BUS 211 Notes
Chapter 1 Introduction and Data Collection
Categorical Variables – responses are a selection i.e. Gender (male or female), Class (freshman,
sophomore, junior, senior), Smoke (yes or no), etc.
Numerical Variables – responses are numbers i.e. Income ($30,000), Age (25), etc.
Can be Discrete (Integer) or Continuous (fractional parts),
Chapter 2 Presenting Data in Tables and Charts
Sort Data – Data | Sort
Stem-and-Leaf Graph – PHStat | Descriptive Statistics | Stem-and-Leaf Display
Frequency Distribution - PHStat | Descriptive Statistics | Frequency Distribution
Set up classes then array (bin) the upper limit of the desired frequency distribution
Be sure to include a label for the array (use Upper Limit)
Relative Frequency distribution – Divide the frequency distribution by the total
Percentage Distribution - Divide the frequency distribution by the total and multiply by 100
Or use Format | Cells… | Percentage
Cumulative Distribution – Sum the frequencies from top to bottom listing each total as you go.
Graphs - PHStat does not work well for most graphs use the chart wizard in Excel
Histogram also known as a Vertical Bar Chart or Column Chart -
Set up the frequency distribution then use the midpoints for labels
Double click the chart icon and select a column graph type
Select the frequency without labels as the data
Select the Series tab, mouse into the X-axis label box then select the midpoints
Select Next to insert the title and axis labels and make any other changes
Select Next to pick a location for the chart then Finish
Double click a bar and select Options, set gap width to 0
Polygon also known as a line graph -
Set up the frequency distribution then use the midpoints for labels.
Insert a class with O frequency and an appropriate label at the top and the bottom.
Double click the chart icon and select a line graph type
Select the frequency without labels as the data
Select the Series tab, mouse into the X-axis label box then select the midpoints
Select Next to insert the title and axis labels and make any other changes
Select Next to pick a location for the chart then Finish
Ogive also known as a cumulative line graph or cumulative polygon
Set up the cumulative frequency distribution use the upper class limit for labels.
Insert a class with O frequency and an appropriate label at the top but not the bottom.
Double click the chart icon and select a line graph type and complete the steps
XY ScatterSet up the data in columns with the X values first and the Y in the second column
Double click the chart icon and select XY Scatter graph
Select both columns as the data, do not select the labels, and complete the steps
Bar ChartSame as Histogram but for categorical data.
Use the category labels: if not numerical values they can be selected with the data.
Pie ChartSame as above. Be sure to remove legend, select Data Labels, check Category name
Pareto Chart Raw Data: use line chart on 2 axis or
Select Descriptive Statistics | One-Way Tables & Charts…
Be sure to select labels as the model will not work otherwise
Check table of frequencies and Pareto Diagram
Bivariate Categorical Tables and Charts Use PHStat (also available in Excel - Data | Pivot Wizard)
In PHStat select Descriptive Statistics | Two-Way Tables & Charts
Chapter 3 Numerical Descriptive Measures
Use Tools | Data Analysis | Descriptive Statistics, check the Summary statistics box to get the following:
sample mean, median, mode, standard deviation, variance, range
population mean, median, mode, range
Use fx the individual functions for the following measures
geometric mean (GEOMEAN), population variance (VARP) and standard deviation (STDEVP)
approximate quartiles (QUARTILE), approximate percentiles (PERCENTILE)
Coefficient of variation: Divide the standard deviation by the mean and multiply by 100%
Box-and-Whisker Plot and Five-Number Summary
PHStat | Descriptive Statistics | Box-and-Whisker Plot then check Five-Number Summary
Gives the exact quartiles not approximations
Coefficient of Correlation: fx (CORREL), or Tools | Data Analysis | Correlation
Chapter 4 Basic Probability
Probability of A or B:If A and B are Mutually Exclusive:
Conditional probability of A given B:If A and B are Independent:
Joint Probability of A and B:If A and B are Independent:
Bayes' Theorem
Chapter 5 Some Important Discrete Probability Distributions
Combinations:
Binomial distribution: (for an infinite population)
PHStat | Probability & Prob. Distributions | Binomial then check Cumulative Probabilities
Hypergeometric distribution: (for a finite population)
PHStat | Probability & Prob. Distributions | Hypergeometric no cumulative probabilities available
Poisson distribution:
PHStat | Probability & Prob. Distributions | Poisson then check Cumulative Probabilities-
Chapter 6 The Normal Distribution and Other Continuous Distributions
Normal Distribution
PHStat | Probability & Prob. Distributions | Normal then check the desired calculation
To check the normality assumption construct a stem-and-leaf, box-and-whisker, histogram or a
Normal probability plot PHStat | Probability & Prob. Distributions | Normal Probability Plot
Uniform Distribution
where a and b are the endpoints of the uniform distribution.
