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Chapter 5: Q39E (page 229)

It is known that\({\rm{80\% }}\)of all brand A extremal hard drives work in a satisfactory manner throughout the warranty period (are "successes"). Suppose that\({\rm{n = 15}}\)drives are randomly selected. Let\({\rm{X = }}\)the number of successes in the sample. The statistic\({\rm{X/n}}\)is the sample proportion (fraction) of successes. Obtain the sampling distribution of this statistic. (Hint: One possible value of\({\rm{X/n is }}{\rm{.2}}\), corresponding to\({\rm{X = 3}}\). What is the probability of this value (what kind of\({\rm{rv}}\)is\({\rm{X}}\))?)

Short Answer

Expert verified

\({\rm{X is binomial random variable}}{\rm{. }}\)

Step by step solution

01

Definition

Probability simply refers to the likelihood of something occurring. We may talk about the probabilities of particular outcomes鈥攈ow likely they are鈥攚hen we're unclear about the result of an event. Statistics is the study of occurrences guided by probability.

02

Obtaining the sampling distribution

Obviously, \({\rm{X}}\)is binomial random variable with parameters

\({\rm{n = 15 and p - 0}}{\rm{.8}}{\rm{. }}\)

All possible values that random variable \({\rm{X/n}}\) can take are

\(\frac{{\rm{x}}}{{{\rm{15}}}}{\rm{,}}\;\;\;{\rm{x脦 \{ 0,1,2, \ldots ,15\} }}{\rm{.}}\)

The probabilities can be calculated as

\(\begin{array}{l}{\rm{P}}\left( {\frac{{\rm{X}}}{{{\rm{15}}}}{\rm{ = }}\frac{{\rm{x}}}{{{\rm{15}}}}} \right)\rm &= P(X = x)\\\rm &= b(x;15,0{\rm{.8),}}\;\;\;{\rm{x脦 \{ 0,1,2, \ldots ,15\} }}\end{array}\)

Theorem: \({\rm{b(x;n,p) = }}\left\{ {\begin{array}{*{20}{l}}{\left( {\begin{array}{*{20}{l}}{\rm{n}}\\{\rm{x}}\end{array}} \right){{\rm{p}}^{\rm{x}}}{{{\rm{(1 - p)}}}^{{\rm{n - x}}}}}&{{\rm{,x = 0,1,2, \ldots ,n}}}\\{\rm{0}}&{{\rm{, otherwisc}}{\rm{. }}}\end{array}} \right.\)

The following table represents the pmf of random variable \({\rm{X/n}}\)

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Most popular questions from this chapter

Refer back to Example, Two cars with six-cylinder engines and three with four-cylinder engines are to be driven over a \(300\)-mile course. Let \({X_1}, . . . {X_5}\)denote the resulting fuel efficiencies (mpg). Consider the linear combination

\(Y = \left( {{X_1} + {X_2}} \right)/2 - \left( {{X_3} + {X_4} + {X_5}} \right)/3\)

which is a measure of the difference between four-cylinder and six-cylinder vehicles. Compute \(P\left( {0 \le Y} \right)\)and\(P(Y > - 2)\).

Let\({{\rm{X}}_{\rm{1}}}{\rm{,}}{{\rm{X}}_{\rm{2}}}{\rm{,}}{{\rm{X}}_{\rm{3}}}{\rm{,}}{{\rm{X}}_{\rm{4}}}{\rm{,}}{{\rm{X}}_{\rm{5}}}\), and \({{\rm{X}}_{\rm{6}}}\) denote the numbers of blue, brown, green, orange, red, and yellow M\&M candies, respectively, in a sample of size\({\rm{n}}\). Then these \({{\rm{X}}_{\rm{i}}}\) 's have a multinomial distribution. According to the M\&M Web site, the color proportions are\({{\rm{p}}_{\rm{1}}}{\rm{ = }}{\rm{.24,}}{{\rm{p}}_{\rm{2}}}{\rm{ = }}{\rm{.13}}\), \({{\rm{p}}_{\rm{3}}}{\rm{ = }}{\rm{.16,}}{{\rm{p}}_{\rm{4}}}{\rm{ = }}{\rm{.20,}}{{\rm{p}}_{\rm{5}}}{\rm{ = }}{\rm{.13}}\), and\({{\rm{p}}_{\rm{6}}}{\rm{ = }}{\rm{.14}}\).

