Probability and Statistics

4% of exam

Discrete, continuous, normal, and binomial distributions, expected value in decision making, hypothesis testing and design of experiments (t-test, ANOVA, outlier testing), measures of central tendency and dispersion with confidence intervals, regression and curve fitting, and statistical process-control limits.

3 concepts

Exam areas

A. Probability distributions

Distributions, Confidence Intervals and Control Limits

Discrete and continuous distributions, the binomial and normal, mean/median/mode and spread, the confidence interval for a mean, and X-bar/R control limits.

Probability and statistics is worth four to six FE-Chemical questions, and almost every one reduces to a small number of moves: name the distribution, compute a center and a spread, and convert a tolerance into a probability or a probability into a limit. The handbook hands you the formulas and the unit-normal table, so the lost points are never algebra — they are basis errors (population versus sample, a count versus a fraction) and table-reading errors (area to the left versus the right of

z

). This concept builds the fluency the NCEES FE Reference Handbook section 'Engineering Probability and Statistics' assumes you already have when it simply prints

f(x)

and a

z

-transform with no worked example.

Discrete versus continuous random variables

A random variable maps each outcome to a number. If it can take only separated values — number of off-spec batches, particle counts, defects per reel — it is discrete and carries a probability mass function

f(x_k)=P(X=x_k)

that sums to one. If it varies continuously — a concentration, a fill weight, a temperature — it carries a probability density function

f(x)

, and probability is the area under the curve:

P(a\le X\le b)=\int_a^b f(x)\,dx

. The single most common slip is asking for

P(X=x)

of a continuous variable; that area is zero, so continuous questions are always about intervals.

P(a\le X\le b)=\int_a^b f(x)\,dx,\qquad \sum_k f(x_k)=1

Center and spread: mean, median, mode, variance

Three measures of center answer different questions. The mean

\bar{X}=\frac{1}{n}\sum x_i

is the balance point and the one the algebra runs on; the median is the middle value once the data are ordered (the average of the two middle values when

n

is even), and it resists outliers; the mode is the most frequent value, and for the symmetric normal all three coincide. Spread is the variance and its square root, the standard deviation. Watch the denominator: a full population divides the squared deviations by

N

, but a sample divides by

n-1

to stay unbiased — the FE exam tests this distinction directly.

s^2=\frac{1}{n-1}\sum_{i=1}^{n}\left(X_i-\bar{X}\right)^2,\qquad s=\sqrt{s^2}

The binomial distribution

When a fixed number

n

of independent trials each succeed with the same probability

p

, the count of successes is binomial. This is the natural model for sampling inspection: how many of

n

pulled units are defective, given a defect rate

p

. The mass function multiplies the number of arrangements

C(n,x)

by the probability of any one such arrangement

p^x q^{n-x}

, with

q=1-p

. Its mean is

np

and its variance is

npq

— note the variance is a count, not a fraction.

P(x)=C(n,x)\,p^{x}q^{\,n-x},\qquad \mu=np,\qquad \sigma^{2}=npq

The normal distribution and the z-transform

The normal (Gaussian) is the default model for measurement scatter, and by the Central Limit Theorem the mean of a sample tends toward it regardless of the parent shape. You never integrate its bell curve by hand; instead you standardize, sliding and scaling any normal variable to the unit normal with

z=(x-\mu)/\sigma

, then read the tabulated areas

F(z)

R(z)=1-F(z)

, and

W(z)

. The discipline that prevents errors is sketching the curve, shading the area you want, and matching that shaded region to the correct column of the table.

z=\frac{x-\mu}{\sigma},\qquad f(x)=\frac{1}{\sigma\sqrt{2\pi}}\,e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^{2}}

Confidence interval for the mean

A confidence interval brackets the unknown population mean from a single sample. The center is always

\bar{X}

; the half-width is a multiplier times the standard error of the mean,

\sigma/\sqrt{n}

. Which multiplier you use depends on what you know: if the population

\sigma

is known you use the normal value

z_{\alpha/2}

(1.960 for 95%), and if you must estimate the spread from the sample with

s

you use Student's

t_{\alpha/2,\,n-1}

, which is wider for small

n

and converges to

z

n

grows. The interval narrows only as

\sqrt{n}

, so halving it costs four times the samples.