Exponential distribution
PHStat | Probability & Prob. Distributions | Exponential
Only returns results for X, for > x use 1-probability, for results between two values find the
probability for each and subtract the smaller from the larger
Sampling distribution of the mean
Calculate the standard deviation of the sampling distribution also called the Standard error of the mean then use the Normal Distribution calculator if the population is normally distributed or
the sample size is > 30 or the population distribution is symmetrical and the sample size is > 15
Infinite population Finite population
Sampling distribution of the proportion:
Calculate the standard deviation of the sampling distribution (Standard Error of the Mean) then
If np > 5 and n(1-p) > 5 use the Normal Distribution calculator PHStat | Probability & Prob. Distributions | Normal
ps = sample proportion p = population proportion
Infinite population Finite population
Chapter 7 Confidence Interval Estimation
Interval estimate of the population mean (x) with x unknown:
PHStat | Confidence Intervals | Estimate for the Mean, sigma unknown
be sure to check the finite box for finite populations
Interval estimate of the population proportion:
PHStat | Confidence Intervals | Estimate for the Proportion
be sure to check the finite box for finite populations
Interval estimate of the population total:
PHStat | Confidence Intervals | Estimate for the Population Total
Sample size (n) for estimating a mean:
PHStat | Sample Size | Determination for the Mean
be sure to check the finite box for finite populations
Estimate of parameters would be from a preliminary sample
Sample size for estimating a proportion:
PHStat | Sample Size | Determination for the Proportion
be sure to check the finite box for finite populations
Estimate of True Proportion would be the proportion from a preliminary sample
If a preliminary sample is not available use .5
Chapter 8 Fundamentals of Hypothesis Testing: One-Sample Tests
One Sample numerical data unknown
Hypothesis
Ho: x = value a two tail testHo: x value Ha: x value upper tail test
Ha: x valueHo: x value Ha: x value lower tail test
Test Statistict
ProcedureSummary Data: PHStat | One-Sample Tests | t Test for the Mean, sigma unknown
Decision Rule If the p-value is less than alpha Reject the Hypothesis
If the p-value is greater than or equal to alpha Fail to Reject the Hypothesis
ConclusionIf rejected – There is sufficient evidence that (Question asked)
If not rejected – There is not sufficient evidence that (Question asked).
Parentheses indicate information to be taken from the problem
One Sample Categorical Data
Hypothesis
Ho: p = value a two tail testHo: p value Ha: p value upper tail test
Ha: p valueHo: p value Ha: p value lower tail test
Test StatisticZ
ProcedureSummary Data: PHStat | One-Sample Tests | Z Test for the Proportion
Raw Data: No Tests available, calculate p and use PHStat
Decision Rule If the p-value is less than alpha Reject the Hypothesis
If the p-value is greater than or equal to alpha Fail to Reject the Hypothesis
ConclusionIf rejected – There is sufficientevidence that (Question asked)
If not rejected – There is not sufficient evidence that (Question asked)
Parentheses indicate information to be taken from the problem
Chapter 9 Two-Sample Tests
Procedure to determine the proper two sample mean test for numerical data:
Two Sample test of Means with Paired numerical data
HypothesisHo: 1 = 2 a two tail testHo: 12 Ha: 12 upper tail test
Ha: 12Ho: 12 Ha: 12 lower tail test
ProcedureSummary Data:no PHStat calculation available
Raw Data: Data Analysis | t Test: Paired Two Sample for Means
Test Statistict
Decision Rule If the p-value is less than alpha Reject the Hypothesis
If the p-value is greater than or equal to alpha Fail to Reject the Hypothesis
ConclusionIf rejected – There is sufficient evidence that (Question asked)
If not rejected – There is not sufficient evidence that (Question asked)
Interval estimate of the difference To get t use function TINV(1-Confidence, df)
Use Descriptive Statistics to get D and sd
Or PhStat | Confidence Intervals | Estimate for the Mean, sigma unknown - Select the differences as the data
Two Sample test of Variances with numerical data
HypothesisHo: 21 = 22 a two tail testHo: 2122 Ha: 2122 upper tail test
Ha: 2122 Ho: 2122 Ha: 2122 lower tail test
ProcedureSummary Data:PHStat | Two-Sample Tests | F Test for the Difference in Two Variances
Raw data: Data Analysis | F Test Two Sample for Variances Do not use only gives lower tail value
Test StatisticF
Decision Rule If the p-value is less than alpha Reject the Hypothesis
If the p-value is greater than or equal to alpha Fail to Reject the Hypothesis
ConclusionIf rejected – There is sufficient evidence that (Question asked)
If not rejected–There is not sufficient evidence that (Question asked)
Two Sample test of Means with numerical data2’s not proven unequal with the F test
HypothesisHo: 1 = 2 a two tail testHo: 12 Ha: 12 upper tail test
Ha: 12Ho: 12 Ha: 12 lower tail test
ProcedureSummary Data:PHStat | Two-Sample Tests | t Test for Differences in Two Means
Raw Data: Data Analysis | t Test: Two Sample Assuming Equal Variances
Test Statistict
Decision Rule If the p-value is less than alpha Reject the Hypothesis
