1. A 1200 kg roller coaster is given an initial velocity of 28 m/s (not the answer to problem 7) and travels uphill to a height of 30m above the ground. What is the velocity at the top of the hill?

2. A 1200 kg roller coaster is given an initial velocity of 28 m/s (not the answer to problem 7) and travels uphill to a height of 30m above the ground, then drops to a height of 13m above the ground. What is the final velocity?

3. The speed of a 1200 kg roller coaster is clocked at 15 m/s while at the top of hill of unknown height. To stop the ride, it runs into a large spring (k = 10000) and compresses it a distance of 10m before locking in place (to keep it from springing back the other way). What was the height of the hill when its speed was registered?

4. A 1200 kg roller coaster is traveling at 10 m/s. What is the magnitude of its momentum?

5. A 1200 kg roller coaster is traveling at 10 m/s when the train breaks mid-ride. One third of it (400 kg) slows down to 6 m/s. What is the velocity of the other two thirds (800 kg)?

6. A 1200 kg roller coaster train is traveling at 10 m/s when it collides with another broken train of mass 1000kg (no people on it) that is at rest on the tracks. The two trains stick together (perfectly inelastic). What is the resulting velocity of the two-train combination?

7. A 1200 kg roller coaster is traveling at 10 m/s when it collides with another broken train of mass 1000kg (no people on it) that is at rest on the tracks. The two trains bounce off each other (perfectly elastic). If the broken train (1000kg train) leaves at 11 m/s, what is the resulting velocity of the 1200 kg train?

A set of solar batteries is used in a research satellite. The satellite can run on only one battery, but it runs best if more than one battery is used. The variance σ² of lifetimes of these batteries affects the useful lifetime of the satellite before it goes dead. If the variance is too small, all the batteries will tend to die at once. Why? If the variance is too large, the batteries are simply not dependable. Why? Engineers have determined that a variance of σ² = 23 months (squared) is most desirable for these batteries. A random sample of 26 batteries gave a sample variance of 15.8 months (squared). Using a 0.05 level of significance, test the claim that σ² = 23 against the claim that σ² is different from 23.

a.)Find the value of the chi-square statistic for the sample. (Round your answer to two decimal places.)

b.)Find a 90% confidence interval for the population variance. (Round your answers to two decimal places.)

c.) Find a 90% confidence interval for the population standard deviation. (Round your answers to two decimal places.)

2. The data set (Concrete1.xlsx) contains the compressive strength, in thousands of pounds per square inch (psi), of 30 samples of concrete taken two and seven days after pouring. (You need to use The Paired Difference Test).

(a) At the 0.10 level of significance, is there evidence of a difference in the mean strengths at two days and at seven days?

(b) Find the p-value in (a) and interpret its meaning.

(c) At the 0.10 level of significance, is there evidence that the mean strength is lower at two days than at seven days?

(d) Find the p-value in (c) and interpret its meaning.

The data set contains the compressive strength, in thousands of pounds per square inch (psi), of 30 samples of concrete taken two and seven days after pouring.

(a) At the 0.10 level of significance, is there evidence of a difference in the mean strengths at two days and at seven days?

(b) Find the p-value in (a) and interpret its meaning.

(c) At the 0.10 level of significance, is there evidence that the mean strength is lower at two days than at seven days?

(d) Find the p-value in (c) and interpret its meaning.

Use the HousePrice data and via multiple regression select the two variables that predict the house selling price the best. Make another table with these two variables and answer the questions. Numerical answers are rounded so choose the answer that matches the best:

9. Identify the negative coefficient. What is its value and what is the interpretation of this number? (Choose the most appropriate answer. Note: numbers are truncated.)

• a. -0.037; This is a negative number and serves no statistical interpretation in this case.
• b. -0.037; This is how much the selling price of the house will decrease in hundred thousand as the house age by a year.
• c. -0.077; This show how much selling price correlates with the age of the house
• d. -1.92; For each additional bathroom the selling price of the house decrease by 1.92 hundred thousand dollars.

10. Which of the two variables has better P-value and what is this P-value? (Note: numbers are truncated.)

• a. Age has the best P-value; P-value = 0.067
• b. The higher the P-value, the better; so a P-value = 1.08E-09 is very good.
• c. #Bathroooms has the best P-value; P-value = 1.08E-09
• d. All P-values are poor since they are allhigher 5%.

11. Using this second table predict the selling price of a housethat is 10 years old, has 2 bathrooms and 3 rooms.

• a. 10.5 hundred thousands
• b. 12.57 hundred thousands
• c. 9.06 hundred thousands
• d. 11.58 hundred thousands

12. Based on the table would you characterize the Regression fit and the prediction as Poor, Good, Very Good, or Excellent?

