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business
small business management

Questions and Answers of Small Business Management

Apply the method of maximum likelihood to estimate the parameters of a uniform distribution on the interval [a, b].
Consider an exponential distribution with rate λ. On the basis of a random sample of size n, apply the method of moments to estimate λ.
Consider a sequence of random variables 0 with probability 1 Xn n with probability —~n Does this sequence converge in probability to a number? What about convergence in quadratic mean?
Consider the simulation of a continuous review (Q, R) inventory control policy. Define the relevant events for the system, and outline a procedure for the management of each event. To deal with a
Define an algorithm to generate pseudorandom variables characterized by the following density function:/(*) = 4 x - 1, if 1 < x < 2 3-x, if 2 < a; < 3 0, otherwise
A m-Erlang distribution with rate λ is obtained when summing m independent exponential random variables with rate λ. This distribution may be used to model more realistic random service times in
Apply one-way ANOVA to check equality of means for the following sample:i = 1 82.31 160.98 230.84 522.06 449.25 t = 2 240.80 228.27 278.73 278.16 172.16 t = 3 181.55 188.83 334.07 326.81 327.55
In one-way ANOVA we define the sum of squares SS& and SS™. Prove the identity s s - = Σ Σ xij -nm*2- -SSb i j
In order to estimate the fraction of defective parts, you take a sample of size 1000 and find that 63 are not acceptable. Find a 99% confidence interval for the fraction of defective parts.
The following dataset is a random sample from a normal distribution:103.23, 111.00, 86.45, 105.17, 101.91 92.15, 97.40, 102.06, 121.47, 116.62 Find a 95% confidence interval for variance.
A study was done to measure the impact of fatigue on human performance when carrying out a certain task. The performance is measured by an appropriate index, the larger the better, which is measured
You want to compare the reliability of two machines that insert chips onto electronic cards. The main problem is the occurrence of jams in the feeding mechanism, as this requires stopping production
Air quality is measured by the concentration of a dangerous pollutant.The mayor of a city has engaged in a program to improve traffic conditions in order to decrease the concentration ofthat
TakeltEasy produces special shoes for runners, whose average life is 1250 km. In order to improve the product, they experiment with a new design, and test prototypes with a sample of 30 runners. The
In standard confidence intervals, you use the sample mean as an estimator of expected value. Now suppose that a friend of yours suggests the following alternative estimator:-^"= Τ7 ρ-^1 "t" TT-^2 +
Find the 97% confidence interval, given a sample mean of 128.37, sample standard deviation of 37.3, and sample size of 50. What is the width of the confidence interval? Suppose that you want to cut
You have to compute a confidence interval for the expected value of a random variable. Using a standard procedure, you take a random sample of size N = 20, and the sample statistics are X = 13.38 and
The director of a Masters' program wants to assess the average IQ of her students. A sample of 18 students yields the following results:130, 122, 119, 142, 136, 127, 120, 152, 141 132, 127, 118, 150,
Consider two random variables X and Y, not necessarily independent.Prove that Cov(X -Y,X + Y)=0.
You have invested $10,000 in IFM stock shares and $20,000 in Peculiar Motors stock shares. Compute the one-day value at risk, at 95% level, assuming normally distributed daily returns. Daily
You are in charge of component inventory control. Your firm produces end items P\ and P2, which share a common component C. You need two components C for each piece of type P\ and three components
You have to decide how much ice cream to buy in order to meet demand at two retail stores. Demand is modeled as follows:£>i = 100X + eu D2= 120X + e2 where X, e\, and ei are independent normal
Batteries produced by a company are known to be defective with a probability of 0.02. The company sells batteries in packages of eight and offers a money-back guarantee that at most one of them is
According to an accurate survey, 40% of people checked at the exit of a well-known pub have made excessive use of alcoholic drinks. If we take a random sample of 25 persons, what is the probability
You are about to launch a new product on the market. If it is a success you will make $16 million; otherwise, you lose $5 million. The probability of success is 65%. You could increase chances of
Using the binomial expansion formula (2.3), prove that the PMF of the binomial distribution [see Eq. (6.16)] adds up to one.
Using Eq. (6.15), prove that the expected value of a geometric random variable X with parameter p is E[X] = 1/p. (Hint: Use the result in Example 2.40.)
Consider a generalization of the Bernoulli random variable, i.e., a variable taking values X\ with probability p and X2 with probability I—p. Which values of p maximize and minimize variance?
