What percent of inchworms are more than 2 standard deviations away from the mean? By the Empirical Rule, we know that about 95% of inchworms have lengths within 2 standard deviations of the mean. That is, between 1−2(0.15)=0.7 and 1+2(0.15)=1.3 inches. The remaining 5% of inchworms have lengths that are more than 2 standard deviations away from the mean. That is, approximately 5% of inchworms have lengths that are less than 0.7 or greater than 1.3 inches. b. What percent of inchworms are more than 2 standard deviations above the the mean? Since the distribution of inchworm lengths is symmetric, about half of the remaining 5% will lie to either extreme. So about 2.5% have lengths that are less than 0.7 inches, or below the mean, and about 2.5% have lengths that are greater than 1.3 inches, or above the mean.
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
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What percent of inchworms are more than 2 standard deviations away from the By the The remaining 5% of inchworms have lengths that are more than 2 standard deviations away from the mean. That is, approximately 5% of inchworms have lengths that are less than 0.7 or greater than 1.3 inches.
b. What percent of inchworms are more than 2 standard deviations above the the mean? Since the distribution of inchworm lengths is symmetric, about half of the remaining 5% will lie to either extreme. So about 2.5% have lengths that are less than 0.7 inches, or below the mean, and about 2.5% have lengths that are greater than 1.3 inches, or above the mean. |
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