\bar{X}\pm z_{\alpha/2}\,\frac{\sigma}{\sqrt{n}}\quad(\sigma\text{ known}),\qquad \bar{X}\pm t_{\alpha/2,\,n-1}\,\frac{s}{\sqrt{n}}\quad(\sigma\text{ unknown})

Statistical process-control limits

A control chart asks a different question than a confidence interval: not 'where is the mean?' but 'is the process still behaving as it did?' You take small subgroups of size

n

, plot each subgroup average on an X-bar chart and each subgroup range on an R chart, and set three-sigma limits from tabulated constants. The center lines are the grand average

\overline{\overline{X}}

and the average range

\bar{R}

; the limits use

A_2

D_3

D_4

from the handbook table. A point outside the limits, or a nonrandom run inside them, signals an assignable cause — these limits come from the process itself, never from the engineering tolerance.

\text{UCL}_{\bar X}=\overline{\overline{X}}+A_2\bar{R},\quad \text{LCL}_{\bar X}=\overline{\overline{X}}-A_2\bar{R};\qquad \text{UCL}_R=D_4\bar{R},\quad \text{LCL}_R=D_3\bar{R}

Exam strategy

Classify first. 'How many out of

n

' with a fixed rate is binomial (

\mu=np

). A continuous spec question is normal: standardize, sketch, shade, then read

F

R

, or

W

from the table — and remember

P(\text{within }\pm z)=2F(z)-1

. For an interval, decide

z

versus

t

by whether

\sigma

is given or only

s

is computed, and never forget the

\sqrt{n}

in the standard error. For control charts, pull

A_2

D_3

D_4

for the right

n

and use

\overline{\overline{X}}

and

\bar{R}

, not the spec window. Carry units through, and quote final answers to three significant figures.

Key equations

Sample mean

\bar{X}=\frac{1}{n}\sum_{i=1}^{n}X_i

Arithmetic mean (balance point) of $n$ observations; the statistic every interval and chart centers on.

Sample variance and standard deviation

s^2=\frac{1}{n-1}\sum_{i=1}^{n}\left(X_i-\bar{X}\right)^2,\qquad s=\sqrt{s^2}

Spread of a sample; divide by $n-1$ (use $N$ for a full population). Units of $s$ match the data; $s^2$ is squared units.

Binomial mass function

P(x)=C(n,x)\,p^{x}(1-p)^{\,n-x}

Probability of exactly $x$ successes in $n$ independent trials, each with success probability $p$ . $C(n,x)=n!/[x!(n-x)!]$ .

Binomial mean and variance

\mu=np,\qquad \sigma^{2}=npq,\quad q=1-p

Expected count and its variance (a count, not a fraction). Standard deviation $\sigma=\sqrt{npq}$ .

Normal probability density

f(x)=\frac{1}{\sigma\sqrt{2\pi}}\,e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^{2}}

Gaussian curve, mean $\mu$ , standard deviation $\sigma$ . Never integrated by hand on the FE; standardize instead.

Standard-normal transform

z=\frac{x-\mu}{\sigma}

Converts any normal value to the unit normal so the handbook table gives $F(z)$ (left area), $R(z)$ (right area), $W(z)$ (within $\pm z$ ).

Within-band probability

P(-z\le Z\le z)=W(z)=2F(z)-1

Fraction of a normal population inside $\mu\pm z\sigma$ ; e.g. $W(1.96)\approx0.95$ .

Standard error of the mean

\sigma_{\bar X}=\frac{\sigma}{\sqrt{n}}\;\;(\text{or }s/\sqrt{n})

Spread of the sample mean; shrinks only as $\sqrt{n}$ . Drives both the confidence interval and the X-bar chart width.

