THE CITADEL DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING

ELEC105-01 Engineering Fundamnetals II Spring 2003

|1| (20 points) A table of 24 data points is given below. Complete the frequency distribution table and plot the histogram on the graph provided.

92 71 89 91 53 93 98 76 96 94 68 91 88 87 93 78 85 98 78 70 87 88 89 95

Classes Frequency FREQUENCY 10 48 - 54 1 8 55 - 61 0 62 - 68 1 6

69 - 75 2 4 76 - 82 3 2 83 - 89 7 0 90 - 96 8 48 55 62 69 76 83 90 97 104 97 - 103 2 CLASSES

Frequency Distribution |2| (20 points) The 10 data points listed under x in the table below are a sample, selected at random, of the highest daily temperatures recorded in Philadelphia. Complete the table and find the mean of the data m and its standard deviation s .

x x ! m (x ! m)2 67 11.9 141.61 2 ! 53.1 2819.61 93 37.9 1436.41 97 41.9 1755.61 23 ! 32.1 1030.41 83 27.9 778.41 35 ! 20.1 404.01 38 ! 17.1 292.41 33 ! 22.1 488.41 80 24.9 620.01

m = ______55.1______

s = ______32.94______

j xi 551 m ' ' n 10

2 j (xi ! m) 9766.9 s 2 ' ' (n ! 1) 9

s ' 1085.21 ' 32.94 |3| (10 points)

Write the equation for the mean µ of a population.

j xi µ ' n

Write the equation for the standard deviation F of a population.

2 1/2 j (xi ! µ) F ' n

The mean is a meaure of ___central tendency______

The standard deviation is a measure of ___variation______|4| (20 points) A manufacturer of electronic components produces a run of 100,000 molded carbon resistors. The mean of the resistors in the run is 1 kS and the standard deviation is 40S . The resistor values in the run follow a normal distribution. (a) Find the expected number of resistors that have values between 950S and 1050S . (b) Find the expected number of resistors that have values between 990S and 1010S .

Given: N = 100,000 resistors µ = 1 kS F = 40 S normal distribution

(a) Find: n = ? such that 950 S # R # 1050 S

Solution:

950 ! 1000 z ' ' ! 1.25 1 40

1050 ! 1000 z ' ' 1.25 2 40

From the table of normal distribution,

P(0 # z # 1.25) ' 0.3944

P(! 1.25 # z # 1.25) ' 2 (0.3944) ' 0.7888

n ' P×N ' 0.7888 (100,000)

Answer: n = 78,880 resistors (b) Find: n = ? such that 990 S # R # 1010 S

Solution:

990 ! 1000 z ' ' ! 0.25 1 40

1010 ! 1000 z ' ' 0.25 2 40

From the table of normal distribution,

P(0 # z # 0.25) ' 0.0987

P(! 0.25 # z # 0.25) ' 2 (0.0987) ' 0.1974

n ' P×N ' 0.1974 (100,000)

Answer: n = 19,740 resistors |5| (10 points)

List the two parameters that completely describe a normal distribution.

______mean, _µ______

______standard deviation, F______

List two of the three properties of a normal distribution.

______symmetrical about the mean______

______extends to infinity in both directions______

______total area = 1______|6| (20 points) A normal distribution has a mean of 26 and a standard deviation of 8. (a) Determine the probability of values between 26 and 36. (b) Determine the probability of values greater than 40. (c ) Determine the probability of values less than 22 .

Given: µ = 26 F = 8

(a) Find: P(x) = ? Such that 26 # x # 36

Solution:

26 ! 26 z ' ' 0 1 8

36 ! 26 z ' ' 1.25 2 8

From the table of normal distribution,

P(0 # z # 1.25) ' 0.3944

Answer: P(x) = 0.3944

(b) Find: P(x) = ? such that x $ 36

Solution:

40 ! 26 z ' ' 1.75 8

From the table of normal distribution,

P(0 # z # 1.75) ' 0.4599 P(z $ 1.75) ' 0.5 ! 0.4599

Answer: P(x) = 0.0401

(c ) Find: P(x) = ? such that x # 22

Solution:

22 ! 26 z ' ' ! 0.5 8

From the table of normal distribution,

P(0 # z # 0.5) ' 0.1915

P(z # ! 0.5) ' 0.5 ! 0.1915

Answer: P(x) = 0.3085