normal distributions answers

7/21/2019 Normal Distributions Answers

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Normal Distributions Answers

1.2. The heights of all 10-year old children are normally distributed, with a mean of

138 cm and a standard deviation of 5 cm.

a. Find the percent of heights greater than 133 cm.

84.13%

b. Find the percent of heights less than 130 cm.

5.48%

c. Find the percent of heights between 131 cm and 142 cm

70.74%

d.

Find the percent of heights between 125 cm and 135 cm

26.96%

e. Find the approximate height of a 10-year old that is taller than at than 80% ofall 10-year olds.

142.21 cm

f.

Find the approximate height of a 10-year old that has 30% of the other 10-yearolds taller than him/her.

140.62 cm

g.

Find the heights of the middle 70%.

[132.82 cm, 143.18 cm]


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3. A school district administered IQ tests to all students in the district and found thedistribution to be normal with a mean of 102 and the standard deviation of 12.

a. Find the percent of scores between 96 and 110.

43.9%

b. Find the percent of scores between 105 and 120.

33.45%

c. Find the percent of scores less than 120.

93.32%

d. Find the percent of scores greater than 114.

15.87%

e. Find the approximate IQ score of a student that has a score less than 65% ofother students IQ scores.

97.38

f. What are the approximate IQ scores for the top 5% of the students?

Greater than or equal to 121.73

g.

What is the smallest range containing 20% of the scores

105.04 98.96 = 6.09


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4. Adult female Dalmatians weigh an average of 50 pounds with a standarddeviation of 3.3 pounds. Adult female Boxers weigh an average of 57.5 poundswith a standard deviation of 1.7 pounds. One statistics teacher owns anunderweight Dalmatian and an underweight Boxer. The Dalmatian weighs 45pounds, and the Boxer weighs 52 pounds. Which dog is more underweight?Explain using percents and z-scores.

5. Marks on a Chemistry test follow a normal distribution with a mean of 65 and astandard deviation of 12.

a. Approximately what percentages of the students have scores below 50?

10.565%

b.What is the approximate 90th percentile of the mark distribution?

80.379

c. If the standard deviation calculated was wrong and actually 20% of the scoreswere below 50, what would be the correct standard deviation.

17.823


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6. The length of human pregnancies from conception to birth is approximatelynormal with a mean of 266 days and a standard deviation of 16 days.

a. What percent of pregnancies last between 240 and 280 days?

75.71%

b. What percent of pregnancies last between 230 and 245 days?

8.25%

c. What percent of pregnancies last more than 266 days?

50%

d. What percent of pregnancies last more than 255 days?

75.41%

e. What percent of pregnancies last less than 270 days?

59.87%

f. What percent of pregnancies last less than 260 days?

35.38%

g. Find the approximate length of a pregnancy that is longer than 40% ofthe other pregnancies.

261.95 days

h. Find the approximate length of a pregnancy that is shorter than 20% ofthe other pregnancies.

279.47 days

i. Find the quartiles for pregnancy length.

255.21 days 276.79 days


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7. There are two different teachers for a certain class that you want to take duringyour first semester at college. Professor Fisher and Professor Savage each giveAs to those scoring above 90 on the final and Fs to those scoring below 60.You also know that Professor Fishers final is approximately normal with a meanof 74 and a standard deviation of 7 and Professor Savages final is approximatelynormal with a mean of 78 and a standard deviation of 18.

a. For what teacher is an A more impressive? Explain using percents and z-scores.

b. For what teacher is an F more discouraging? Explain using percents and z-scores.

c. What Professors class are you going to take? Explain.

8.

The cost of treatment per patient for a certain medical problem was modeled byone insurance company as a normal distribution with mean $775 and standarddeviation $150.

a. What percent of treatments cost are less than $1,000?

93.3%

b. What percent of patient pay between $700 and $900 for their treatment?

48.913%

c. What is the cost of the most expensive 5% of treatments?

1021.73 and above

d.What are the costs of the middle 60% of treatments?

[648.76, 901.24]


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9. Landrum Airlines flies the route between Chicago and Pittsburgh. Thedistribution is normal with a mean number of passengers per flight of 160 and astandard deviation of 20. The aircraft used for the route has 180 seats.

a. What percent of flights are sold out?

b. The airline must sell 150 seats to break even and the crew likes it when thereare at least 5 open seats on the plane. On what percent of flights are both theairline company and the flight crew happy with the number of passengers?

c.