Confidence interval, sigma known

\bar{X}\pm z_{\alpha/2}\,\frac{\sigma}{\sqrt{n}}

Interval for $\mu$ when the population $\sigma$ is known; $z_{0.025}=1.960$ for 95%, $1.645$ for 90%.

Confidence interval, sigma unknown

\bar{X}\pm t_{\alpha/2,\,n-1}\,\frac{s}{\sqrt{n}}

Interval for $\mu$ using sample $s$ ; $t$ has $n-1$ degrees of freedom and is wider than $z$ for small samples.

X-bar chart limits

\text{UCL}=\overline{\overline{X}}+A_2\bar{R},\qquad \text{LCL}=\overline{\overline{X}}-A_2\bar{R}

Three-sigma limits on subgroup averages; $\overline{\overline{X}}$ is the grand mean, $\bar{R}$ the mean range, $A_2$ from the handbook table for subgroup size $n$ .

R chart limits and process sigma

\text{UCL}_R=D_4\bar{R},\quad \text{LCL}_R=D_3\bar{R},\qquad \hat{\sigma}=\frac{\bar{R}}{d_2}

Limits on subgroup range; $D_3,D_4,d_2$ from the table. $\hat{\sigma}=\bar{R}/d_2$ estimates the process standard deviation.

Worked examples

Binomial sampling inspection

Problem. A filling line produces off-spec bottles at a rate of $p=0.08$ . An inspector pulls $n=10$ bottles at random. Find the probability that exactly two are off-spec, the probability that at most one is off-spec, and the expected number off-spec with its standard deviation.

Solution. Exactly two:

P(2)=C(10,2)\,(0.08)^2(0.92)^8=45(0.0064)(0.5132)=0.148

. At most one:

P(0)+P(1)=(0.92)^{10}+10(0.08)(0.92)^9=0.4344+0.3777=0.812

. Expected count:

\mu=np=10(0.08)=0.800

bottles; variance

\sigma^2=npq=10(0.08)(0.92)=0.736

, so

\sigma=\sqrt{0.736}=0.858

bottles. Sanity check:

P(2)=0.148

is plausible since the mean count is below 1, so two defects is moderately unlikely but not rare. Answers:

0.148

0.812

\mu=0.800

\sigma=0.858

P(2)=C(10,2)(0.08)^2(0.92)^8=0.148

Normal fraction outside spec

Problem. A reactor's product concentration is normally distributed with $\mu=250\ \text{g/L}$ and $\sigma=4\ \text{g/L}$ . What fraction of product exceeds $256\ \text{g/L}$ , and what fraction falls within $\pm 6\ \text{g/L}$ of the mean?

Solution. Standardize the upper bound:

z=(256-250)/4=1.50

. Fraction above:

R(1.50)=1-F(1.50)=1-0.9332=0.0668

, i.e.

6.68\%

. Within

\pm 6\ \text{g/L}

corresponds to

\pm z=1.50

W(1.50)=2F(1.50)-1=2(0.9332)-1=0.866

, i.e.

86.6\%

. Sanity check:

\pm 1.5\sigma

should capture most but not nearly all of a normal population, and

86.6\%

sits sensibly between the

68\%

\pm1\sigma

and the

95\%

\pm2\sigma

. Answers:

6.68\%

above

256

86.6\%

within band.

z=\frac{256-250}{4}=1.50,\qquad R(1.50)=0.0668

Confidence interval on conversion

Problem. Sixteen batches give a mean fractional conversion $\bar{X}=0.740$ with sample standard deviation $s=0.050$ . Construct a 95% confidence interval for the true mean conversion.

