Using the Empirical Rule In Exercises 29–34, ...

Question

Using the Empirical Rule In Exercises 29–34, use the Empirical Rule.

Use the sample statistics from Exercise 29 and assume the number of vehicles in the sample is 75.

a. Estimate the number of vehicles whose speeds are between 63 miles per hour and 71 miles per hour.

Accepted Answer

Hi, everyone. Let's take a look at this practice problem. This problem says, using the empirical rule, estimate the number of trees in a sample of 75 whose heights are between 18 ft and 26 ft. Assume the sample mean height is 22 ft, and the standard deviation is 2 ft. And we give 4 possible choices as our answers. For choice A, we have 62 trees. For choice B, we have 71 trees. For choice C, we have 75 trees, and for choice D, we have 68 trees. So, in this problem, we're gonna actually be using the empirical rule. And recall that the empirical rule states that for a normal distribution, approximately 68% of the data falls within one standard deviation from the mean, approximately 95% of the data falls within two standard deviations of the mean, and approximately 99.7% of the data falls within three standard deviations from the mean. So the first thing we want to do is actually calculate our range in terms of the standard deviation. So, we're gonna begin by looking at the upper bound of our range, which we'll label as UB. And we actually want to write this in terms of the number of standard deviations. So for upper bound, it's going to be equal to the upper bound in feet, which is going to be 26 that we're looking at, but we need to subtract off our sample mean, which is going to be 22, since we're looking at the number of standard deviations where we are from the mean. And then we need to take that quantity and divide it by our standard deviation, which in this case is 2 ft, so we'll divide it by 2. And so when we evaluate this expression, this is going to be equal to 2. So, our upper bound of 26 ft is two standard deviations away from our sample mean of 22 ft. So next we're going to look at our lower bound. So we'll label that as LB, and so this is going to be equal to our sample mean height, which is going to be 22, minus our lower bound of 18 ft. And we're gonna divide that quantity by our standard deviation, which is 2 ft. And when we evaluate this expression, this is going to be equal to 2. So our lower bound is 2 standard deviations away from the sample mean. Now, since our upper bound and lower bound are both 2 standard deviations away, that means that by the empirical rule that 95% of our data fall within this interval. So we can now use that to calculate the number of trees in our samples, whose heights fall between 18 ft and 26 ft. So that means that our number of trees, which we'll label as N, is going to be equal to 95%. We'll need that as a decimal. So we'll divide it by 100, so we'll get 0.95. And want to multiply that by our sample size, which is 75. And when we calculate this quantity, this is going to be equal to 71. So that is the answer that we're looking for for this problem, which is going to be 71 trees. And if we look at our answer choices, we see that the correct answer choice corresponds to answer choice B. So, I hope that this explanation has been useful, and I'll see you in the next video.

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Using the Empirical Rule In Exercises 29–34, use the Empirical Rule.

Use the sample statistics from Exercise 29 and assume the number of vehicles in the sample is 75.

a. Estimate the number of vehicles whose speeds are between 63 miles per hour and 71 miles per hour.

Key Concepts

Empirical Rule

Normal Distribution

Sample Statistics

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Using the Empirical Rule In Exercises 29–34, use the Empirical Rule.Use the sample statistics from Exercise 29 and assume the number of vehicles in the sample is 75.a. Estimate the number of vehicles whose speeds are between 63 miles per hour and 71 miles per hour.