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Unit 4The Normal Curve and Normal

ApproximationFPP Chapter 5

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- 1. Symmetric about zero
- 2. Area under the curve 100
- 3. Always above the horizontal axis
- 4. Area between -1 and 1 is about 68.
- Area between -2 and 2 is about 95.

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Standard units say how many SDs above or below

the average a value is. They allow us to compare

different normal curves.

Normal Approximation

For many lists, the entries falling into an

interval can be estimated using the normal

curve. (1) Convert the interval in question to

standard units. (2) Find the area above this

interval, under the normal curve. That area is

approximately the entries from the list falling

into the interval.

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Standard Units

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Example Exam Scores

- Average score 20 points SD 5 points
- You score 25 points.
- Is your score above or below average?
- How many points above or below average?
- How many SDs is that?
- 25 points is ___________ in standard units.
- Convert the score points to standard

units. - What is the score 20 points in standard units?

Reading the Normal Table

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Normal Curve Arithmetic

Normal Curve Arithmetic

Service Times

- The time to complete the 40,000 mile service at a

local automobile dealership follows a normal

curve with average 100 minutes and SD 10 minutes. - What is the probability that it will take between

100 and 115 minutes? - You bring your car in for service, but you need

it to be done in 120 minutes or less. What is

the chance that your service will be done in 120

minutes or less?

Percentiles

- Refer back to the Exam Scores example
- What is the 90th percentile?
- What is the 10th percentile?

Some Normal Curve Problems

- 1. The diameters of metal rods manufactured by a

certain supplier follow a normal distribution

with mean 4.0 centimeters and SD 0.2 centimeters. - (a) What proportion of the rods have diameters

less than 3.8 cm? - (b) What proportion of the rods have diameters

greater than 4.2 cm? - (c) What proportion of the rods have diameters

between 3.9 and 4.1 cm? - 2. A consultant states that her uncertainty

about the time needed to complete a construction

project can be represented by a normal random

variable with mean 60 weeks and SD 8 weeks. - (a) What is the probability that the project

will take more than 70 weeks to complete? - (b) What is the probability that the project

will take less than 52 weeks to complete? - (c) What is the probability that the project

will take between 52 and 70 weeks to complete?

More Normal Curve Problems

- 3. A company services gas central-heating

furnaces. A review of its records indicates that

the time taken for a routine maintenance service

call can be represented by a normal distribution

with mean 60 minutes and SD 10 minutes. - (a) What proportion of such service calls take

more than 45 minutes? - (b) What proportion of such service calls take

less than 75 minutes? - (c) Sketch a graph to illustrate the reason for

the coincidence in the answers to (a) and (b). - 4. On average, graduates of a particular

university earn 59,000 five years after

graduation. The standard deviation is 4,000.

What percent earn less than 54,000? What

percent earn more than 70,000? What assumptions

did you make? Are those assumptions reasonable?

Even More Practice Problems

- 5. Trucks at a certain warehouse are loaded with

200 boxes each. It is known that, on average, 8

of all boxes are damaged during loading. What

percent of the time do 21 or more boxes get

damaged if the SD is 3.8 boxes? What assumptions

did you make? - 6. I am considering two alternative investments.

In both cases, I am unsure about the percentage

return but believe that my uncertainty can be

represented by normal distributions with means

and SD's as follows. For Investment A, the mean

is 10.4, the SD is 1.2. For Investment B, the

mean is 11.0, the SD is 4.0. I want to make the

investment that is more likely to produce a

return of at least 10. - Which should I choose?

More Practice Problems

- The mean GPA of University of Washington's last

graduating class was 2.7 with SD 0.4. What GPA

did the 90th percentile have? - 8. In a 3 year period, 665,281 people took the

GMAT (including repeaters). The distribution of

scores approximately follows a normal curve. The

mean score was 492 and the SD was 103. What was

the 80th percentile for these GMAT scores? What

proportion of scores were above 550?

More Normal Curve Problems

- 9. It is estimated that major league baseball

game times to completion follow a normal

distribution with mean 132 minutes and SD 12

minutes. - (a) What proportion of all games last between

120 and 150 minutes? - (b) Thirty-three percent of all games last

longer than how many minutes? - (c) What is the 67th percentile of game times to

completion? - (d) What proportion of games last less than 120

minutes? - 10. The weights of the contents of boxes of a

brand of cereal have a normal distribution with

mean 24 ounces and SD 0.7 ounces. - (a) What is the probability that the contents of

a randomly chosen box weigh less than 23 ounces? - (b) The contents of 10 of all boxes weight more

than how many ounces? - (c) What proportion of boxes have contents

weighing between 23.5 and 24.5 ounces?

- 11. A management consultant found that the amount

of time per day spent by executives performing

tasks that could be done equally well by

subordinates followed a normal distribution with

mean 2.4 hours. It was also found that 10 of

executives spent over 3.5 hours per day on such

tasks. Find the standard deviation of the

distribution of daily time spent by executives on

tasks of this type. (Newb228) - 12. The cereal manufacturer of problem 10 wants

to adjust the production process so that the mean

weight of the contents of the boxes of cereal is

still 24 ounces, but only 3 of the boxes will

contain less than 23 ounces of cereal. What SD

for the weights of the contents is needed to

attain this objective?

- 13. A video display tube for computer graphics

terminals has a fine mesh screen behind the

viewing surface. During assembly the mesh is

stretched and welded onto a metal frame. Too

little tension at this stage will cause wrinkles,

while too much tension will tear the mesh. The

tension is measured by an electrical device with

output readings in millivolts (mV). At the

present time, the tension readings for successive

tubes follows a normal distribution with mean 275

and standard deviation 43 mV. - (a) The minimum acceptable tension corresponds

to a reading of 200 mV. What proportion of the

tubes exceed this limit? - (b) The mean tension can be adjusted in the

production process, but the SD remains at 43 mV

regardless of the mean tension setting. What

mean tension setting should be used to make it so

that 2 of the tubes have tension readings below

the limit of 200 mV? - (c) In production, tension above 375 mV will

usually tear the mesh. Thus the acceptable range

of tension readings is actually 200 mV to 375

mV. What proportion of tubes are in this

acceptable range?