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Chapter 6: Problem 33

Let \(x\) be a continuous random variable that is normally distributed with a mean of 25 and a standard deviation of 6 . Find the probability that \(x\) assumes a value a. between 29 and 36 b. between 22 and 35

Short Answer

Expert verified
To find the probabilities required, convert the given x values to z-scores using the formula \(Z=\frac{X-\mu}{\sigma}\), then use the Z-table to find the desired probabilities. The exact answers depend on the specific values in the Z-table.

Step by step solution

01

Normal Distribution Parameters

The given continuous random variable \(x\) follows a normal distribution where the mean \(\mu\) is 25 and the standard deviation \(\sigma\) is 6.
02

Conversion to Standard Normal Distribution

The values from a normal distribution can be standardized and converted to a standard normal distribution using the formula \(Z=\frac{X-\mu}{\sigma}\).
03

Calculate Z-scores for part a

In part a, we are interested in the probability of \(x\) assuming a value between 29 and 36. Find the associated Z-scores by substituting the given X values (29 and 36) in our formula. The calculations will be done as follows: \For 29, it's \(Z1=\frac{29-25}{6}\) and for 36, it's \(Z2=\frac{36-25}{6}\).
04

Lookup Z-table for Probabilities for part a

After calculating the Z-scores (Z1 and Z2) from Step 3, we use the standard normal distribution Z-table to find the associated probabilities. Let's denote the looked up values as \(P1\) and \(P2\) respectively. The probability \(x\) lying between \(29\) and \(36\) (\(P_{29 < x <36}\)) equals \(P2 - P1\).
05

Calculate Z-scores for part b

In part b, we are interested in the probability of \(x\) assuming a value between 22 and 35. Find the associated Z-scores by substituting the given X values (22 and 35) in our formula. The calculations will be done as follows: \For 22, it's \(Z1'=\frac{22-25}{6}\) and for 35, it's \(Z2'=\frac{35-25}{6}\).
06

Lookup Z-table for Probabilities for part b

After calculating the Z-scores (Z1' and Z2') from Step 5, we use the standard normal distribution Z-table to find the associated probabilities. Let's denote the looked up values as \(P1'\) and \(P2'\) respectively. The probability \(x\) lying between \(22\) and \(35\) (\(P_{22 < x <35}\)) equals \(P2' - P1'\).

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Most popular questions from this chapter

According to an article published on the Web site www.PCMag.com, Facebook users spend an average of 190 minutes per month checking and updating their Facebook pages (Source: http://www.pcmag,com/ article \(2 / 0,2817,2342757,00\).asp). Suppose that the current distribution of time spent per month checking and updating a member's Facebook page is normally distributed with a mean of 190 minutes and a standard deviation of \(53.4\) minutes. For a randomly selected Facebook member, determine the probability that the amount of time that he or she spends per month checking and updating his or her Facebook page is a. more than 300 minutes b. between 120 and 180 minutes

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