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Unit 4

The Normal Curve and Normal Approximation

FPP Chapter 5

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%.

A.05

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fX(x) = 1σ 2Π

exp[- 12(x-μσ

)2] for -∞ < x < ∞

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-30-20-10 0 10 20 30

X

5040

X

602010 30 700 80

Standard units say how many SD’s above or below the average a value is.

They allow us to compare different normal curves.

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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

? ? ? ? ? ? ? ? ? Original 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 SD’s is that?

25 points is ___________ in standard units.

Convert the score points to standard units.

What is the score 20 points in standard units?

1712

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Reading the Normal Table

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Normal Curve Arithmetic

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-4 -3 -2 -1 0 1 2 3 40.0

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Normal Curve Arithmetic

-4 -3 -2 -1 0 1 2 3 40.0

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-4 -3 -2 -1 0 1 2 3 40.0

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-4 -3 -2 -1 0 1 2 3 40.0

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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?

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Percentiles

Refer back to the Exam Scores example:

What is the 90th percentile?

What is the 10th percentile?

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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?

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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?

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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?

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More Practice Problems

7. 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?

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More Normal Curve Problems9. 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?

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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?

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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?