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Chapter 4: Q74E (page 247)

Arrivals at an emergency room. Applications of the Poisson distribution were discussed in the Journal of Case Research in Business and Economics (December 2014). In one case involving a hospital emergency room, 2 patients arrive on average every 10 minutes. Let x = number of patients arriving at the emergency room in any 10-minute period. Assume that x has a Poisson distribution with mean 2. What is the probability that more than 4 patients arrive at the emergency room in the next 10 minutes?

Short Answer

Expert verified

The probability that more than 4 patients arrive at emergency room in next 10 minutes is 0.22.

Step by step solution

01

Given Information

Let x be the number of patients arriving at the emergency room on average every 10 minutes with mean 2

The p.m.f of the Poisson distribution is given by

pX=x=e-λλxx!;x=0,1,2,...0 otw

02

 Compute probability 

The probability that more than four patients arrive is calculated as:

px⩾4=1-px<4=1-e-2200!+e-2211!+e-2222!+e-2233!=1-0.13+0.27+0.27+0.18=1-0.78=0.22

Therefore, the probability is 0.22.

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Most popular questions from this chapter

Blood diamonds. According to Global Research News (March 4, 2014), one-fourth of all rough diamonds produced in the world are blood diamonds, i.e., diamonds mined to finance war or an insurgency. (See Exercise 3.81, p. 200.) In a random sample of 700 rough diamonds purchased by a diamond buyer, let x be the number that are blood diamonds.

a. Find the mean of x.

b. Find the standard deviation of x.

c. Find the z-score for the value x = 200.

d. Find the approximate probability that the number of the 700 rough diamonds that are blood diamonds is less than or equal to 200.

The binomial probability distribution is a family of probability distributions with every single distribution depending on the values of n and p. Assume that x is a binomial random variable with n = 4.

  1. Determine a value of p such that the probability distribution of x is symmetric.
  2. Determine a value of p such that the probability distribution of x is skewed to the right.
  3. Determine a value of p such that the probability distribution of x is skewed to the left.
  4. Graph each of the binomial distributions you obtained in parts a, b, and c. Locate the mean for each distribution on its graph.\
  5. In general, for what values of p will a binomial distribution be symmetric? Skewed to the right? Skewed to the left?

Find a value of the standard normal random variable z, call itz0 such that

a.P(-z0≤z≤z0)=.98b.P(z≥z0)=.05c.P(z≥z0)=.70d.P(z≥z0)=.025e.P(z≤z0)=.70

Choosing portable grill displays. Refer to the Journal of Consumer Research (Mar. 2003) marketing study of influencing consumer choices by offering undesirable alternatives, Exercise 3.109 (p. 204). Recall that each of 124 college students selected showroom displays for portable grills. Five different displays (representing five different-sized grills) were available. Still, the students were instructed to select only three displays to maximize purchases of Grill #2 (a smaller-sized grill). The table shows the grill display combinations and the number of times each was selected by the 124 students. Suppose one of the 124 students is selected at random. Let x represent the sum of the grill numbers selected by this student. (This value indicates the size of the grills selected.)

a. Find the probability distribution for x.

b. What is the probability that x exceeds 10?

Checkout lanes at a supermarket. A team of consultants working for a large national supermarket chain based in the New York metropolitan area developed a statistical model for predicting the annual sales of potential new store locations. Part of their analysis involved identifying variables that influence store sales, such as the size of the store (in square feet), the size of the surrounding population, and the number of checkout lanes. They surveyed 52 supermarkets in a particular region of the country and constructed the relative frequency distribution shown below to describe the number of checkout lanes per store, x.

a. Why do the relative frequencies in the table represent the approximate probabilities of a randomly selected supermarket having x number of checkout lanes?

b. FindE(x) and interpret its value in the context of the problem.

c. Find the standard deviation of x.

d. According to Chebyshev’s Rule (Chapter 2, p. 106), what percentage of supermarkets would be expected to fall withinμ±σ? withinμ±2σ?

e. What is the actual number of supermarkets that fall within? ? Compare your answers with those of part d. Are the answers consistent?
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