Specialty Toys, Inc. sells a variety of new and innovative children’s toys. Management learned that the preholiday season is the best time to introduce a new toy, because many families use this time to look for new ideas for December holiday gifts. When Specialty discovers a new toy with good market potential, it chooses an October market entry date. In order to get toys into its stores by October, Specialty places one-time orders with its manufacturers in June or July of each year. Demand for children’s toys can be highly volatile. If a new toy catches on, a sense of shortage in the marketplace often increases the demand to high levels and large profits can be realized. However, new toys can also flop, leaving Specialty stuck with high levels of inventory that must be sold at reduced prices. The most important question the company faces is deciding how many units of a new toy should be purchased to meet anticipated sales demand. If too few are purchased, sales will be lost; if too many are purchased, profits will be reduced because of low prices realized in clearance sales. For the coming season, Specialty plans to introduce a new product called Weather Teddy. This variation of a talking teddy bear is made by a company in Taiwan. When a child presses Teddy’s hand, the bear begins to talk. A built-in barometer selects one of five responses that predict the weather conditions. The responses range from “It looks to be a very nice day! Have fun” to “I think it may rain today. Don’t forget your umbrella.” Tests with the product show that, even though it is not a perfect weather predictor, its predictions are surprisingly good. Several of Specialty’s managers claimed Teddy gave predictions of the weather that were as good as those of many local television weather forecasters. As with other products, Specialty faces the decision of how many Weather Teddy units to order for the coming holiday season. Members of the management team suggested order quantities of 15,000, 18,000, 24,000, or 28,000 units. The wide range of order quantities suggested indicates considerable disagreement concerning the market potential. The product management team asks you for an analysis of the stock-out proba- bilities for various order quantities, an estimate of the profit potential, and help with mak- ing an order quantity recommendation. Specialty expects to sell Weather Teddy for $24 based on a cost of $16 per unit. If inventory remains after the holiday season, Specialty will sell all surplus inventory for $5 per unit. After reviewing the sales history of similar products, Specialty’s senior sales forecaster predicted an expected demand of 20,000 units with a .95 probability that demand would be between 10,000 units and 30,000 units.

Prepare a managerial report that addresses the following issues and recommends an order quantity for the Weather Teddy product.

1. Use the sales forecaster’s prediction to describe a normal probability distribution that can be used to approximate the demand distribution. Sketch the distribution and show its mean and standard deviation.

2. Compute the probability of a stock-out for the order quantities suggested by members of the management team.

3. Compute the projected profit for the order quantities suggested by the management team under three scenarios: worst case in which sales = 10,000 units, most likely case in which sales = 20,000 units, and best case in which sales = 30,000 units.

4. One of Specialty’s managers felt that the profit potential was so great that the order quantity should have a 70% chance of meeting demand and only a 30% chance of any stock-outs. What quantity would be ordered under this policy, and what is the projected profit under the three sales scenarios?

5. Provide your own recommendation for an order quantity and note the associated profit projections. Provide a rationale for your recommendation.

1. (Binomial) The probability that a patient recovers from a delicate heart operation is 0.85. Of the next 7 patients, what is the probability that

   1. (a) exactly 5 survive?

   2. (b) between 3 and 6 survive (inclusive)?

   3. (c) What is the probability that 4 or more patients will NOT recover from the heart operation?

According to a Yale program on climate change communication survey, 71% of Americans think global warming is happening.†

(a)

For a sample of 16 Americans, what is the probability that at least 13 believe global warming is occurring? Use the binomial distribution probability function discussed in Section 5.5 to answer this question. (Round your answer to four decimal places.)

(b)

For a sample of 140 Americans, what is the probability that at least 90 believe global warming is occurring? Use the normal approximation of the binomial distribution to answer this question. (Round your answer to four decimal places.)

(c)

As the number of trials in a binomial distribution application becomes large, what is the advantage of using the normal approximation of the binomial distribution to compute probabilities?

As the number of trials becomes large, the normal approximation simplifies the calculations required to obtain the desired probability.As the number of trials becomes large, the normal approximation gives a more accurate answer than the binomial probability function.   

(d)

When the number of trials for a binomial distribution application becomes large, would developers of statistical software packages prefer to use the binomial distribution probability function shown in Section 5.5 or the normal approximation of the binomial distribution discussed in Section 6.3? Explain.

Please answer the following:

1. United Dairies, Inc., supplies milk to several independent grocers throughout Dade County, Florida. Managers at United Dairies want to develop a forecast of the number of half gallons of milk sold per week. Sales data for the past 12 weeks are:

.

1. Show the exponential smoothing forecasts using α = 0.1.
   1. Applying the MSE measure of forecast accuracy, would you prefer a smoothing constant of α=0.1 or α= 0.2 for the United Dairies sales time series?
   2. Are the results the same if you apply MAE as the measure of accuracy?
   3. What are the results if MAPE is used?

2.    Use exponential smoothing with a α = 0.4 to develop a forecast of demand for week 13. What is the resulting MSE?

A survey asked parents of children aged ten and under how many birthday parties they attended last year. Let X represent the number of birthday parties. The probability distribution is given below. Find the mean and the standard deviation of the probability distribution using Excel. Round the mean and standard deviation to three decimal places.

x P(x) 1 0.0303 2 0.0639 3 0.0197 4 0.003 5 0.0164 6 0.0454 7 0.0981 8 0.0648 9 0.0657 10 0.0124 11 0.0118 12 0.0539 13 0.0497 14 0.0648 15 0.0373 16 0.0475 17 0.0224 18 0.0191 19 0.0088 20 0.0406 21 0.0445 22 0.1202 23 0.0597

Provide a rationale as to why the three (3) aforementioned concepts or skills are important to someone in the field of business statistics.

