of the automobiles. The sample showed an average of 51.5 miles per gallon. The population standard deviation is 6 miles per gallon.

1. With a 0.025 level of significance, test to determine whether or not the manufacturer's advertising campaign is legitimate.

1. Use critical value approach

2. Use p value approach

3. State a type 1 error for this problem

4. State a type 2 error for this problem

1. Toyota has been manufacturing small automobiles that have averaged 50 miles per gallon of gasoline in highway driving. The company has developed a more efficient engine for its small cars and now advertises that its new small cars average more than 50 miles per gallon in highway driving. An independent testing service road-tested 36 of the automobiles. The sample showed an average of 51.5 miles per gallon. The population standard deviation is 6 miles per gallon.

1. With a 0.025 level of significance, test to determine whether or not the manufacturer's advertising campaign is legitimate.

1. Use critical value approach

2. Use p value approach

3. State a type 1 error for this problem

4. State a type 2 error for this problem

A researcher is testing a new painkiller that claims to relieve pain in less than 15 minutes, on average. A random sample of 49 painkillers were tested and the mean time was 13 minutes. Suppose the population standard deviation is 7 and the test is conducted at α=0.05.

1. State the null and alternate hypothesis

2. Test this hypothesis
   1. Use the critical value approach

Conclusion:

2. Use the p value approach

3. What is your conclusion?

4. What is a type 1 error?

5. What is a type 2 error?

Chapter 6, Section 2-CI, Exercise 111

What Influences the Sample Size Needed?

In this exercise, we examine the effect of the value of the estimated standard deviation on determining the sample size needed.

Find the sample size needed to give, with 95% confidence, a margin of error within ±5 , if the estimated standard deviation is ά= 50. If the estimated standard deviation is ά=20. If the estimated standard deviation is ά=10. Round your answers up to the nearest integer.

ά= 50: n=________

ά= 20: n= _________

ά= 10: n=_______

Here is a sample of amounts of weight change​ (kg) of college students in their freshman​ year: 12​,7​,0​,−8​, where −8 represents a loss of 8 kg and positive values represent weight gained. Here are ten bootstrap​ samples:

{12, 12, 12, 0}​,

{12, −8, 0, 12}​,

{12, −8, 7, 0}​,

{7, −8, 0, 12}​,

{0, 0, 0, 7}​,

(7, −8, 7, −8}​,

{12, 7, −8, 0}​,

{−8, 7, −8, 7}​,

{−8, 0, −8, 7}​,

{7, 12, 12, 12}.

Complete parts​ (a) and​ (b) below.

(a) Using only the ten given bootstrap​ samples, construct an 80​% confidence interval estimate of the mean weight change for the population.

(b) Using only the ten given bootstrap​ samples, construct an 80​% confidence interval estimate of the standard deviation of the weight changes for the population.

An owner of a football stadium is determining how much fertilizer is needed to keep the grass in the football stadium green. Thus, the owner collected the following data on how much fertilizer was used in 10 football stadiums from throughout the country. The unit of measurement is number of pounds per 1,000 square feet.

Field Fertilizer

1 1.35

2 1.46

3 1.51

4 1.42

5 1.46

6 1.35

7 1.59

8 1.37

9 1.33

10 1.66

Prior to the collection of these above data, the owner thought that the population mean amount of fertilizer applied to football fields was μ = 1.3 pounds per 1,000 square feet. Do these data suggest that it the actual population mean amount is greater than 1.3 pounds per 1,000 square feet? Show your work

The ideal (daytime) noise-level for hospitals is 45 decibels with a standard deviation of 12 db; which is to say, this may not be true. A simple random sample of 70 hospitals at a moment during the day gives a mean noise level of 47 db. Assume that the standard deviation of noise level for all hospitals is really 12 db. All answers to two places after the decimal.

(a) A 99% confidence interval for the actual mean noise level in hospitals is (___ db, ___ db)

(c) Unless our sample (of 81 hospitals) is among the most unusual 2% of samples, the actual mean noise level in hospitals is between ___db and ___ db

(d) A 99.9% confidence interval for the actual mean noise level in hospitals is (42.28db, ___ db)

(g) How many hospitals must we examine to have 95% confidence that we have the margin of error to within 1 db?

(h) How many hospitals must we examine to have 99.9% confidence that we have the margin of error to within 1 db?