a. If\({\rm{n = 12}}\), what is the probability that there are exactly two M\&Ms of each color?

b. For\({\rm{n = 20}}\), what is the probability that there are at most five orange candies? (Hint: Think of an orange candy as a success and any other color as a failure.)

c. In a sample of\({\rm{20M \backslash Ms}}\), what is the probability that the number of candies that are blue, green, or orange is at least \({\rm{10}}\) ?

Suppose the expected tensile strength of type-A steel is \({\rm{105ksi}}\)and the standard deviation of tensile strength is \({\rm{8ksi}}\). For type-B steel, suppose the expected tensile strength and standard deviation of tensile strength are \({\rm{100ksi}}\)and \({\rm{6ksi}}\), respectively. Let \({\rm{\bar X = }}\)the sample average tensile strength of a random sample of \({\rm{40}}\) type-A specimens, and let \({\rm{\bar Y = }}\)the sample average tensile strength of a random sample of \({\rm{35}}\)type-B specimens.

a. What is the approximate distribution of \({\rm{\bar X ? of \bar Y?}}\)

b. What is the approximate distribution of \({\rm{\bar X - \bar Y}}\)? Justify your answer.

c. Calculate (approximately) \(P( - 1拢\bar X - \bar Y拢1)\)

d. Calculate. If you actually observed , would you doubt that \({{\rm{\mu }}_{\rm{1}}}{\rm{ - }}{{\rm{\mu }}_{\rm{2}}}{\rm{ = 5?}}\)

We have seen that if\(E\left( {{X_1}} \right) = E\left( {{X_2}} \right) = . . . = E\left( {{X_n}} \right) = \mu \), then\(E\left( {{X_1} + . . . + {X_n}} \right) = n\mu \). In some applications, the number of\({X_i} 's\)under consideration is not a fixed number n but instead is a rv N. For example, let N\(5\)be the number of components that are brought into a repair shop on a particular day, and let Xi denote the repair shop time for the\({i^{th}}\)component. Then the total repair time is\({X_1} + {X_2} + . . . + {X_n}\), the sum of a random number of random variables. When N is independent of the\({X_i} 's\), it can be shown that

\(E\left( {{X_1} + . . . + {X_N}} \right) = E\left( N \right) \times \mu \)

a. If the expected number of components brought in on a particular day is\({\bf{10}}\)and the expected repair time for a randomly submitted component is\({\bf{40}}\)min, what is the expected total repair time for components submitted on any particular day?

b. Suppose components of a certain type come in for repair according to a Poisson process with a rate of\(5\)per hour. The expected number of defects per component is\(3.5\). What is the expected value of the total number of defects on components submitted for repair during a four-hour period? Be sure to indicate how your answer follows from the general result just given.

A shipping company handles containers in three different sizes:\(\left( 1 \right)\;27f{t^3}\;\left( {3 \times 3 \times 3} \right)\)\(\left( 2 \right) 125 f{t^3}, and \left( 3 \right)\;512 f{t^3}\). Let \({X_i}\left( {i = \;1, 2, 3} \right)\)denote the number of type i containers shipped during a given week. With \({\mu _i} = E\left( {{X_i}} \right)\)and\(\sigma _i^2 = V\left( {{X_i}} \right)\), suppose that the mean values and standard deviations are as follows:

\(\begin{array}{l}{\mu _1} = 200 {\mu _2} = 250 {\mu _3} = 100 \\{\sigma _1} = 10 {\sigma _2} = \,12 {\sigma _3} = 8\end{array}\)

a. Assuming that \({X_1}, {X_2}, {X_3}\)are independent, calculate the expected value and variance of the total volume shipped. (Hint:\(Volume = 27{X_1} + 125{X_2} + 512{X_3}\).)

b. Would your calculations necessarily be correct if \({X_i} 's\)were not independent?Explain.
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