1. Common Cause vs Special Cause Variation
2. SIPOC Model
3. Basic Improvement & Problem Solving Frameworks

1. The data below are survival times (in years) for cancer patients receiving experimental therapy (TREAT) or standard of care (SOC) in a time-to-mortality study.

1. By hand, calculate the Kaplan-Meier estimate of the survival function for each therapy:
2. By hand, calculate the pointwise 95% confidence intervals for the Kaplan-Meier survival estimate for each therapy.
3. By hand, calculate the point and 95% confidence interval estimates of the 25^th, 50^th, and 75^th percentiles of the survival distribution for each therapy.

1. QD Corporation makes paints known for their quality and quick drying features. QD has developed a new wall paint code named Mickey that it is testing in a controlled environment. The firm is testing for drying time of the paint. Drying time is affected by factors like humidity, air pollution, ventilation and temperature. The test results show that the drying time of Mickey is normally distributed with a mean of 10 hours and a standard deviation of 1.8 hours.

Compute the following:

1. What is the probability that the drying time is less than 7 hours?
2. What is the probability that the drying time is between 6 hours and 12 hours?
3. What should the label on the paint can say if the firm wants to be 95% sure that the drying time will be the claimed time or less?
4. A customer returns a can saying that the drying time was 15 hours. What is the z score corresponding to that observed value?

4.3. Referring to the previous problem, again suppose that a uniform prior is placed on the proportion π, and that from a random sample of 327 voters, 131 support the sales tax. Also suppose that the newspaper plans on taking a new survey of 20 voters. Let y∗ denote the number in this new sample who support the sales tax.

1. Find the posterior predictive probability that y∗ = 8.

2. Find the95% posterior predictive interval for y∗.Do this by finding the predictive probabilities for each of the possible values of y∗ and ordering them from largest probability to smallest. Then add the most probable values of y∗ into your probability set one at a time until the total probability exceeds 0.95 for the first time.

In the average month there is about 150,000 trucks sold across the board. There is about 350,000 vehicles sold every month. Trucks make up about 43% of the sales every month in the USA. At a 95% confidence level there are between ____% and ____% trucks sold every month.

Dixie Showtime Movie Theaters, Inc., owns and operates a chain of cinemas in several markets in the southern U.S. The owners would like to estimate weekly gross revenue as a function of advertising expenditures. Data for a sample of eight markets for a recent week follow.

(a) Use the data to develop an estimated regression with the amount of television advertising as the independent variable. Let x represent the amount of television advertising. If required, round your answers to three decimal places. For subtractive or negative numbers use a minus sign even if there is a + sign before the blank. (Example: -300)

(b) How much of the variation in the sample values of weekly gross revenue does the model in part (a) explain? If required, round your answer to two decimal places.

(c) Use the data to develop an estimated regression equation with both television advertising and newspaper advertising as the independent variables. Let x1 represent the amount of television advertising. Let x2 represent the amount of newspaper advertising. If required, round your answers to three decimal places. For subtractive or negative numbers use a minus sign even if there is a + sign before the blank. (Example: -300)

(d) How much of the variation in the sample values of weekly gross revenue does the model in part (c) explain? If required, round your answer to two decimal places. %

Two players (player A and player B) are playing a game against each other repeatedly until one is bankrupt. When a player wins a game they take $1 from the other player. All plays of the game are independent and identical. a) Suppose player A starts with $2 and player B starts with $1. If player A wins a game with probability p, what is the probability that player A wins all the money? b) Suppose player A starts with $6 and player B starts with $6. If player A wins a game with probability 0.5, what is the probability the game ends (someone loses all their money) on exactly the 10th play of the game?

For the random variables below, indicate whether you would expect the distribution to be best described as geometric, binomial, Poisson, exponential, uniform, or normal. For each item, give a brief explanation of your answer.

a) The number of heads in 13 tosses of a coin.

b) The number of at-bats (attempts) required for a baseball player to get his first hit.

c) The height of a randomly chosen adult female.

d) The time of day that the next major earthquake occurs in Southern California.

e) The number of automobile accidents in a town in one week.

f) The amount of time before the first score in a lacrosse game.

g) The number of times a die needs to be rolled before a 3 appears.

h) The number of particles emitted by a radioactive substance in five seconds.

The quality control manager of Ridell needs to estimate the mean breaking point of a large shipment of helmets sent to the Philadelphia Eagles. Given the production process, the known standard deviation of the population of breaking points is 15.5 lbs. A random sample of 49 helmets were selected and subjected to increasing pressure until every one of them broke. The breaking point of each helmet was recorded, and average breaking point of the sample was 150 lbs.

What are the critical values from the z distribution associated with a 95% confidence interval?

n a study designed to test the effectiveness of magnets for treating back​ pain, 3535 patients were given a treatment with magnets and also a sham treatment without magnets. Pain was measured using a scale from 0​ (no pain) to 100​ (extreme pain). After given the magnet​ treatments, the 3535 patients had pain scores with a mean of 12.0 and a standard deviation of 2.2. After being given the sham​ treatments, the 3535 patients had pain scores with a mean of 10.2 and a standard deviation of 2.6

Construct the 90​% confidence interval estimate of the mean pain score for patients given the sham treatment.

What is the confidence interval estimate of the population mean μ​?