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TAMU MATH 166 - wir5-II

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WIR Math 166-copyright Joe Kahlig, 09C Page 1Week in Review–Additional Material sections 3.3 and 3.4Section 3.3: Measures of Spread.• variance( var) and standard deviation (st. dev.) measure the spread of the data about the mean.• var = (st. dev)2• Chebychev’s Inequality.• Population versus Samples• A population is the entire group.• µ is the mean of a population• σ is the standard deviation of a population.• A sample is a subgroup of the population. This is usually used when it is difficultto get d ata from the entire population. This information is then usedto estimate the information about the population.•x is the mean of the sample• sxis the standard deviation of the sample1. A consum er organization evaluates new cars and reports the number of major defects(X) oneach car.X 0 1 2 3 4 5 6 7 8 9 10cars 40 100 209 223 128 114 91 28 11 4 1Find th e variance and the standard deviation.2. The entire class of two hundred students is asked, “To the nearest hours, how many hours(X)did you study yesterday?”X 0 1 2 3 4 5 6 7 8students 16 28 43 43 26 22 14 6 2Find th e variance and the standard deviation.3. A certain type of surgery has a success rate of 84%. Assume that the success of the surgerybetween different patients is independent. This surgary is to be preformed on 500 patients inthe next year. Let X be the number of patients for who this surgery is a success.(a) What values of X are within 1.2 standard deviations of the mean?(b) What values of X are more than .75 standard deviations above the mean?(c) Find the probability that the number of people hav ing a successful surgery is within twostandard deviations of the mean.4. A hair dryer has a mean lifespan of 24 months and a standard deviation of 3 months, assumingtypical usage. Estimate the probability th at one of th ese hair dryers w ill last between 20 and28 months.5. It h as been found that with normal usage, the expected lifetime of a product is 3 years witha standard deviation of 4 months. Estimate the number of items in a shipment of 9000 thatwill last between 30 months and 42 months.Section 3.4: The Normal Distribution• continuous probability distribution.• probability density function• f (x) ≥ 0 for values of x• area under the curve is 1• normal distribution(bell cur ve)WIR Math 166-copyright Joe Kahlig, 09C Page 2• standard normal curve, Z• µ = 0, σ = 1• To convert values of any normal curve, X, withmean µ and σ to Z-values use z =x−µσ• calculator commands• normalcdf(left cutoff, right cutoff, µ, σ)• invNorm(area, µ , σ)6. Compute the following(a) P (0.3 < Z < 1.83) =(b) P (Z < 1.5) =(c) P (Z = 1.25) =7. Find the values of A and B for the following.(a) P (Z < A) = .68(b) P (−B < Z < B) = .488. Suppose X is a normal random variable with µ = 40 and σ = 8.(a) P (32 < X < 53) =(b) P (X > 45) =(c) Find the value of A such that P (X > A) = .759. The random variable X is normally distributed with a mean of 83 and a standard deviationof 5. Find the percent of the area that is less than 1.3 standard deviations above the mean.10. The tread life of a tire is normally d istributed with a mean of 40,000 miles and a standarddeviation of 2000 miles.(a) What is the probability that a tire selected at random will have a tread life of more than35,000 miles?(b) In a group what 800 tires, approximately how many of them will last more than 35,000miles.(c) What is the probability that a tire selected at random will have a tread life between38,000 miles and 44,000 miles?WIR Math 166-copyright Joe Kahlig, 09C Page 3(d) If 4 tires are installed in a car and experience even wear, determine the probability thatall 4 tires will have a tread life between 38,000 miles and 44,000 miles.(e) If 4 tires are installed in a car and experience even wear, determine the probability thatexactly 3 tires will have a tread life between 38,000 miles and 44,000 miles.11. It has been found that the ammount of chees on a p izza is normally distributed with a meanof 8 oz of cheese with a standard deviation of 0.5 oz.(a) What is the probability that a pizza selected at random will have less than 7.2 oz ofcheese?(b) Out of 300 pizzas, approx imately how many of them w ill h ave less than 7.2 oz of


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TAMU MATH 166 - wir5-II

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