Answered: Anita's, a fast-food chain specializing…

Calculus For The Life Sciences

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter12: Probability

Section12.3: Conditional Probability; Independent Events; Bayes' Theorem

Problem 82E

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Question

Anita's, a fast-food chain specializing in hot dogs and garlic fries, keeps track of the proportion of its customers who decide to eat in the restaurant (as opposed to ordering the food "to go") so it can make decisions regarding the possible construction of in-store play areas, the attendance of its mascot Sammy at the franchise locations, and so on. Anita's reports that

45% of its customers order their food to go. Suppose that this proportion is correct and that a random sample of

50 individual customers is taken.

Answer the following.

(If necessary, consult a list of formulas.)

(a)	Estimate the number of customers in the sample who order their food to go by giving the mean of the relevant distribution (that is, the expectation of the relevant random variable). Do not round your response.

(b)	Quantify the uncertainty of your estimate by giving the standard deviation of the distribution. Round your response to at least three decimal places.

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

Expert Solution

Step 1

Given information:

Anita's, a fast-food chain specializing in hot dogs and garlic fries, keeps track of the proportion of its customers who decide to eat in the restaurant (as opposed to ordering the food "to go").

Anita's reports that:

Percentage of customers who order their food to go = 45%
A random sample of n = 50 customers is taken.

It is required to compute:

The number of customers in the sample who order their food to go by giving the mean of the relevant distribution.
the standard deviation of the distribution for quantifying the uncertainty of the estimate.

Introduction:

The binomial distribution is a discrete type of probability distribution with parameters n and p given by:

$\begin{array}{rcl} f (x) & = & {}^{n}C_{x} {(p)}^{x} {(1 - p)}^{x}; x = 0, 1, 2, 3 . . . . . . . . . ., n \\ = & 0; otherwise \end{array}$

The mean and standard deviation of this distribution is np and $\sqrt{n p q}$ respectively.

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Calculus For The Life Sciences

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ISBN:

9780321964038

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GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:

Pearson Addison Wesley,