Fuel economy estimates for automobiles built one year...

Fuel economy estimates for automobiles built one year predicted a mean of 25.1 mpg and a standard deviation of 7.07 mpg for highway driving. Assuming the gas mileages for automobiles are normally distributed, use the Empirical Rule to answer the following: The middle 99.7% of automobiles can be expected to have gas mileages between ___ and ___. Approximately ___% of automobiles will have a gas mileage more than 18.03 mpg. Approximately ___% of automobiles will have gas mileages between 32.17 and 39.24. Which of the statements below describes the gas mileages of the worst 2.5% of cars? They get more than 39.24 mpg. They get less than 3.89 mpg. They get less than 10.96 mpg. They get more than 46.31 mpg.

Transcript

00:01 For this exercise, we are told that theal economy for automobiles, that's called that random variable x, is normally distributed with a mean of 25 .1 miles per gallon and a standard deviation of 7 .07 miles per gallon.

00:18 And we are asked to use the empirical rule to answer the following questions.

00:23 The first question is the middle 99 .7 % of automobiles can be expected to have gas mileage.

00:32 Between what two numbers.

00:34 Now the empirical rule says that 99 .7 % of a bell -shaped distribution, or approximately 99 .7%, lies within three standard deviations of the mean.

00:47 So this is the mean plus or minus three sigma.

00:52 So for this random variable, that's 25 .1 plus or minus 3 times 7 .07.

01:03 And this gives us a range from 3 .89 miles per gallon.

01:08 To 46 .31 miles per gallon.

01:17 And then for the second question, we're asked approximately what percent of automobiles will have a gas mileage of more than 18 .03.

01:27 Now, 18 .03 is equal to 25 .1 minus 7 .07.

01:41 This is the mean minus one standard deviation.

01:45 Now, if we look at our bell -shaped distribution or normal distribution, the empirical rule tells us that approximately 68 % of the distribution lies within one standard deviation of the mean.

02:04 So that means 68 % lies within the range from the mean minus 1 sigma to the mean plus 1 sigma.

02:13 And so for our random variable, the mean minus sigma is 18 .03, and the mean plus 1 sigma is 32 .17.

02:46 So if 68 % of the distribution lies between these two values, so the area under the curve between these two numbers, this blue -shaded region, has an area of 0 .68.

02:59 And since the total area under the curve is 1, that means that the area outside of these two numbers must be 1 minus .68, which is .32.

03:17 And since the distribution is symmetrical, that means that we'd have an area here to the left of the mean minus 1 standard deviation of .32 divided by 2, which is .16, and the same here.

03:36 So between these two values, .68 and on each side, .16 for a total of 1.

03:45 And so if it's .16 to the left of this value, then it's 1 minus .16 to the right of it, which is .84...

Question

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