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Chapter 7: Problem 50

A breeder of show dogs is interested in the number of female puppies in a litter. If a birth is equally likely to result in a male or a female puppy, give the probability distribution of the variable \(x=\) number of female puppies in a litter of size 5 .

Short Answer

Expert verified
The probability distribution of \( x \) = number of female puppies in a litter of size 5 is given by: \( P(X=k) \) for \( k=0 \) to \( 5 \), calculated using the binomial distribution formula.

Step by step solution

01

Prepare your Inputs

First, identify values for n, p, and the possible values for k. Here, n = 5 (litter size), p = 0.5 (probability of getting a female), and k varies from 0 to 5 (count of females in the litter). Therefore, we need to calculate probabilities for k=0, k=1, k=2, k=3, k=4, and k=5.
02

Use Binomial Distribution Formula

Apply the binomial distribution formula for k=0 to 5. The formula is given by: \[ P(X=k) = C(n, k) \cdot (p^k) \cdot ((1-p)^{n-k}) \] where \( C(n, k) \) is the number of combinations of n items taken k at a time, calculated as \[ n! / (k!(n-k)!) \] Here, '!' denotes the factorial of a number.
03

Calculate Individual Probabilities

Calculate the individual probabilities for k=0 to 5. These will form the probability distribution of x.
04

Ensure Probabilities Add to 1

Finally, ensure that all of these individual probabilities add up to 1. This is a characteristic of any probability distribution.

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Most popular questions from this chapter

Suppose that the distribution of the number of items \(x\) produced by an assembly line during an 8 -hr shift can be approximated by a normal distribution with mean value 150 and standard deviation \(10 .\) a. What is the probability that the number of items produced is at most 120 ? b. What is the probability that at least 125 items are produced? c. What is the probability that between 135 and 160 (inclusive) items are produced?

Suppose that \(20 \%\) of the 10,000 signatures on a certain recall petition are invalid. Would the number of invalid signatures in a sample of size 1000 have (approximately) a binomial distribution? Explain.

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The paper "The Load-Life Relationship for M50 Bearings with Silicon Nitride Ceramic Balls" (Lubrication Engineering [1984]: \(153-159\) ) reported the following data on bearing load life (in millions of revolutions); the corresponding normal scores are also given: $$ \begin{array}{cccc} \boldsymbol{x} & \text { Normal Score } & \boldsymbol{x} & \text { Normal Score } \\ \hline 47.1 & -1.867 & 240.0 & 0.062 \\ 68.1 & -1.408 & 240.0 & 0.187 \\ 68.1 & -1.131 & 278.0 & 0.315 \\ 90.8 & -0.921 & 278.0 & 0.448 \\ 103.6 & -0.745 & 289.0 & 0.590 \\ 106.0 & -0.590 & 289.0 & 0.745 \\ 115.0 & -0.448 & 367.0 & 0.921 \\ 126.0 & -0.315 & 385.9 & 1.131 \\ 146.6 & -0.187 & 392.0 & 1.408 \\ 229.0 & -0.062 & 395.0 & 1.867 \\ & & & \\ \hline \end{array} $$ Construct a normal probability plot. Is normality plausible?
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