Probability and Statistics

Probability and Statistics · Study · FE Chemical · FE → PE Prep

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}

Binomial mass function

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

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), $R(z)$

Within-band probability

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

Fraction of a normal population inside $\mu\pm z\sigma$

Standard error of the mean

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

Confidence interval, sigma known

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

Confidence interval, sigma unknown

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

X-bar chart limits

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

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}

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 $\pm 6\ \text{g/L}$

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

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 $\bar{R}=4.0\%$

A. Probability distributions

Distributions, Confidence Intervals and Control Limits

Discrete versus continuous random variables

Center and spread: mean, median, mode, variance

The binomial distribution

The normal distribution and the z-transform

Confidence interval for the mean

Statistical process-control limits

Exam strategy

Key equations

Worked examples

Common pitfalls

References

C. Hypothesis testing and design of experiments

Expected Value, Hypothesis Testing and DOE

E. Regression and curve fitting

Regression and Curve Fitting