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ISDS 361A - Final Cheat Sheet
Business Analytics I (California State University, Fullerton)
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Statistics – a way to get information from data
Scales of measurement:
Nominal – qualitative or categorical and labels are used to denote the categories
Ordinal – Same as nominal but there is ordering or ranking that is meaningful
Interval – Quantitative or numerical in nature (sat scores, grades etc)
Descriptive statistics – summary of important aspects of a data set (includes
collecting
data, organizing the data, presenting with charts etc.
Business Statistics is analyzing the data
Parameter (Variable) – A descriptive measure of the population that is of interest
Statistic – A descriptive measure that is calculated from the sample
Sampling:
Examine part of the whole or population (impractical, prohibitive, costly)
Randomized sample (every item in the population has an equal change of being in the
sample
Non-Random Sampling Errors:
Selection Bias (one subset has unequal change of being selected)
Non-response Bias (When data is unavailable or unattainable)
Measurement errors (Inaccuracies in getting/recording data, ambiguous questions etc)
Symmetrical
Normal Distribution
Single Mode
Mean is recommended
Mean = Median = Mode
50% of the probability on each side of the mean
Skewed to the left
Negatively Skewed
Single Mode
Median is better measure
Mean < Median
Q2 close to Q3
Bi-Modal – 2 modes
Empirical Rule (Symmetrical Distribution)
68% within 1 SD
95% within 2 SD
100% within 3 SD (99.7%)
If 2 population means are not equal,
we use coefficient of variation (C.V.)
Data is facts and figures. Decision Science – Getting info from data
Cross Sectional – Data collected at the same time
Time series data – Collected over several time periods
Inferential Statistics – goes beyond data to draw conclusions about population based
on sample data.
Population – a set of items under the study
Random Sample – A random subset chosen from the population
Purpose of Inferential Stats To make inferences about a parameter of a population
based on information obtained from a statistic of the sample.
Sources of statistical data:
Designed experiment
Public Source
Survey
Observation studies
Shape (Distribution) -Histogram (symmetry, skewness, modality)
Location – Central tendency (mean, median, mode)
Variability / Spread – range, IQR, variance, standard deviation
Skewed to the right
Positively Skewed
Single Mode
Median is better measure
Mean > Median
Q2 close to Q1
Range (Max – Min) – Best for limited data i.e. under 10 data only uses 2 values
Interquartile Range – IQR (Q3 – Q1) Measure variablitily of the middle 50% of data
Variance (Not meaningful)
Standard Deviation (σ) =
uses all of the data (most efficient)
Outliers Formula = Q1 - 1.5(IQR) and Q3 + 1.5(IQR) (IQR Formula = Q3-Q1)
Chebyshev’s Rule (Non-Symmetrical Distribution)
Matched Pair – W/Pop Std.
=(Xbar1-Xbar2)/SQRT((std1^2/samp1)+(std2^2/samp2))
p-value =1-NORM.S.DIST(Test Stat, True)
Continuous – Fractions (time, height, weight). (361 always uses continuous)
Any value within an interval CAN NEVER BE EXACTLY ANYTHING
Random Variable – A numerical description of the outcome of an event
Probability Distribution – The collection of all possible values of the random
variable X and the associated probabilities P(X=x)
Discrete – No fraction, Whole #’s, P(X=1)
A discrete number of possible values
Higher Standard Deviation – Wider Spread
Lower Standard Deviation – Narrower Spread
Standard Normal Distribution – mean is 0 and standard deviation is 1
Location:
Mean = average (range of data)
Median =median (range of data)
Mode (Single) = mode.sngl (range of data)
Mode (Multiple) =mode.mult (range of data)
If Test Stat is < C.V. Do not reject H0
3 treatments 10 observations, SSE=399.6 MSE? 399.6 / 3*10 – 3
EXCEL COMMANDS
Variability
Range =Max(range of data)-Min(range of data)
IQR =(Quartile.exc(range of data),3)- (Quartile.exc(range of data),1)
Standard Deviation =stdev.p (range of data) for population =stdev.s (range of data)
for sample
For a random variable X with mean = 10, variance =25, P(X>20)? (x-mean)/sqrt(var)
Inferential stats goes beyond data at disposal, descriptive does not.
