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According to the Sleep Foundation, the average night's sleep is 6.8 hours ( Fortune, March$20,2006 ) .$ Assume the standard deviation is. 6 hours and that the probability distributionis normal.a. What is the probability that a randomly selected person sleeps more than 8 hours?b. What is the probability that a randomly selected person sleeps 6 hours or less?c. Doctors suggest getting between 7 and 9 hours of sleep each night. What percentageof the population gets this much sleep?

a. 0.0228b. 0.0918c. 37.07$\%$

Intro Stats / AP Statistics

Chapter 6

Continuous Probability Distributions

Descriptive Statistics

Probability Topics

Continuous Random Variables

Temple University

University of St. Thomas

Idaho State University

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all right. According to the Sleep Foundation, the mean number of hours asleep people get it's 6.8 hours. And standard deviation for that we're assuming is 0.6 hours. But x b the random variable. That is the number of hours of sleep one gets right. So Part asks us to find the probability that somebody gets more than eight hours of sleep. All right, what? Z score eight against thes statistics. Uh, and that is, uh, two standard. You just two point. Oh, and that gives us according to our table 0.9772 So up here, the secrets, the probability that Z is greater than 2.0 So we're gonna take one, minus the probability that Z, it's less than or equal to 2.0 So that's one minus 0.9772 which is 0.228 All right, part B. We need to find the probability that someone get six hours of sleep or less, which is not healthy. But, hey, it happens. All right? What's the score? Six. All right, this is gonna be well, this up here is gonna be negative. 0.8 over. That's negative. Forth. Uh, 0.6 which is negative for third. So this is 1.33 We checked this with our table. We get that this is equal to 0.0 918 This is the probability and sees list some 1.33 I'm sorry. Negative. 1.33 So this is 0.918 All right, so we transcribe this information just so it's in front of me. Ah, for part c were given that doctors recommend between seven and nine hours of sleep every night. So it asked us to find the proportion of people who actually get that amount of sleep. All right, let's find our Z lower and the upper So Z lower is gonna be seven minus 6.8. Also, overs zero points expired away. I'd like to point out the lower bound of this already greater than our mean. I don't think this looks good already. So this is 0.2 of reserve when six. That's 1/3 so 0.33 from the probability that Z is less than or equal to 0.33 is that we check our table. Um what? See three 0.62 93 right here we have I'm minus 6.8 all over 0.6, and ah, well, this is 3.67 because this is greater than our Z value of 3.50 Where is it going to say that the probability 0.9999 So anyway, this is gonna be the probability that ah, sorry Z is between, uh 0.33 and 3.867 And that is the probability that Z is less than or equal to 3.67 minus the probability that Z is less than or equal to 0.33 That's 0.9999 minus 0.6293 Ah, doing some mental subtraction. That 0.37 06 So 37.6 people percent of people are getting a proper amount of sleep. That's not great. But hey, such a CZ life

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