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Chapter 6: Problem 18

Obtain the following probabilities for the standard normal distribution. a. \(P(z>-.98)\) b. \(P(-2.47 \leq z \leq 1.29)\) c. \(P(0 \leq z \leq 4.25)\) d. \(P(-5.36 \leq z \leq 0)\) e. \(P(z>6.07)\) f. \(P(z<-5.27)\)

Short Answer

Expert verified
a. 0.8365, b. 0.8946, c. 0.5, d. 0.5, e. 0, f. 0

Step by step solution

01

- Understand the standard normal distribution table

The standard normal distribution table shows the probability that a normally distributed random variable Z, with mean equal to 0 and variance equal to 1, is less than or equal to z. It is used to find the probability that a statistic is observed below, above, or between values.
02

- Calculate Probability for \(P(z>-.98)\)

The table gives us the probability that a Z-score is less than -0.98 (i.e. \(P(Z<-0.98)\)). And because the total probability under the bell curve is 1, the probability that a Z-score is greater than -0.98 is \(1 - P(Z<-0.98)\). Using the Z-score table, we find that \(P(Z<-0.98) = 0.1635\). Therefore, \(P(Z > -0.98) = 1 - 0.1635 = 0.8365\).
03

- Calculate Probability for \(P(-2.47 \leq z \leq 1.29)\)

To find this, we need to find \(P(Z \leq 1.29)\) and \(P(Z \leq -2.47)\), and then subtract the two. From the table, \(P(Z \leq 1.29) = 0.9015\) and \(P(Z \leq -2.47) = 0.0069\). Therefore, the answer is \(0.9015 - 0.0069 = 0.8946\).
04

- Calculate Probability for \(P(0 \leq z \leq 4.25)\)

This is equivalent to finding \(P(Z \leq 4.25)\). Since the standard normal distribution is symmetric about zero, the probability that Z is less than 0 is 0.5. Thus, we find from the table that \(P(Z \leq 4.25) = 1\), so the answer is \(1 - 0.5 = 0.5\).
05

- Calculate Probability for \(P(-5.36 \leq z \leq 0)\)

The probability that Z is less than 0 is 0.5. And the probability that Z is less than -5.36 is basically 0, because -5.36 is many standard deviations away from the mean. Therefore, the answer is \(0.5 - 0 = 0.5\).
06

- Calculate Probability for \(P(z > 6.07)\) and \(P(z < -5.27)\)

For any Z-score that is more than 5 standard deviations away from the mean, the probabilities are practically 0. Therefore, \(P(Z > 6.07) = 0\) and \(P(Z < -5.27) = 0\)

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Most popular questions from this chapter

Find the following areas under a normal distribution curve with \(\mu=20\) and \(\sigma=4\) a. Area between \(x=20\) and \(x=27\) b. Area from \(x=23\) to \(x=26\) c. Area between \(x=9.5\) and \(x=17\)

What are the parameters of the normal distribution?

An office supply company conducted a survey before marketing a new paper shredder designed for home use. In the survey, \(80 \%\) of the people who tried the shredder were satisfied with it. Because of this high satisfaction rate, the company decided to market the new shredder. Assume that \(80 \%\) of all people are satisfied with this shredder. During a certain month, 100 customers bought this shredder. Find the probability that of these 100 customers, the number who are satisfied is a. exactly 75 b. 73 or fewer c. 74 to 85

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