Consider the following extension of the EOQ model, often labeled economic manufacturing quantity (EMQ). Most of the assumptions of the EOQ and EMQ models are the same, but in the latter, rather than
Assume that you work from age 25 to age 65. At the end of each year, you contribute C to your pension fund, which is invested at a yearly rate r = 5%. At age 65 you retire and plan to consume $20,000
Consider a stream of constant cash flows Ft= C, for t = 1,..., T.From Example 2.39, we know how to find its present value at time t = 0.Now imagine that these cash flows are the amount you invest to
Prove that the intersection of two convex sets Si and S2 is a convex set.
Consider the following functions and tell if they are convex, concave, or neither:fi{x) = e'a+\ / 2 (x) = ln(x + l), / 3 (x) = x 3- x 2+ 2
In Example 2.33 we assume discrete-time compounding of interest. This results in the need for introducing modified duration, in Eq. (2.19), to get rid of an annoying factor 1 + y, where y is yield to
Consider function and find linear (first-order) and quadratic (second-order) approximations around points xo — 0 and xo = 10. Check the quality of approximations around these points.
Consider the following functions defined on piecewise domains:j-x, x0'h{x) = Ì3X, *>0 'h{x)~-3 2 In other cases, one should define an integral by considering the value of the function at the
Consider functions /(x) = x3— x and g(x) — x3+ x. Use derivatives to sketch the function graphs, and look for maxima and minima.
Find the first-order derivative of the following functions:
Find the equation of a line• With slope -3 and intercept 10• With slope 5 and passing through point (-2,4)• Passing through points (1,3) and (3,-5)
Find the domain of functions 1 1 f(x) = ,, —7> 9(x) y/l - x2- 1 ' Vx2+ 1 - x
Use the concepts that we have introduced in Chapter 2 to check the qualitative properties of the logistic function of Eq. (16.11).
Apply the formulas of multiple regression to the case of a single regressor, and verify that the familiar formulas for simple regression are obtained.
Prove the identity in Eq. (14.26)
Consider the fraction È of defective items in a batch of manufactured parts. Say that the prior distribution of È is a beta distribution with parameters «i = 5 and ct2 = 10 (see Section 7.6.2 for
Consider again the Bayesian coin flipping experiment of Example 14.14, where the prior is uniform. If we use Eq. (14.21) to find the Bayesian estimator, what is the estimate of È after the first
Consider the data of the Braess' paradox example in Section 14.5, but imagine that a central planner can assign routes to drivers, in order to minimize total travel cost. Check that adding the new
Two firms have a production technology involving a fixed cost and constant marginal cost, as represented by the cost function:TCi(f t) = {f +C| * *« > 0 . i =1, 2 1^0 if
Two firms have the same production technology, represented by the cost function:T Cite ) = ^ + 2«·>» , «_i, S[0 if ft = 0 3 5 There is also an alternative way of interpreting Black-Litterman
Consider the Cournot competition outcome of Eqs. (14.12) and (14.13).Analyze the sensitivity of the solution with respect to innovation in production technology, i.e., how a reduction in production
Find the Nash equilibria in the games in Tables 14.2 and 14.4. Are they unique?
Consider a point-to-point transportation network consisting of M nodes. By "point-to-point" we mean that, given transportation requirements between all pair of nodes, there is a direct transportation
Consider the plant location model of Section 12.4.5 [see Eqs. (12.54-12.55)]. Adapt the model to cope with uncertain demand scenarios, building a two-stage stochastic linear programming model with
We know that VaR, in general, is not a subadditive risk measure.Consider a portfolio of two assets, with jointly normal returns.• Show that, in this specific case, VaR is a subadditive risk
You have invested $150,000 in stock shares of Doom and $200,000 in stock shares of Mishap. Assume that daily returns follow a multivariate normal distribution; daily volatilities for the two stock
An investor has an initial wealth WQ that must be allocated between a risk-free asset, with certain return r{, and a risky asset. We assume a simple binomial model of uncertainty, like we did in
You own a plant whose value is $100,000. In case of a fire, the value of your property might be significantly reduced or even destroyed, depending on how severe the accident is. Let us represent risk
A decision maker with a quadratic utility function of the form (13.12)is offered the following lottery:Probability Payoff 0.20 $10,000 0.50 $50,000 0.30 $100,000 If the risk aversion coefficient is
You are the manager of a pension fund, and your fee depends on the return attained. You can play it safe and allocate wealth to a risk-free portfolio earning 4% per year. Alternatively, you can
The Research and Development (R&D) division of your firm has developed a new product that could be immediately launched on the market. If so, the probability of success is 60%, in which case profit
Consider again the "fancy coin flipping" example of Section 1.2.3, i.e., the decision of producing a movie or not. Formalize the problem with a proper decision tree.
A firm sells a perishable product, with a time window for sales limited to 1 month. The product is ordered once per month, and the delivery lead time is very small, so that the useful shelf life is
Given the observed data x 45 50 55 60 65 70 75 y 24.2 25.0 23.3 22.0 21.5 20.6 19.8 build a 95% confidence interval for the slope.
