MATH 1530

Unit 1 Formula Sheet

Chapter 2

Standard Deviation:

z-Score:

Empirical Rule:

(68-95-99.7 Rule) For symmetric, bell-shaped data sets

Class Width:

Mean

Interquartile Range and Outliers

Unit 2 Formula Sheet

Chapter 3

Classical (or Theoretical) Probability:

\(P(E)=\frac{ \text {Number of Outcomes in Event E} }{ \text {Total number of outcomes in the sample space }}\)

Empirical (or Statistical) Probability:

\(P(E)=\frac{ \text {Frequency of event E} }{ \text {Total frequency} }=\frac fn\)

Probability of a Complement:

\(P(E')=1-P(E)\)

Probability of occurrence of both events A and B:

\(P(A\; \text {and} \;B)=P(A)\cdot P(\left.B\right|A)\)

\(P(A\; \text {and} \;B)=P(A)\cdot P(B)\) if \(A\; \text {and} \;B\) are independent

Probability of occurrence of either A or B:

\(P(A \; \text {or} \; B)=P(A)\;+\;P(B)\;-\;P(A \; \text {and} \; B)\)

\(P(A\; \text {or} \;B)=P(A)\;+\;P(B)\) if A and B are mutually exclusive

Chapter 4

Mean of a Discrete Random Variable:

\(\mu=\Sigma x\;P(x)\)

Variance and Standard Deviation of a Discrete Random Variable:

Variance: \(\sigma^2=\Sigma\left(x-\mu\right)^2\;P(x)\)

Standard Deviation: \(\sigma=\sqrt{\sigma^2}=\sqrt{\Sigma\left(x-\mu\right)^2\;P(x)}\)

Expected Value:

\(E(x)=\mu=\Sigma x\;P(x)\)

Binomial Probability of x successes in n trials:

\(P(x)={}_nC_x\;p^xq^{n-x}=\frac{n!}{(n-x)!x!}p^xq^{n-x}\)

Population Parameters of a Binomial Distribution:

Unit 3 Formula Sheet

Chapter 5

Standard Score or z-Score:

\(z=\frac{ \text {Value}- \text {Mean} }{ \text {Standard Deviation} }=\frac{x-\mu}\sigma\)

Transforming a z-Score to an x-Value:

\(x=\mu+z \cdot \sigma\)

Central Limit Theorem:

(\(n\geq30\) or population is normally distributed)

Mean of the Sampling Distribution: \(\mu_x=\mu\)

Standard Deviation of the Sampling Distribution (Standard Error):

\(\sigma_x=\frac\sigma{\sqrt n}\)

z-Score \(=\frac{{\displaystyle\overset\_x}-\mu_x}{\sigma_x}=\frac{\displaystyle\overset\_x-\mu}{ \frac {\sigma}{\sqrt n}}\)

Chapter 6

Confidence Interval for the Mean:

\(\overset-x-E<\mu<\overset-x+E\)

\(E=t_c\frac s{\sqrt n}\)

Minimum Sample Size to Estimate the Mean:

\(n=\left(\frac{z_c\sigma}E\right)^2\)

Population Proportion:

\(\widehat p=\frac xn\)

Confidence Interval for Population Proportion:

\(\widehat p-E\;<\;p\;<\;\widehat p+E\)

\(E=z_c\sqrt{\frac{\displaystyle\widehat p\widehat q}n}\)

Minimum Sample Size to Estimate:

\(n=\widehat p\widehat q\left(\frac{z_c}E\right)^2\)

Unit 4 Formula Sheet

Chapter 7

Test Statistics:

Mean: \(t=\frac{{\displaystyle\overset\_x}-\mu}{ \frac {s}{\sqrt n}}\)

Proportion: \(z=\frac{{\displaystyle\widehat p}-\mu_{\displaystyle\widehat p}}{\sigma_{\displaystyle\widehat p}}=\frac{\widehat p-p}{\sqrt{ \frac {pq}{n}}}\)

Chapter 9

Correlation Coefficient:

\(r=\frac{n\sum xy-\left(\sum x\right)\left(\sum y\right)}{\sqrt{n\sum x^2-\left(\sum x\right)^2}\sqrt{n\sum y^2-\left(\sum y\right)^2}}\)

t-Test for Linear Correlation:

\(t=\frac r{\sqrt{\displaystyle\frac{1-r^2}{n-2}}}\) (\(\text {degrees of freedom} = n-2\))

Equation of a Regression Line:

\(\widehat y=mx+b\)

Chapter 10:

Chi-Square

\(\chi^2={\textstyle\sum }\frac{\left(O-E\right)^2}E\)

Goodness of Fit Test:

degrees of freedom = \(k-1\)

Independence Test:

degrees of freedom = \((r-1)(c-1)\)

r=number of rows

c=number of columns

Table of Critical Values Pearson Correlation

2-tailed: Degrees of Freedom \(= n-2\)