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Sampling Distribution, Sample Mean (x̄ )
1. Convert x̄ into: Z = (x̄ - x) / SE
a. SE of x̄ = σ /
2. Solve for probability
a. P(x̄ < x1) = NORM.S.DIST(Z,TRUE)
b. P(x̄ > x1) = 1-NORM.S.DIST(Z,TRUE)
c. (between) P(x1 < x- < x2) =
NORM.S.DIST(Z2,TRUE) –
NORM.S.DIST(-Z1,TRUE)
or
NORM.S.DIST(X/SE,TRUE) –
NORM.S.DIST(-X/SE,TRUE)
Interval Estimation of p (proportions %)
1. Find: CV = -NORM.S.INV(a/2)
2. Find: SE of p̂ =
3. Find confidence interval (UL and LL) of
µ with p̂ +/- ME
a. ME = CV * SE of p̂
Sampling Distribution, Sample Proportion (p̂ )
1. Find: p̂ = x/n
2. Find SD =
3. Check distrib: when n*p ≥ 5 or n(1-p) ≥
5; then distrib is normal
4. Same “P rules” as Finding Prob Norm
Distrib = NORM.S.DIST functions
Sample size in interval estimation of p
1. Find: Za/2 = NORM.S.INV(a/2)
2. Determine ME (desired)
Solve: Solve: n
=(((NORM.S.INV(0.05/2)/0.06)^2)*(0.47*(1-0.47)))
Type I Error – Incorrectly rejected H0 when H0 is
true
Hypothesis Testing (known SD )
1. Determine H0 and Ha
2. Find: SE of x̄ = /
a. =( /SQRT(n))
3. Find: Z = x̄ -mean / SE of x̄
4. Find: CV =NORM.S.INV(a) (left) or
=-NORM.S.INV(a) (right) (a/2) for two tailed
5. Find: p-value =1-NORM.S.DIST(6.45,TRUE)
6. Reject H0 if Z > CV or Z < -CV
Null Hypothesis (H0): tentative assumption, can
possibly be disproved using sample evidence,
always is =, ≥, ≤
Interval Estimation of µ with known population σ
1. Find: CV = -NORM.S.INV(a/2)
2. Find: SE of x̄ = pop σ /
a. = σ /SQRT(n)
3. Find confidence interval (UL and LL) of
µ with x̄ +/- ME
a. ME = CV * SE of x̄
b. **ME = confidence.norm(α, ,n)
TIM! LOOK IN BOX TO LEFT!
<<<<<<<<<<<<<<<<<<<
Recommended sample size, in interval
estimation when σ is known
1. Find: Za/2 = NORM.S.INV(a/2)
2. Determine ME (desired margin of error)
3. Solve: n =
=SD^2*(z/a)^2
Interval Estimation of µ with unknown
population σ
1. Find: CV = -T.INV(a/2, n-1)
2. Find: SE of x̄ = s /
=σ /SQRT(n)
3. Find confidence interval (UL and LL) of µ with
x̄ +/- ME
a. ME = confidence.norm(α, ,n)
Alternative Hypothesis (Ha): opposite of null,
often what the test attempts to establish.
If x̄ > H0, Everything to right in Excel
If x̄ < H0, Everything to the left in Excel
Type II Error – Incorrectly accepted / not rejected
H0 when H0 is false.
Hypothesis Testing (P-Value Method average)
1. Determine H0 and Ha
2. Find: SE of x̄ = /
3. Test Statistic: x̄ -Ha/SE
a. = (x̄ -Ha)/SE)
4. Find:
a. P-Value = 1-NORM.S.DIST
(Z,True) for P (Z>Zobs) test
b. P-Value = NORM.S.DIST (Z,Ture)
for P (Z<Zobs) test
5. Reject H0 if P < a
Reject H0 if Z > Standard C.V.
Reject H0 if P-value < a
***If P-value is less than α, Reject
Hypothesis Testing (Population Proportion %)
1. Determine H0 and Ha
2. Find: SE of p =
a. =SQRT((*(1-))/n)
3. Test Statistic: -Ha/S
= (-Ha)/SE)
4. Find:
a. P-Value = 1-NORM.S.DIST (Z,True)
for P (Z>Zobs) test (lower)
b. P-Value = NORM.S.DIST (Z,True) for
P (Z<Zobs) test (upper)
Reject H0 if P < a
Hypothesis Testing (µ Differences Known SD)
(matched samples)
Find Critical Value (Za/2) =NORM.S.INV(a/2)
i.e. for 95% confidence it’s =NORM.S.INV(.05/2)

At 90% confidence, it will always be +/- 1.65

At 95% confidence, it will always be +/- 1.96

At 98% confidence, it will always be +/- 2.33

At 99% confidence, it will always be +/- 2.58
Step 2 – Find Margin of Error
a.
