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Chapter 6: Problem 12
For the standard normal distribution, what is the area within three standard deviations of the mean?
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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
Let \(x\) be a continuous random variable that follows a normal distribution with a mean of 550 and a standard deviation of \(75 .\) a. Find the value of \(x\) so that the area under the normal curve to the left of \(x\) is \(.0250\). b. Find the value of \(x\) so that the area under the normal curve to the right of \(x\) is \(.9345\). c. Find the value of \(x\) so that the area under the normal curve to the right of \(x\) is approximately .0275. d. Find the value of \(x\) so that the area under the normal curve to the left of \(x\) is approximately \(.9600\). e. Find the value of \(x\) so that the area under the normal curve between \(\mu\) and \(x\) is approximately \(.4700\) and the value of \(x\) is less than \(\mu\). f. Find the value of \(x\) so that the area under the normal curve between \(\mu\) and \(x\) is approximately \(.4100\) and the value of \(x\) is greater than \(\mu\).
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 \(x=17\)
Tommy Wait, a minor league baseball pitcher, is notorious for taking an excessive amount of time between pitches. In fact, his times between pitches are normally distributed with a mean of 36 seconds and a standard deviation of \(2.5\) seconds. What percentage of his times between pitches are a. longer than 39 seconds? b. between 29 and 34 seconds?
Suppose you are conducting a binomial experiment that has 15 trials and the probability of success of \(.02\). According to the sample size requirements, you cannot use the normal distribution to approximate the binomial distribution in this situation. Use the mean and standard deviation of this binomial distribution and the empirical rule to explain why there is a problem in this situation. (Note: Drawing the graph and marking the values that correspond to the empirical rule is a good way to start.)
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