FE Review ‐ Statistics, Study Guides, Projects, Research of Statistics

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FE Review ‐ StatisticsDR. GERRY KNAPP, P.E. (LOUISIANA)

Statistics Topics on FE^ Number of questions:◦^

Chemical

◦^

Civil

(part of math)

◦^

Electrical

◦^

Environmental

◦^

Industrial

◦^

Mechanical

◦^

Other

Measures of central tendencies and dispersions (e.g., mean, mode, standarddeviation), measurement uncertainty, confidence intervals. IE, ChE: controllimits

All

Basic laws of probability & probability distributions (e.g., discrete,continuous, normal, binomial, empirical). EE & IE: Conditional probability;IE: permutations & combinations, sets

All

Expected value (weighted average)

All

Regression & goodness of fit (linear, multiple, curve fitting, correlationcoefficient, R2, least squares). IE: Residual analysis

All

Sampling & hypothesis testing (central limit, sampling distributions, standarderror, normal, t, chi‐square, types of error, sample size, outlier testing,significance)

ChE, EE, IE,Other

Analysis of Variance (ANOVA). IE: Factorial Design

Che, IE, EE

Measures of centraltendencies anddispersions

Common Statistics^ STATISTIC

SYMBOL

FORMULA

PopulationSymbol

Mean

ത^ 𝑋

∑ 𝑋

௜ 𝑛^



Median

m

Order data smallest to largestIf n is odd, middle value;if n is even, midpoint between middle 2 values

Mode

‐^

Value which repeats most often (its possible to have zero or multiple modes)

Range

R^

Max(X

) – Min(Xi^

)^ i^

‐

Variance

(^2) s

∑ ௑

௑ ത೔ି^

మ ௡ିଵ

for population



2 , divide by n

(^2) 

Standard Deviation

s^

Sqrt(s

2 )^



Pg. 63, Reference book ver. 10 Others: Coefficient of variation (CV), geometric mean, root mean square (RMS)

What is the median of the following data?10, 7, 12, 4, 6, 8a)

4 b)

c)^

8 d)

10 SOLN:Put in ascending (sort) order: 4,6,7,8,10,12If n odd, middle item; if even, average of middle two itemsm = (7+8) / 2 = 7.5 (b)Follow‐up: What is the mode of this data?

Confidence Intervals Provide a bracket on the location of the true population mean given a sample

ത 𝑋 and S. Based on

Student‐T distribution◦ If standard deviation

^

is unknown:

ത െ 𝑡ഀ𝑋

,௡ିଵమ^

ௌ ௡^

൏ 𝜇 ൏ 𝑋

ത ൅ 𝑡ഀ

,௡ିଵమ^

ௌ ௡

◦ If standard deviation

^

is known:

ത െ 𝑍𝑋

ఈ/ଶ

ఙ ௡^

൏ 𝜇 ൏ 𝑋

ത ൅ 𝑍

ఈ/ଶ

ఙ ௡

May also be stated as:

ത േ 𝑡𝑋

ఈ/ଶ

ௌ ௡

=significance level (type 1 error) = 1 – confidence level Z values: use table on page 44, t‐values: use student‐t table pg. 46IEs: also refresh on formulas for CI on difference means and on variance

Pg. 74, Reference book ver. 10

Standard error

Half width

A random sample is selected from a normal distribution with variance 46.12336. If the width of a 95%confidence interval about the sample mean is 4.5, what is the size of the sample?SOLN:Variance is known, so will use Z rather than t: Z

0.05/

= 1.

If width of CI is 4.5, half width is 4.5/2 = 2.25CI halfwidth = Z

/

*

/sqrt(n) , so have 2.25 = 1.96*sqrt(46.12336)/sqrt(n)

Solve for n:

sqrt(n) = 1.96sqrt(46.12336)/2.25n = (1.96^2)46.12336 / (2.25^2) =

^35

Values of Z

a/

CI

Za/

80%

90%

95%

96%

98%

99%

Control Limits/Charts

Pg 82‐83 in Ref.handbook ver. 10

Measurement Uncertainty See also expectation later this review and "Combinations of Random Variables" in the FEReference Handbook for formulas for mean and variance of linear combinations of variables

Example Given function y =

𝑥

ଶ^ ଵ

െ 𝑥

ଶ^ 𝑥ଶ

where measured values xଵ

= 4+/‐0.1 and x 1

= 16+/‐0.5 (+/‐ values 2

indicating standard deviations of each), what is y and the uncertainty of y,



?y

SOLN:y =

4

ଶ^

െ 16

ଶ^

∗ 4 ൌ െ1,

y

=^

భ

𝜎௫ ଶ^ ൅^ భ

మ

𝜎௫

ଶ^ మ

=^

2𝑥

െ 𝑥ଵ

ଶ^ ଶ

ଶ^

∗ 0.

ଶ^ ൅

0 െ 2𝑥

∗ 𝑥ଶ

ଵ^

ଶ^ ∗ 0.

=^

2 ∗ 4 െ 16

ଶ^

ଶ^ ∗ 0.

ଶ^

൅

0 െ 2 ∗ 16 ∗ 4

∗ 0.

= 68.6y = ‐1,008 +/‐ 68.

Equally Likely Outcomes If an experiment can result in N equally likely outcomes, and n of those outcomes make up eventA, the probability of event A is

𝑃ሺ𝐴ሻ ൌ

Example: Single die toss; let A={2, 4, 6}, so◦ P(A) = 3/6 = 1/2 = 0.5Example: Single coin toss, let A={H}◦ P(A) = 1/2= 0.

Probability characteristics Probability of an event is a number between zero and one, 0<= P(A) <=1.

P(A) = 1, means event A occurs with certainty.P(A) = 0, means event A is impossible.

P(A

) = 1 – P(A), that is, the probability of not A is simply 1 minus the chance of A happening.

If the events are mutually exclusive (A1, A2,…)

𝑃ሺ𝐴

൅ 𝐴ଵ

൅... ሻ ൌ 𝑃ሺ𝐴ଶ^

ሻ ൅ 𝑃ሺ𝐴ଵ

ሻ൅.. .ଶ^

If the events A

..A 1

n^ are a partition of S,

𝑃ሺ𝐴

൅ 𝐴ଵ

൅ ⋯ ൅ 𝐴ଶ^

ሻ௡^

ൌ 𝑃ሺ𝐴

ሻ ൅ 𝑃ሺ𝐴ଵ

ሻ൅... ൅𝑃ሺ𝐴ଶ^

ሻ ൌ 1௡^

If A and A

^ are complementary,

P(A)+P(A

)=

Examples Two students are working independently on a problem. Their respective probabilities of solving the problemare 1/3 and 3/4. What is the probability that at least one of them will solve the problem?a)

b)^

c)^

d)^

SOLN:Given P(A)=1/3, P(B)=3/4. Looking for P(A+B).= P(A) + P(B) – P(A,B) = P(A) + P(B) – P(A)*P(B) (since A and B independent)= 1/3 + 3/4 ‐ (1/3)(3/4) = 5/

(d)

A coin is flipped and a 6‐sided die thrown. What is the probability of getting heads and a 5?

a)

1/ b)

1/ c)^

1/ d)

1/

SOLN:12 possible outcomes (2*6), all equally likely. Only one produces the outcome of interest.Pr = 1/12 (d)