Exercise 1: Defective Light Bulbs

This is a Statistic subject.

The data set lists the number of defective 60-watt light bulbs found in samples of 100 bulbs selected over 25 days from a manufacturing process. Assume that during these 25 days, the manufacturing process was not producing an excessively large fraction of defectives.

1.Plot a p chart to monitor the manufacturing process. What do you conclude? Is the process out of control?

2. How large must the fraction of defective items be in a sample selected from the manufacturing process before the process is suspected to be out of control?

3. During a given day, suppose a sample of 100 items is selected from the manufacturing process and 15 defective bulbs are found.

(a) Obtain an upper bound on the probability of observing such a sample? (hint: here you can assume that the real proportion p is indeed less than the UCL and thus UCL (upper control limit) is an upper bound for p).

(b) If a decision is made to shut down the manufacturing process in an attempt to locate the source of the implied controllable variation, what is the probability that this decision is based on erroneous conclusions?

13) Calculate the present value of a perpetuity with a $6.50 payment per year and a 6.5% annual interest rate.

To calculate the present value of a perpetuity, you just divide the payment by the interest rate. Therefore, the present value would be:

a) You are offered an investment that will pay the following cash flows at the end of each of the next five years:

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How much would you be willing to pay for this investment if your required rate of return is 12% per year?

Use Excel’s =NPV(B1,B5:B9) function. Note that we did not include the period 0’s cash flow in the function. Excel’s NPV function doesn't really calculate net present value. Instead, it simply calculates the present value of uneven cash flows. It does not take the cost of the initial outlay into account.

B) How much you will get if you invest the following cash flows at 12% per year?

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There is no function to calculate the future value of uneven cash flows. Therefore, we need to find the future value of each of the cash flows individually and then add them all together.

The cash flow in period 1 needs to be taken four periods forward (moved from period 1 to 5) so the formula in C5 is: =FV($B$1,$A$9-A5,0,B5). Notice that NPer is calculated by taking the period of the last cash flow (5, in A9) minus the period of the current cash flow (1, in A5). Also, note that the dollar signs serve to freeze the reference so that when you copy the formula down those addresses won't change (i.e., they are absolute references). Copy and then paste that formula into A6:A9. To find the future value of the cash flows in B11, use the formula: =SUM(C5:C9).

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Another way to find the future value of any set of cash flows is to first find the present value of those cash flows and then to find the future value of that present value. The picture, below, demonstrates the process:

We already saw that we can calculate the present value of uneven cash flows using the NPV function, so we will use the NPV function for the PV argument in the FV function. The formula becomes: =FV(B1,A9,0,NPV(B1,B5:B9)).

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A firm produces two products, the respective quantities of which
are ?1 and ?2. The firm has agreed to produce a total of 100 units of
both products for a client. The total cost TC for the firm is
??(?1, ?2) = 2?1 + ?1?2 + 4?2
with demand functions ?1 = 20 − ?1 + ?2 and ?2 = 30 + 2?1 − ?2.
Show that the firm’s maximum profit is 2,204, where profit is
?(?1, ?2) = ?1?1 + ?2?2 − ??(?1, ?2)

I need to make sure that my answers are correct

please review.

The Case as following:

Focus Drilling Supplies has been growing steadily over the last 20 years. With increased exploration in the mining sector, the company has decided to expand their facilities for supplies and custom drill bit production to meet the increased demand.

The expansion will occur over 4 years and is expected to require $2.8 million.

Management has developed a payment plan for carrying out this expansion. The plan requires a cash input of $300,000 now, $700, 000 one year from now, $800,000 two years from now, and finally, $1,000,000 four years from now.

While Focus Drilling is able to allocate $2.6 million dollars from the cash reserves to fund the project, there is no other contingency fund for cost overruns or construction delays. However, any interest earned on the expansion fund during the four-year time frame will be set aside for the project.

Before the final decision on implementation, the company treasurer is asked to assess the plan to determine if the current $2.6 million allocation will meet the $2.8 million in payment obligations of the plan over the four-year period. The Treasurer has predicted interest rates over the next four years to be as follows:

Year 1: interest rate of 4.5% p.a. compounded semi-annually

Year 2: interest rate of 5.0% p.a. compounded semi-annually

Year 3: interest rate of 5.0% p.a. compounded semi-annually

Year 4: interest rate of 5.5% p.a. compounded semi-annually

As an alternative, the Treasurer has found that the $2.6 million cash allocation could be invested at a fixed rate of 5.2% p.a. compounded quarterly for a period of five years.