• a. Poor
• b. Good
• c. Very Good
• d. Excellent

Age     #Bathrooms      #Rooms  #BedRooms       #FirePlaces     sellingPrice in $100000
42      1       7       4       0       4.9176
62      1       7       4       0       5.0208
40      1       6       3       0       4.5429
54      1       6       3       0       4.5573
42      1       6       3       0       5.0597
56      1       6       3       0       3.891
51      1       7       3       1       5.898
32      1       6       3       0       5.6039
32      1       6       3       0       5.8282
30      1       6       3       0       5.3003
30      1       5       2       0       6.2712
32      1       6       3       0       5.9592
32      1       6       3       0       5.6039
50      1.5     8       4       0       8.2464
17      1.5     6       3       0       7.7841
23      1       7       3       0       9.0384
22      1.5     6       3       0       7.5422
44      1.5     6       3       0       6.0931
3       1       7       3       0       8.14
31      1.5     8       4       0       9.1416
42      2.5     10      5       1       16.4202
14      2.5     9       5       1       14.4598
46      1       5       2       1       5.05
22      1.5     7       3       1       6.6969
40      1       6       3       1       5.9894
50      1.5     8       4       1       8.7951
48      1.5     8       4       1       8.3607
30      1.5     6       3       1       12

1. TopGarment is a boutique store specializing in high-end female apparel. The store must decide on the quantity of silk scarfs to order from China for the upcoming holiday selling season, which will last 14 weeks. The unit cost of each scarf is c=$40, and the scarf is sold at p=$150. A local discount store agrees to buy any leftover scarfs at the end of the season for s=$30 each. The store manager forecasts the weekly demand to be normally distributed with a mean D=20 and a standard deviation of σD= 15.

a. What is the optimal ordering quantity for TopGarment if it has to stock all the inventory before the selling season starts? What is the optimal expected profit?

b. TopGarment worries that silk scarfs may not be appreciated by local customers. After negotiating with the supplier, the supplier commits on a lead time of 6 weeks. This allows TopGarment to place two orders, one at least 6 weeks before the season starts and another one at the end of week 1 after observing the sales of the first week. Thus, inventory ordered from the 2nd order will arrive right before week 8 and can fulfill the demand for week 8 to week 14.

(i) How much inventory should the store manager order for the 1st order? Note that the 1 st order only needs to cover the demand in weeks 1-7.

(ii) Due to the recent overwhelming workload, the store manager forgot to update her demand forecast by the end of week 1 and had to decide on the 2 nd order quantity based on her old forecast. How much inventory would she order? What is the total expected profit of the entire selling season? (You can assume that unmet demand in the first 7 weeks is lost, but leftover inventory from the 1st order can be carried over for sales in weeks 8-14.)

(iii) Fortunately, the product manager paid enough attention and did another forecast based on the sales data of week 1. She found that the demand for the product had less uncertainty than what the store manager initially thought. The new forecasted standard deviation of the weekly demand drops to 3, whereas the mean weekly demand stays at 20. The product manager reported the new forecasts to the store manager. Now, how much inventory should the store manager order for the 2nd order? What is the total expected profit of the entire selling season? (Again, unmet demand in the first 7 weeks is lost and leftover inventory from the 1st order will be carried over.)

Wang Life Insurance Company issues a three year annuity that pays 40,000 at the end of each year. Wang uses the following three bonds to absolutely match the cash flow under this annuity:

1. A zero coupon bond which matures in one year for 1,000.
2. A two year bond which matures for 1,200 and pays an annual coupon of 100. This bond is priced using an annual yield of 7%.
3. A three year bond which matures for 2,000 and pays annual coupons of 75. This bond has a price of 1,750.

It costs Wang 104,000 to purchase all three bonds to absolutely match this annuity.

Calculate the one year spot interest rate.

Long two calls at 115, short a put at 130, short a share of stock.

(Please include a table with at least 50 data points for the graph. Graph for stock prices between 100 and 140.)

A small chemical company uses process control charts for the container filling processes within its operations. One of the products is a weed killer, Lawn Order. The company wants to setup process control charts for this product and has the following data.

                                     Observations

Sample            1          2          3          4          X-Bar R

     1                 50        90        40        80        65.0     50

     2                 50        60        80        60        62.5     30

     3                 40        70        40        90        60.0     50

     4                 50        60        100      90        75.0     50

     5                 90        60        50        30        57.5     60

     6                 70        60        90        40        65.0     50

     7                 100      30        80        70        70.0     70

     8                 50        100      60        90        75.0     50

A. What are the center line, lower control limit, and upper control limit equal to for the X-Bar chart? (Use the sample range to calculate the upper and lower control limits)

A consumer finds only three products, X, Y, and Z, are for sale. The amount of utility which their consumption will yield is shown in the table below.

Assume that the prices of X, Y, and Z are $10, $2, and $8, respectively.

The consumer has an income of $74 to spend.

(a)          Complete the table by computing the marginal utility per dollar for successive units of X, Y, and Z to one or two decimal places.

(b)          How many units of X, Y, and Z will the consumer buy when maximizing utility and spending all income?

(c)           Why would the consumer not be maximizing utility by purchasing 2 units of X, 4 units of Y, and 1 unit of Z?