Consider the following sales data:Week 12345678 9 10 Sales 30 20 45 35 30 60 40 50 45 65 Build a linear regression model to predict sales and calculate R2.
In some applications we are interested in the distribution of the maximum among a set of realization of random variables. Let us consider a set of n i.i.d. variables Ui, i = 1,... ,n, with uniform
You are in charge of deciding the purchased amount of an item with limited time window for sales and uncertain demand. The unit purchase cost is $10 per item, the selling price is $16, and unsold
You work for a manufacturing firm producing items with a limited time window for sale. Items are sold by a distributor facing uncertain demand over the time window, which we model by a normal
Let X ~ χ^ be a chi-square variable with n degrees of freedom. Prove that E[X] = n.
You have just issued a replenishment order to your supplier, which is not quite reliable. You have ordered 400 items, but what you will receive is a normal random variable with that expected value,
Let X be a normal random variable with expected value μ = 6 and standard deviation σ = 1. Consider random variable W = 3X2. Find the expected value E[W] and the probability P(W > 120).
A friend of yours is an analyst and is considering a probability model to capture uncertainty in monthly demand of an item featuring high-volume sales. He argues that the central limit applies and,
You are working in your office, and you would like to take a very short nap, say, 10 minutes. However, every now and then, your colleagues come to your office to ask you for some information; the
We should set the reorder point R for an item, whose demand during lead time is uncertain. We have a very rough model of uncertainty - the lead time demand is uniformly distributed between 5000 and
Consider a normal variable with μ = 250 and σ — 20, and find the probability that X falls in the interval between 230 and 260.
A random variable X has normal distribution with μ = 250 and σ = 40.Find the probability that X is larger than 200.
Professors at a rather unknown but large college have developed a habit of heavy drinking to forget about their students. The following data show the number of hangovers since the beginning of
Management wants to investigate the time it takes to complete a manual assembly task. A sample of 12 workers is timed, yielding the following data(in seconds):21.3 15.2 13.6 16.1 15.0 19.2 21.0 14.3
You observe the following data, reporting the number of daily emergency calls received by a firm providing immediate repair services for critical equipment:Day 1 2 3 4 5 6 7 8 N calls 5462382 4 Day 9
The following table shows a set of observed values and their frequencies:Value 1 2 3 4 5 6 7 8 Frequency 5 4 7 10 13 8 3 1• Compute mean, variance, and standard deviation.• Find the cumulated
You are carrying out a research about how many pizzas are consumed by teenagers, in the age range from 13 to 17. A sample of 20 boys/girls in that age range is taken, and the number of pizzas eaten
Consider the optimal mix model of Section 1.1.2. How could we extend the model to cope with• Third-party suppliers offering items at given cost?• The possibility of overtime work at some resource
Consider the optimal mix model of Section 1.1.2. How could we extend the model to cope with • Third-party suppliers offering items at given cost? • The possibility of overtime work at some
Consider again the growth option problem of Section 1.2.2. We want to check the impact of less extreme assumptions about the conditional probabilities for the second movie, but we are unsure which
Identify the three main considerations in setting a price for a product.
Explain what breakeven analysis is and why it is important for pricing in a small business.
Present examples of customer-oriented and internal-oriented pricing.
Explain why and how small businesses extend credit.
Describe the advertising, personal selling, public relations, and sales-promotion tools that a small business owner uses to compile a promotional mix.
What strategies should be considered if a small business is setting prices for a product that is to be exported? How do these strategies differ from those used in a domestic market? LO-3
What advantages and disadvantages are involved for a small business offering sales on credit? LO-3
As the owner of a small, hometown drugstore, how would you prepare for a Wal-Mart being built in your area? LO-3
What can happen if the price of a product does not fit with the three other Ps of the marketing mix? LO-3
Should a small business owner’s judgment be used to determine prices if so many mathematical techniques have been developed for that purpose? LO-3
Discuss the importance of remaining professional and friendly when trying to collect an unpaid bill. LO-3
What factors should be considered when a small business owner decides to advertise? LO-3
Discuss the personality traits that a good salesperson should have. What traits would detract from the personal-selling process? LO-3
Explain the ratchet effect on sales. LO-3
How would promotional mix decisions change for a small business that is expanding into a foreign market? LO-3
Of the pricing techniques described in this chapter, which one do you think is most commonly used by small businesses? Why? LO-3
In this exercise we will walk through the steps of creating a print advertisement.Step 1: Choose a concept. The following words commonly appear in advertising copy. Choose three or four words that
Working in teams of no more than three, choose one of the two examples to work on. Develop an outline for a comprehensive marketing strategy for the company and its product. Be specific in defining

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