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Unit 4 The Normal Curve and Normal Approximation FPP Chapter 5

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For many lists, the % entries falling into an interval can be estimated using the normal curve. ... Some Normal Curve Problems ... – PowerPoint PPT presentation

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Title: Unit 4 The Normal Curve and Normal Approximation FPP Chapter 5


1
Unit 4The Normal Curve and Normal
ApproximationFPP Chapter 5
0.4
0.3
0.2
0.1
0.0
-4
-3
-2
-1
0
1
2
3
4
X
  • 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
2

-30
-20
-10
0
10
20
30
X
50
40
60
20
10
30
70
0
80
X
Standard units say how many SDs above or below
the average a value is. They allow us to compare
different normal curves.
3
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.
0.4
0.3
0.2
0.1
0.0
? ? ? ? ? ? ? ? ? Original
Units
Standard Units
-4
-3
-2
-1
0
1
2
3
4
4
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?

5
Reading the Normal Table
6
(No Transcript)
7
Normal Curve Arithmetic
8
Normal Curve Arithmetic
9
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?

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

11
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?

12
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?

13
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?

14
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?

15
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?

16
  • 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?

17
  • 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?
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