A manufacturer claims that the mean lifetime, u, of its light bulbs is 50 months. The standard deviation of these lifetimes is 8 months. Fifty bulbs are selected at random, and their mean lifetime is found to be 49 months. Can we conclude, at the 0.1 level of significance, that the mean lifetime of light bulbs made by this manufacturer differs from 50 months?

Provide a description of the sample space for each of the following random experiments. Identify if the sample space is discrete or continuous in each case.
Note that there can be more than one acceptable interpretation of each experiment. State any assumptions you may have made.
a) Shuffling a standard deck of cards and revealing the top card.
b) Shuffling a standard deck of cards and revealing the top and bottom cards.

1)The life in hours of a battery is known to be approximately normally distributed, with standard deviation 1.25 hours. A random sample of 10 batteries has a mean life of 40.5 hours. A. Is there evidence to support the claim that battery life exceeds 40 hours ? Use α= 0.05.

B. What is the P-value for the test in part A?

C. What is the β-error for the test in part A if the true mean life is 42 hours?

D. What sample size would be required to ensure that doesnot exceed 0.10 if the true mean life is 44 hours?

2) To study the elastic properties of a material manufactured by two methods, the deflection in mm of 2x15 elastic bars made from the two materials are measured, and the are shown in the table:

A) Construct Relative Frequency Plots for the two sets of data, use 6 bins of increment of 10 starting from 170mm, and then create a Cumulative Distributions of the two data sets. By comparing the slopes of the cumulative distributions, do you think these plots provide support of the assumptions of normality and equal variances for the populations of the two sets?

B) Assuming equal variances of population, do the data support the claim that the mean deflection from method 2 exceeds that from method 1? Use α= 0.05

C) Resolve part B assuming that the variances of populations are not equal.

D)Suppose that if the mean deflection from method 2 exceeds that of method 1 by as much as 5mm, it is important to detect this difference with probability at least 0.90. Is the choice of n1=n2=15 in part A of this problem adequate

E) Make a test that can be used to tell which data set is more accurate than the other (whether method 1 is more accurate or from method two) .

1. Dubois Corporation has four outlets across the country: Boston, Dallas, Los Angeles, and St. Paul. Plans call for the following number of products (in 1000s) to be needed at each location.

Boston             70

Dallas              80

Los Angeles    100

St. Paul            100

            Dubois has two warehouses located in St. Louis and Pittsburgh.

Dubois has three plants capable of producing the product. The plants and production capabilities are as follows:

            Denver            100

            Atlanta            100

            Chicago           150

Transportation costs are as follows:

                                    To       St. Louis          Pittsburgh

            From    Denver                  4                      7

                        Atlanta                  5                      4

                        Chicago                 2                      5

                        To                    Boston             Dallas Los Angeles   St. Paul

            From    St. Louis                4                     4                 6               8

                        Pittsburgh              3                     5                 8               7

1. Develop a network representation of this problem.

2. Set up the objective function and the constraints.

1. The Ace Manufacturing Company has orders for three products. Each product can be produced on one of three different machines. The time (hours) for producing each product on each machine is given below:

Product

Machine                      A                    B                     C

                 1                             11                    12                    10

                 2                             12                    9                     11

                 3                             13                    15                    12       

Each machine can be assigned to only one product. Set up the objective function and the constraints necessary to minimize the total time.

A. Explain which model in the “Data and Graphs” attachment is most accurate, based only on their graphical qualities and R² values, which can both be found in the “Data and Graphs” attachment.

Note: R² is the square of the correlation coefficient between the data and the model.

B. Given that the actual U.S. Population in 2010 was 308.75 million, explain which of the following models is most accurate, including computations of the relative errors, based only on the following U.S. population predictions in millions by each model for the year 2010:

• linear: 242.89

• exponential: 515.34

• quadratic: 304.36

• third-degree polynomial: 308.22

• fourth-degree polynomial: 311.96

A. Explain which model in the “Data and Graphs” attachment is most accurate, based only on their graphical qualities and R² values, which can both be found in the “Data and Graphs” attachment.

Note: R² is the square of the correlation coefficient between the data and the model.

B. Given that the actual U.S. Population in 2010 was 308.75 million, explain which of the following models is most accurate, including computations of the relative errors, based only on the following U.S. population predictions in millions by each model for the year 2010:

• linear: 242.89

• exponential: 515.34

• quadratic: 304.36

• third-degree polynomial: 308.22

• fourth-degree polynomial: 311.96

A. Explain which model in the “Data and Graphs” attachment is most accurate, based only on their graphical qualities and R² values, which can both be found in the “Data and Graphs” attachment.

Note: R² is the square of the correlation coefficient between the data and the model.

B. Given that the actual U.S. Population in 2010 was 308.75 million, explain which of the following models is most accurate, including computations of the relative errors, based only on the following U.S. population predictions in millions by each model for the year 2010:

• linear: 242.89

• exponential: 515.34

• quadratic: 304.36

• third-degree polynomial: 308.22

• fourth-degree polynomial: 311.96

A. Explain which model in the “Data and Graphs” attachment is most accurate, based only on their graphical qualities and R² values, which can both be found in the “Data and Graphs” attachment.

