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Chapter 7: Problem 67

Let \(z\) denote a random variable having a normal distribution with \(\mu=0\) and \(\sigma=1\). Determine each of the following probabilities: a. \(P(z<0.10)\) b. \(P(z<-0.10)\) c. \(P(0.40-1.25)\) g. \(P(z<-1.50\) or \(z>2.50)\)

Short Answer

Expert verified
The short answer will be the specific values calculated for each of the probabilities from parts a through g using the standard normal distribution table or calculator.

Step by step solution

01

Understanding the Normal Distribution CDF

The cumulative distribution function (CDF) gives the probability that a random variable is less than or equal to a certain value. For a normally distributed random variable, it is typically represented as \(P(z < x)\), where \(z\) is the standard normal variable and \(x\) is the value of interest.
02

Calculating Probabilities using the CDF

Using the normal distribution tables or a calculator with a built-in normal distribution function, the probabilities can be calculated as follows: a. \(P(z<0.10) = F(0.10)\) b. \(P(z<-0.10) = F(-0.10)\) c. \(P(0.40-1.25) = 1 - F(-1.25)\) g. \(P(z<-1.50\) or \(z>2.50) = F(-1.50) + (1 - F(2.50))\)\nNote: F(x) represents the value of the cumulative distribution function at x.
03

Using the Cumulative Distribution Function (CDF) Values

Replace each instance of F(x) with the appropriate value from the standard normal distribution table or calculator. Follow the order of operations to carry out the arithmetic.

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Most popular questions from this chapter

Bob and Lygia are going to play a series of Trivial Pursuit games. The first person to win four games will be declared the winner. Suppose that outcomes of successive games are independent and that the probability of Lygia winning any particular game is .6. Define a random variable \(x\) as the number of games played in the series. a. What is \(p(4)\) ? (Hint: Either Bob or Lygia could win four straight games.) b. What is \(p(5) ?\) (Hint: For Lygia to win in exactly five games, what has to happen in the first four games and in Game \(5 ?\) ) c. Determine the probability distribution of \(x\). d. How many games can you expect the series to last?

A sporting goods store has a special sale on three brands of tennis balls - call them D, P, and W. Because the sale price is so low, only one can of balls will be sold to each customer. If \(40 \%\) of all customers buy Brand \(\mathrm{W}\), \(35 \%\) buy Brand \(\mathrm{P}\), and \(25 \%\) buy Brand \(\mathrm{D}\) and if \(x\) is the number among three randomly selected customers who buy Brand \(\mathrm{W}\), what is the probability distribution of \(x\) ?

A gas station sells gasoline at the following prices (in cents per gallon, depending on the type of gas and service): \(315.9,318.9,329.9,339.9,344.9\), and 359.7. Let \(y\) denote the price per gallon paid by a randomly selected customer. a. Is \(y\) a discrete random variable? Explain. b. Suppose that the probability distribution of \(y\) is as follows: $$ \begin{array}{lrrrrrr} y & 315.9 & 318.9 & 329.9 & 339.9 & 344.9 & 359.7 \\ p(y) & .36 & .24 & .10 & .16 & .08 & .06 \end{array} $$ What is the probability that a randomly selected customer has paid more than $$\$ 3,20$$ per gallon? Less than $$\$ 3.40$$ per gallon? c. Refer to Part (b), and calculate the mean value and standard deviation of \(y .\) Interpret these values.

A contractor is required by a county planning department to submit anywhere from one to five forms (depending on the nature of the project) in applying for a building permit. Let \(y\) be the number of forms required of the next applicant. The probability that \(y\) forms are required is known to be proportional to \(y ;\) that is, \(p(y)=k y\) for \(y=1, \ldots, 5\). a. What is the value of \(k ?\) (Hint: \(\sum p(y)=1 .\) ) b. What is the probability that at most three forms are required? c. What is the probability that between two and four forms (inclusive) are required? d. Could \(p(y)=y^{2} / 50\) for \(y=1,2,3,4,5\) be the probability distribution of \(y\) ? Explain.

Determine the value of \(z^{*}\) such that a. \(-z^{*}\) and \(z^{*}\) separate the middle \(95 \%\) of all \(z\) values from the most extreme \(5 \%\) b. \(-z^{*}\) and \(z^{*}\) separate the middle \(90 \%\) of all \(z\) values from the most extreme \(10 \%\) c. \(-z^{*}\) and \(z^{*}\) separate the middle \(98 \%\) of all \(z\) values from the most extreme \(2 \%\) d. \(-z^{*}\) and \(z^{*}\) separate the middle \(92 \%\) of all \(z\) values from the most extreme \(8 \%\)
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