The airline would like to reduce the number of flight attendants on 20percent of these flights. This will be done on the flights with the fewestpassengers. Below what number of passengers on a flight will the airlinereduce the number of flight attendants?

d. With an aggressive advertising campaign the company was able to increasetheir average number of passengers from 160 to 170 but they still have thesame percentages of sold out flights. What is the new standard deviation forthis distribution?


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10.Heights of American men are approximately normal with a mean of 70 inchesand a standard deviation of 3 inches. Heights of American women are alsoapproximately normal with a mean of 65 inches and a standard deviation of 3inches.

a.

What percent of American men are shorter than 56 tall?

b. What percent of American men are taller than 63 tall?

c. What percent of American men are between 58 tall and 62 tall?

e. Who is relatively shorter, an American man that is 55 tall or an Americanwoman who is 51 tall? Answer question using both z-scores and percents?

f. Who is relatively taller, an American man that is 6 tall or an American

woman who is 59 tall? Answer question using both z-scores and percents?


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11.A z-score describes the number of standard deviations a value is from the mean.

a. What percent of the values in a normal curve are greater than z = 0.65?

b. What percent of the values in a normal curve are less than z = 1.72?

c. What percent of the values in a normal curve are between z = -2.3 andz = -1.2?

d. What percent of the values in a normal curve are between z = -2.25 andz = 1.9?

e.

What is the z-score of a value in a normal curve that has 65% of the scoresless than it?

f. What is the z-score of a value in a normal curve that is greater than 85% ofthe other scores?

g.What is the z-score of a value in a normal curve that has 60% of the scoresgreater than it?

h.What is the z-score of a value in a normal curve that is less than 30% of theother scores?

i.

The middle 90% of the values in a normal curve are between what z-scores.


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12.The times during a 5k race are normally distributed with a mean of 31.34 minutesand a standard deviation of 4.86 minutes?

a. What percent of the runners took more than 40 minute to complete the race?

3.87%

b. What percent of the runners took between 25 and 35 minutes to complete therace?

67.83%

c. What are the times of the fastest 15% of the runners?

26.3 seconds and below

13.A manufacturer of photocopy machines states that the lifespan of theirphotocopiers are normally distributed with a mean of 42 months and a standarddeviation of 7 months.

a. What is the 3rd

quartile of copier lifespans?

46.72 months

b.

What percent of the copiers will fail before 36 months?

19.57%

c. The manufacture of the photocopy machines wants to reduce the 36-monthfailure rate to only 10%. Assuming the mean lifespan stays the same, whatstandard deviation must the manufacture achieve?


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14.The lifespan of a standard white light bulb is normally distributed with a mean of

675 hours and a standard deviation of 50 hours. The lifespan of a soft white lightbulb is normally distributed with a mean life of 700 hours and a standarddeviation of 35 hours.

a. For which type of light bulb is a light lasting 750 hours more unusual?Explain your answer using z-scores.

b. A standard white light bulb lasting 650 hours is equal to a soft white light bulblasting how long? Show work

15.The costs for a veterinary visit at a local animal hospital follow a Normalmodel with a mean of $80 and a standard deviation of $20.

a. What percent of the visits cost less than $90?

69.14%

b. What percent of the visits costs between $50 and $100?

77.45%

c. What are the costs of the most expensive 30% of the visits?

$90.49 and above


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16.A machine for a soda company fills soda cans according to a normal modelwith a mean of 12.1 ounces and a standard deviation of 0.05 ounces.

a. If the company claims to have at least 12 ounces of soda in eachcan, what percent of the cans are under-filled?

b. Management of the company wants to ensure that only 1% of thecans are under-filled but still keeping the mean at 12.1 ounces.What standard deviation does the filling machine need to have toachieve this goal?

17.The Chipmunks, a minor league baseball team, believes that a Normal modelis useful in projecting the number of fans who will attend home games. Theirmean is 8500 fans and a standard deviation of 600 fans. Their rival, thePossums, also believes that a Normal model is useful in projecting the numberof fans who will attend home games. Their mean is 9250 fans and a standarddeviation of 1000 fans. For each team to break even the must have anattendance of at least 7500 fans. Which team is more likely to break even?Explain.

18.The posted speed limit sign on a certain road is 35 mph. The distribution for

car speeds on this road is normally distributed with a standard deviation of3.56 mph. It is known that 4.2% of the drivers drive at least 10 mph over theposted speed limit. What is the average speed of the cars on this road?

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