The company can withdraw part of the money from either investment at any time without penalty to meet the cash payment requirements.

Questions (total of 40 marks)

1-Can Focus Drilling meet the cash payment requirements of the expansion given the variable interest rates given above? Use today as a focal date. Show your calculations.

as per the above calculations Focus drilling is able to meet the cash paymet requirements given the variable interest rates

2-What, if any, is the accumulated value of the difference between the allocated cash available and the total cash payments required for the expansion?

Accumulated value of the difference between the allocated cash available and total cash payments made is $99180.40.

3-Can Focus Drilling meet the cash payment requirement by investing with the fixed five-year interest rate given above? Show your calculations.

Focus drill can meet its payment requirement by investing with th fixed 5 year interest rate.

4-What, if any, is the accumulated value of the difference between the allocated cash available and the total cash payments required for the expansion?

Accumulated value of the difference between allocated cash available and total cash payment is $130,106.48

5-Which investment strategy should the company select and why?

In both of the options the company is able to meet the cash payments. However under fixed rate option the balnce at the end of year 4 is more as compared to variable interest rates.The company can withdraw the balance at any time without incurring any penalty. Hence they should go with the fixed rate strategy for investment as it gives higher return.

An investor is very confident that a stock will change significantly over the next few months; however, the direction of the price changes is unknown. Which pair of strategies is most likely to produce a profit if the stock price moves as expected?

I. Short butterfly spread

2. Bearish calendar spread

3. Long at-the-money straddle

4. Short strangle

A. 2 and 3

b. 2 and 4

C. 1 and 4

D. 1 and 3

PROBLEM 5 – HOUSING PRICES

Situation:

Real Estate One conducted a recent survey of house prices for properties located on the shores of Tawas Bay. Data on 26 recent sales, including the number of bathroom, square feet and bedrooms are below.

Action:

Use the data above and multiple regression to produce a model to predict the average sale price from other variables. Comment on the following:

a. Regression equation

b. R, R2 and 1-R2, adjusted R2

c. Standard error of estimate

d. Report the t's for each value and the corresponding p-values

e. Overall test of hypothesis and decision

f. Use a .05 level of significance. Cite which variables are significant and which are not significant, based on the t values and p values for each independent variable.

Quiz 5

1.   Use the following definitions to “fill” the arrays. Write in both the data (in the grid) and the index (below the grid). Cross out any unused locations. [6 each]

            a.          final int SIZE = 7;

            int nums [SIZE] = {5, 1, 3, 8, 4};

nums

            b.         final int SIZE = 5;

            float values[SIZE] = {5.7, 3.2, 9.4, 2.5, 8.5};

values

            c.          char letters[] = {'Z', 'Y', 'X', 'W', 'V', 'U'};

letters

2.   Follow the segment of code, and use the grid to keep track of the array. Below, state only the requested values in the array at the conclusion of the code segment.

x

final int SIZE = 6;

int x[SIZE] = {5, 7, 6, 2};

x[2] = x[1];

x[5] = 28;

x[1] = 15;

x[3] = x[1] * 2;

x[1] = 32;

x[2] = x[4];

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

            a.          x[1] = _____________ [1]

            b.         x[2] = _____________ [1]

            c.          x[3] = _____________ [1]

            d.         x[4] = _____________ [1]

            e.          x[5] = _____________ [1]

            f.          x[8] = _____________ [1]

eigenvalues of the matrix A = [1 3 0, 3 ?2 ?1, 0 ?1 1] are 1, ?4 and 3. express the equation of the surface x^2 ? 2y^2 + z^2 + 6xy ? 2yz = 16. How should i determine the order of the coefficient in the form X^2/A+Y^2/B+Z^2/C=1?

Find the relative extrema, if any, classify as absolute max/min.

a.) f(x)= x+1/x-2 on [2,4]

b.) f(x)= x^(2) -2x-3 on [-2,3]

c.) f(x)= x^(2/3) (x^2-4) on [-1,3]

Solve for x:

a.) 6^(2x) =36

b.) 2^(2x) -4 * 2^(x) +4=0

c.) 3^(x-x^2) =1/9x

Determine if the following statements are true or false. If it is true, explain why. If it is false, provide an example.

a.) If a and b are positive numbers, then (a+b)^x=a^x+b^x

b.) If x < y, then e^x < e^y

c.) If 0 < b <1 and x < y then b^x > b^y

d.) if e^kx > 1, then k > 0 and x >0