= x̄ +(CV* (/SQRT(n)))
CV =
=-NORM.S.INV(0.025)
=1.96
Sample proportion of p̂
Normal when:
np ≥ 5 & n(1-p) ≥ 5
Find P Value of Proportion
=Z*(SQRT(P*(1-P)/n))
If sample size increases, probability increases.
If sample size decrease, probability decreases.
1. Find P Value =NORM.S.DIST(Z,TRUE)
2. Find CV =NORM.S.INV(a/2)
3. Test Statistic =( x̄ -µ)/( /SQRT(n))
# on horizonal axis =NORM.INV
Area under the curve = NORM.DIST
If you know SD, it’s a Z test
If you do not know SD, it’s a T test.
In ANOVA, treatments refer to different levels of a
factor
A descriptive measure of linear association
between two variables is the correlation coefficient
If F is less than F Critical Value, Do not reject
P value > alpha, Do not reject
1-F.DIST(F,df1, df2,TRUE)
Test Stat CV F.INV(.95,df(test),df(error)
Mathematical solution procedure is called least
square method.
R^2 is coefficient of determ
Higher R^2 the better the model.
Between Groups – Treatments
Within groups – error
Central Limit Theorem – Even if X does not have a
normal distribution, x̄ will be approximately normal if n
is ≥ 30 The only other way to be normal is if X is normal.
ANOVA – Factor: Sales, Car Waxes (Top Column)
Treatment: 3 Types of car waxes (# of columns)
Experimental Units – Cars
Response Variable - # of washes
ŷ = b0 + b1, x
ŷ is predicted variable b0 is estimate of B0
b1 is estimate of B1 x is independent variable
Regression analysis is a statistical procedure that
describes how one dependent, and one or more
independent variables are related.
1.
2.
Enter the information on excel
Go to Data – Data Analysis (must be enabled)
Choose “t-Test: Paired Two Sample for Means
3.
a.
b.
c.
d.
e.
f.
g.
Var 1 range: choose first column (w/header)
Var 2 range: choose sec column (w/header)
Hypothesized Mean Difference = 0
Click box for Labels (if you included header)
Alpha will be given (.05 or .01)
Output range, click any empty cell in Excel
You have your t stat; determine right tail or left
tail and you have your critical & p-value
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Hypothesis Testing (Unknown SD)
1.
2.
3.
4.
5.
6.
7.
8.
Determine H0 and Ha
Find: SE of x̄ = / = s/SQRT(n)
Find: T = x̄ - mean / SE of x̄ =( x̄ - µ)/SE
Find: CV =T.INV(a,n-1) (left) or =T.INV(a,n-1)
(right) or =T.INV(a/2,n-1) for 2-tailed
Find:
P-Value = T.DIST(T,n-1,TRUE) for P(t < T) test
P-Value = 1-T.DIST(T,n-1,TRUE) for P(t > T) test
P-Value = 2*T.DIST(T,n-1,TRUE) for two-tailed
test
Reject H0 if P < a
SE of =
=( /SQRT(n))
SE of =
= SQRT(P*(1-P)/n)
Hypothesis Testing (µ Differences Unknown SD)
(chart)
1. Use spreadsheet from Canvas with SE
2. Determine H0 and Ha (always 0)
3. SE X1-X2
a. =SQRT((s1^2/n1)/(s2^2/n2)
b. Test stat =(x1-x2)/SE (SE from excel)
c. P-value =1-T.DIST(T,DF,TRUE)
d. The Degree of Freedom (DF) is on
spreadsheet.
Regression: Y=B0 + B1, X + E
Y=Dependent Variable (like to predict)
B0 = Intercept B1=Slope X=Independent Variable
E=Error Term (unexplained variation)
Regression H0: B1 = 0 Ha: B1 ≠ 0
T-Stat = b1 – B1 / se(b1)
p-value= 2*P(t>t-stat) or 2q*(1-T.DIST(F,df-1,True)
Rows (blocks)
Columns (treatments)
Test stat for treatments C.V. = F.INV
P-value = 1-F.DIST > Alpha, do not reject