Note: R² is the square of the correlation coefficient between the data and the model.

B. Given that the actual U.S. Population in 2010 was 308.75 million, explain which of the following models is most accurate, including computations of the relative errors, based only on the following U.S. population predictions in millions by each model for the year 2010:

• linear: 242.89

• exponential: 515.34

• quadratic: 304.36

• third-degree polynomial: 308.22

• fourth-degree polynomial: 311.96

A. Explain which model in the “Data and Graphs” attachment is most accurate, based only on their graphical qualities and R² values, which can both be found in the “Data and Graphs” attachment.

Note: R² is the square of the correlation coefficient between the data and the model.

B. Given that the actual U.S. Population in 2010 was 308.75 million, explain which of the following models is most accurate, including computations of the relative errors, based only on the following U.S. population predictions in millions by each model for the year 2010:

• linear: 242.89

• exponential: 515.34

• quadratic: 304.36

• third-degree polynomial: 308.22

• fourth-degree polynomial: 311.96

A. Explain which model in the “Data and Graphs” attachment is most accurate, based only on their graphical qualities and R² values, which can both be found in the “Data and Graphs” attachment.

Note: R² is the square of the correlation coefficient between the data and the model.

B. Given that the actual U.S. Population in 2010 was 308.75 million, explain which of the following models is most accurate, including computations of the relative errors, based only on the following U.S. population predictions in millions by each model for the year 2010:

• linear: 242.89

• exponential: 515.34

• quadratic: 304.36

• third-degree polynomial: 308.22

• fourth-degree polynomial: 311.96

A. Explain which model in the “Data and Graphs” attachment is most accurate, based only on their graphical qualities and R² values, which can both be found in the “Data and Graphs” attachment.

Note: R² is the square of the correlation coefficient between the data and the model.

B. Given that the actual U.S. Population in 2010 was 308.75 million, explain which of the following models is most accurate, including computations of the relative errors, based only on the following U.S. population predictions in millions by each model for the year 2010:

• linear: 242.89

• exponential: 515.34

• quadratic: 304.36

• third-degree polynomial: 308.22

• fourth-degree polynomial: 311.96

4. ?? , ? = 1, . . . , ? are i.i.d. and has p.d.f. ?(?) = { 0 ? < 0 ??−? + (1 − ?)2? −2? ? ≥ 0 , here 0 ≤ ? ≤ 1. Write down the likelihood function. (10 points) When ? = 1, write down the MLE of ?. (10 points) When ? = 1, write down the bias and variance of the MLE of ?. (10 points)

1. Huntington is considering have police sub-stations to provide better coverage of the city. There are six (6) areas in the city. There are five possible locations for sub-stations. Given the data below, formulate the integer programming model that could be used to find the minimum number of locations necessary to provide coverage to all areas.

Potential Locations                             Areas Covered

                                    A                                                         1, 5

                                    B                                                         2, 3, 5, 6         

                                    C                                                         1, 4

                                    D                                                         1, 2, 4, 6

                                    E                                                          4, 5

1. You would like to estimate the starting salaries of recently graduated business majors (B.S. in any business degree). You randomly select 60 recently graduated business majors and get a sample mean of $43,800 and the population standard deviation is known to be $8,198

A. Construct a 90% confidence interval to estimate the average starting salary of a recently graduated business major (Round to the nearest penny and state the answer as an interval – for example $351.89 to $728.14).

B. Using the same confidence level, you would like the margin of error to be within $500, how many recently graduated business majors should you sample?

2. You would like to estimate the amount of student loan debt a graduating senior will have at the time of repayment which begins in November. You randomly select 72 graduating seniors and get a sample mean of $31,172 with a standard deviation of $6,423. Construct a 98% confidence interval for the amount of debt a graduating senior will have. (Make sure you are careful selecting the correct values and that you round to the nearest penny. You will not get any credit if you are off by more than 1 cent.) (Round to the nearest penny and state the answer as an interval – for example $351.89 to $728.14).

3. You would like to estimate the proportion of student loan debts that are in default. You randomly select 211 people who have student loans and find that 25 are in default.

A. Construct a 95% confidence interval to estimate the proportion of student loans that are in default. (Round your sample proportion to 4 decimal points as well as round your margin of error to 4 decimal points. For example: .23916 would be .2392 and this represents 23.92%. State the answer as an interval – for example 27.36% to 31.43%).

B. Using the same confidence level, you would like the margin of error to be with 3%, how many people with student loans should you sample?

The credit scores for 12 randomly selected adults who are considered high risk borrowers before and two years after they attend a personal finance seminar are given below.

You will run a significance test to check if there is enough evidence to support the claim that the personal finance seminar helps adults increase their credit scores.

You’ll use α = 0.01 for significance test.