Solution. Population

\sigma

is unknown (only

s

is given), so use

t

with

n-1=15

degrees of freedom:

t_{0.025,15}=2.131

. Standard error:

s/\sqrt{n}=0.050/\sqrt{16}=0.0125

. Half-width:

2.131\times 0.0125=0.0266

. Interval:

0.740\pm0.0266=(0.713,\ 0.767)

. Sanity check: the interval is symmetric about

0.740

and only

\pm3.6\%

wide because

n

is moderate; using

z=1.96

instead of

t

would slightly underestimate it. Answer:

95\%\ \text{CI}=(0.713,\ 0.767)

0.740\pm t_{0.025,15}\frac{0.050}{\sqrt{16}}=0.740\pm0.0266

X-bar and R control limits

Problem. A purity measurement is charted in subgroups of $n=5$ . Over many subgroups the grand average is $\overline{\overline{X}}=80.0\%$ and the average range is $\bar{R}=4.0\%$ . With $A_2=0.577$ , $D_3=0$ , $D_4=2.114$ , and $d_2=2.326$ , find the X-bar and R control limits and estimate the process standard deviation.

Solution. X-bar chart:

\text{UCL}=80.0+0.577(4.0)=82.3\%

;

\text{LCL}=80.0-0.577(4.0)=77.7\%

; center

=80.0\%

. R chart:

\text{UCL}_R=2.114(4.0)=8.46\%

;

\text{LCL}_R=0(4.0)=0

; center

=4.0\%

. Process spread:

\hat{\sigma}=\bar{R}/d_2=4.0/2.326=1.72\%

. Sanity check: the X-bar limits sit about

\pm A_2\bar{R}=\pm2.31\%

, which is roughly

3\hat{\sigma}/\sqrt{n}=3(1.72)/\sqrt{5}=2.31\%

— consistent. Answers: X-bar

(77.7,\ 82.3)\%

; R

(0,\ 8.46)\%

;

\hat{\sigma}=1.72\%

\text{UCL}_{\bar X}=80.0+0.577(4.0)=82.3,\qquad \text{LCL}_{\bar X}=77.7

Common pitfalls

•Dividing the sum of squared deviations by $n$ for a sample: use $n-1$ for sample variance, $N$ only for a full population. The FE tests this denominator directly.
•Asking for $P(X=x)$ of a continuous variable. For the normal that probability is zero — every continuous question is about an interval, so standardize the endpoints and read areas.
•Mis-reading the unit-normal table: $F(z)$ is the area to the LEFT, $R(z)=1-F(z)$ to the right. Sketch and shade the region you actually want before picking a column.
•Using $z=1.96$ when only the sample $s$ is known. With $\sigma$ unknown the multiplier is $t_{\alpha/2,n-1}$ , which is larger for small $n$ and makes the interval correctly wider.
•Dropping the $\sqrt{n}$ in the standard error. The interval half-width and the X-bar limits both use $\sigma/\sqrt{n}$ , not $\sigma$ — forgetting it inflates the answer by a factor of $\sqrt{n}$ .
•Building control limits from the engineering spec (tolerance) instead of the process data. UCL/LCL come from $\overline{\overline{X}}$ , $\bar{R}$ , and the chart constants; specs belong to capability, not control.
•Confusing binomial variance with a fraction: $\sigma^2=npq$ is a variance in counts. The variance of the proportion $\hat p$ is $pq/n$ , a different quantity.

References

NCEES FE Reference Handbook — Engineering Probability and Statistics
NCEES FE Reference Handbook — Statistical Quality Control (Average and Range Charts)
Montgomery & Runger, Applied Statistics and Probability for Engineers — distributions, confidence intervals, control charts

C. Hypothesis testing and design of experiments

Expected Value, Hypothesis Testing and DOE

Expected value for decisions, one- and two-sample t-tests and ANOVA, Type I/II error and significance, outlier screening, and the logic of designed experiments.

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E. Regression and curve fitting

Regression and Curve Fitting

Least-squares slope and intercept, the correlation coefficient and R-squared, residuals and the standard error of estimate, and linearizing power and exponential data.

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