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Chapter 5: Problem 57

An office supply company conducted a survey before marketing a new paper shredder designed for home use. In the survey, \(80 \%\) of the people who used the shredder were satisfied with it. Because of this high acceptance rate, the company decided to market the new shredder. Assume that \(80 \%\) of all people who will use it will be satisfied. On a certain day, seven customers bought this shredder. a. Let \(x\) denote the number of customers in this sample of seven who will be satisfied with this shredder. Using the binomial probabilities table (Table I, Appendix C), obtain the probability distribution of \(x\) and draw a graph of the probability distribution. Find the mean and standard deviation of \(x\). b. Using the probability distribution of part a, find the probability that exactly four of the seven customers will be satisfied.

Short Answer

Expert verified
The probability distribution of \(x\) will have a possible range from 0 to 7 with various probabilities for each outcome. The mean (\(渭\)) will be \(5.6\) and the standard deviation (\(蟽\)) will be \(0.843\). The probability that exactly four of the seven customers will be satisfied is about \(0.130\).

Step by step solution

01

Calculate the Probability Distribution

First, need to calculate the binomial probability for each possible outcome, from 0 to 7, using the binomial probability formula: \(P(x) = C(n, x) \times p^x \times (1 - p)^{n - x}\), where \(n = 7\) (the number of trials), \(x\) is the number of successes we are interested in, \(p = 0.8\) (the success probability), and \(C(n, x)\) is the combination of \(n\) items taken \(x\) at a time.
02

Calculate the Mean and Standard Deviation

The mean (\(渭\)) of a binomial distribution can be found using the formula \(渭 = np\) and the standard deviation (\(蟽\)) can be found using the formula \(蟽 = 鈭歯p(1-p)\), where \(n = 7\) is the number of trials and \(p = 0.8\) is the probability of success.
03

Calculate the probability of exactly 4 out of 7 customers being satisfied

Apply the binomial probability formula again, but this time fill in \(x = 4\). This will allow obtaining the specific probability for exactly 4 successes.

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Most popular questions from this chapter

Twenty percent of the cars passing through a school zone are exceeding the speed limit by more than \(10 \mathrm{mph}\). a. Using the Poisson formula, find the probability that in a random sample of 100 cars passing through this school zone, exactly 25 will exceed the speed limit by more than \(10 \mathrm{mph}\). b. Using the Poisson probabilities table, find the probability that the number of cars exceeding the speed limit by more than 10 mph in a random sample of 100 cars passing through this school zone is \(\begin{array}{lll}\text { i. at most } 8 & \text { ii. } 15 \text { to } 20 & \text { iii. at least } 30\end{array}\)

One of the toys made by Dillon Corporation is called Speaking Joe, which is sold only by mail. Consumer satisfaction is one of the top priorities of the company's management. The company guarantees a refund or a replacement for any Speaking Joe toy if the chip that is installed inside becomes defective within 1 year from the date of purchase. It is known from past data that \(10 \%\) of these chips become defective within a 1 -year period. The company sold 15 Speaking Joes on a given day. a. Let \(x\) denote the number of Speaking Joes in these 15 that will be returned for a refund or a replacement within a 1 -year period. Using the appropriate probabilities table from Appendix \(\mathrm{C}\), obtain the probability distribution of \(x\) and draw a graph of the probability distribution. Determine the mean and standard deviation of \(x\). b. Using the probability distribution constructed in part a, find the probability that exactly 5 of the 15 Speaking Joes will be returned for a refund or a replacement within a 1 -year period.

Let \(x\) be a discrete random variable that possesses a binomial distribution. Using the binomial formula, find the following probabilities. a. \(P(x=5)\) for \(n=8\) and \(p=.70\) b. \(P(x=3)\) for \(n=4\) and \(p=.40\) c. \(P(x=2)\) for \(n=6\) and \(p=.30\) Verify your answers by using Table I of Appendix C.

Briefly explain the following. A. A binomial experiment b. A trial c. A binomial random variable

Based on its analysis of the future demand for its products, the financial department at Tipper Corporation has determined that there is a \(.17\) probability that the company will lose \(\$ 1.2\) million during the next year, a \(.21\) probability that it will lose \(\$ .7\) million, a \(.37\) probability that it will make a profit of \(\$ .9\) million, and a \(.25\) probability that it will make a profit of \(\$ 2.3\) million. a. Let \(x\) be a random variable that denotes the profit earned by this corporation during the next year. Write the probability distribution of \(x\). b. Find the mean and standard deviation of the probability distribution of part a. Give a brief interpretation of the value of the mean.
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