1. 1. Write out the null and alternative hypothesis associated with the research question.
2. 2. What type of statistical test will you use to answer the proposed research question? (Note: Is this a z-test or a t-test for independent or dependent samples)
3. 3. What is the critical value at the 0.01 level of significance? Be sure and include whether this critical value is a z or t value and, if appropriate, include the degrees of freedom associated with this statistical test.
4. 4. What is the standardized test statistic?
   5. At 0.01 significance level what decision should be made about the null hypothesis? In other words, should you reject or retain the null hypothesis?
   6. Provide a brief conclusion regarding your findings.
   7. Show your excel output

An agent for a residential real estate company in a suburb located outside of Washington, DC, has the business objective of developing more accurate estimates of the monthly rental cost for apartments. Toward that goal, the agent would like to use the size of an apartment, as defined by the square footage to predict the monthly rental cost. The agent selects a sample of one-bedroom apartments and collects and stores the data.

a. Construct a scatter plot.

b. Use the least-squares method to determine the regression coefficients b0 and b1.

c. Interpret the meaning of b0 and b1 in this problem.

d. Predict the mean monthly rent for an apartment that has 800 square feet.

e. Why would it not be appropriate to use the model to predict the monthly rent for apartments that have 1,500 square feet?

f. Your friends Jim and Jennifer are considering signing a lease for a one-bedroom apartment in this residential neighborhood. They are trying to decide between two apartments, one with 800 square feet for a monthly rent of $1,130 and the other with 830 square feet for a monthly rent of $1,410. Based on (a) through (d), which apartment do you think is a better deal?

A woman who was shopping in Los Angeles had her purse stolen by a young, blonde female who was wearing a ponytail. The blonde female got into a yellow car that was driven by a man with a mustache and a beard. The police located a blonde female named Janet who wore her hair in a ponytail and had a male friend who had a mustache and beard and also drove a yellow car. Based on this evidence the police arrested the two suspects. Because there were no eyewitnesses and no real evidence, the prosecution used probability to make its case against the defendants. The probabilities listed below were presented by the prosecution for the known characteristics of the thieves. Characteristic Probability Yellow car 1/10 Man with mustache 1/4 Woman with ponytail 1/10 Woman with blonde hair 1/3 Man with beard 1/10 Interracial couple in a car 1/1000

(a) Assuming that the characteristics listed above are independent of each other, what is the probability that a randomly selected couple has all these characteristics? That is what is, calculate the probability: P( “yellow car” and “man w/ mustache, beard and … “interracial couple in car”)?

(b) Based on the above result would you convict the defendant? Explain thoroughly.

(c) Now let n represent the number of couples in the Los Angeles area who could have committed the crime. Let p represent the probability that a randomly selected couple has all 6 characteristics listed in the table. Assuming that the random variable X follows the binomial probability function, we have: ?(?) = ?(?,?) ∙ ? ? ∙ (1 − ?) ?−? , ? = 0, 1, 2, … ? Note: Use the calculator link http://stattrek.com/online-calculator/binomial.aspx Assuming there are n = 50,000 couples in the Los Angeles area, what is the probability that more than one of them has the characteristics listed in the table? ?(? > 1) =

(d) Does this result cause you to change your mind regarding the defendant’s guilt? Explain.

(e) The probability that more than one couple has these characteristics assuming there is at least one couple is given by the formula below and each is evaluated with the binomial formula from (c). ?( ? > 1 ∣ ? ≥ 1 ) = ?(? > 1) ?(? ≥1) = (f) Do you think the couple should be convicted “beyond all reasonable doubt” based on the answer from part (e)? Explain why or why not.

7. The following data is the weight of diamond x with the US dollar price y.

x : 0.3 0.4 0.5 0.5 1.0 0.7

y : 510 1151 1343 1410 5669 2277

(a) Find the regression equation for the data points given.

(b) Determine the percentage of variation in price of diamond y that is explained by the weight x.

(c) Is it reasonable to predict the price of a 0.8-carat diamond using this model? If yes, predict it. If no, state why.

(d) Is it reasonable to predict the price of a 1.5-carat diamond using this model? If yes, predict it. If no, state why.

This is strictly to check answers.

The accompanying table shows a portion of a data set that refers to the property taxes owed by a homeowner (in $) and the size of the home (in square feet) in an affluent suburb 30 miles outside New York City.

a. Estimate the sample regression equation that enables us to predict property taxes on the basis of the size of the home. (Round your answers to 2 decimal places.)

Taxes= [ ] + [ ]Size

c. Predict the property taxes for a 1,700-square-foot home. (Round coefficient estimates to at least 4 decimal places and final answer to 2 decimal places.)

Taxes= [ ]

Excel Data File

A psychologist is interested in the relationship between color of food and appetite. To explore this relationship, the researcher bakes small coockies with icing of one of three different colors (green, red or blue). The researcher offers cookies to subjects while they are performing a boring task. Each subject is run individually under the same conditions, except for the color of the icing on the cookies that are available. Six subjects are randomly assigned to each color. The number of cookies consumed by each subject during the 30-minute session is shown in table below.

a. Calculate the F statistic: ***PLEASE show handwritten work for calculating SSbetween.***

b. Find the critical F(a = .01)

c. Present your results in the form of a summary table