If the p-value is greater than or equal to alpha Fail to Reject the Hypothesis
ConclusionIf rejected – There is sufficient evidence that (Question asked)
If not rejected – There is not sufficient evidence that (Question asked)
Interval estimate of the difference
To get t use function TINV(1-Confidence, df)
Two Sample test of Means with numerical data2‘s proven unequal with the F test
HypothesisHo: 1 = 2 a two tail testHo: 12 Ha: 12 upper tail test
Ha: 12Ho: 12 Ha: 12 lower tail test
ProcedureSummary Data: Use spreadsheet downloaded from the Homework web page
Raw Data: Data Analysis | t Test: Two Sample Assuming Unequal Variances
Test Statistict
Decision Rule If the p-value is less than alpha Reject the Hypothesis
If the p-value is greater than or equal to alpha Fail to Reject the Hypothesis
ConclusionIf rejected – There is sufficient evidence that (Question asked)
If not rejected – There is not sufficient evidence that (Question asked)
Interval estimate of the difference
To get t use function TINV(1-Confidence, df)
Two Sample test of a Proportion with categorical data
HypothesisHo: p1 = p2a two tail testHo: p1 p2 Ha: p1 p2 upper tail test
Ha: p1 p2Ho: p1 p2 Ha: p1 p2 lower tail test
Procedure PHStat | Two-Sample Tests | Z Test for the Differences in Two Proportions
Test StatisticZ
Decision Rule If the p-value is less than alpha Reject the Hypothesis
If the p-value is greater than or equal to alpha Fail to Reject the Hypothesis
ConclusionIf rejected – There is sufficient evidence that (Question asked)
If not rejected – There is not sufficient evidence that (Question asked)
Interval estimate of the difference
To get Z use function NORMSINV(two tail)
where two tail=Confidence+(1-Confidence)/2
Chapter 10 Analysis of Variance (Multi (c) Sample tests with numerical data)
Equality of Variances
HypothesisHo: 21 = 22= 23 a two tail test
Ha: not all ’s are equal
ProcedureRaw data:PHStat | Multiple-Sample Tests | Levene’s Test
Test StatisticF
Decision Rule If the p-value is less than alpha Reject the Hypothesis
If the p-value is greater than or equal to alpha Fail to Reject the Hypothesis
ConclusionIf rejected – There is sufficient evidence that (Question asked)
If not rejected–There is not sufficient evidence that (Question asked)
One Factor ANOVA
HypothesisHo: 1 = 2 = 3 … = cc = the number of populations
Ha: not all ’s are equal
ProcedureTools | Data Analysis |Anova: Single Factor
Test StatisticF from the computer printoutP-value = The Probability of
Decision Rule If the p-value is less than alpha Reject the Hypothesis
If the p-value is greater than or equal to alpha Fail to Reject the Hypothesis
ConclusionIf rejected – There is sufficient evidence that (Question asked)
If not rejected – There is not sufficient evidence that (Question asked)
Tukey's multiple comparison method: (determines which of the c means are different from each other).
ProcedurePHStat | Multiple-Sample Tests | Tukey-Kramer Procedure
Test StatisticCritical Range
Input Q found in the Studentized Range Table where column = c and row = n-c
c = number of groups n = total number of data points in all groups
Decision RuleIf the absolute difference between any two pairs of means is greater than the critical range the pair is different.
Two Factor With Replication
HypothesisHo1: A1 = A2 = A3 … = rr = the number of levels in Factor A
Ha1: not all ’s are equal
Ho2: B1 = B2 = B3 … = cc = the number of levels in Factor B
Ha2: not all ’s are equal
Ho3: No Interaction
Ha3: Interaction
ProcedureTools | Data Analysis |Anova: Two Factor With Replication
Test StatisticF from the computer printout. p-value = The Probability of
For differences in rows see p-value for the Sample row of the ANOVA
For differences in columns see p-value for the Columns row of the ANOVA
For interaction between factors see p-value for the Interaction row of the ANOVA
Decision Rule If the p-value is less than alpha Reject the Hypothesis
If the p-value is greater than or equal to alpha Fail to Reject the Hypothesis
ConclusionH1 If rejected – There is sufficient evidence of a difference in (factor A)
H2 If rejected – There is sufficient evidence of a difference in (factor B)
H3 If rejected – There is sufficient evidence of an interaction term
If not rejected – There is not sufficient evidence to make a conclusion about …
Tukey's multiple comparison method for Two Factor ANOVA with replication:
No spreadsheet, hand calculate with the following formulas:
MSW from ANOVA MS Within
Q table column is r the number of levels in Factor A
Q table row is rc(n’-1) where c is the levels in Factor B, and n’ is the number of replications
MSW from ANOVA MS Within
Q table column is c the number of levels in Factor B
Q table row is rc(n’-1) where r is the levels in Factor A, and n’ is the number of replications
Chapter 11 Chi-Square Tests and Nonparametric Tests
Two Sample test of a Proportion with categorical data(Alternate Procedure)
HypothesisHo: p1 = p2Ha: p1 p2 (No <, or > Hypothesis)
Procedure PHStat | Two-Sample Tests | Chi-Square Test for the Differences in Two Proportions
Test Statistic2
Decision Rule If the p-value is less than alpha Reject the Hypothesis
If the p-value is greater than or equal to alpha Fail to Reject the Hypothesis
ConclusionIf rejected – There is sufficient evidence that (Question asked)
If not rejected – There is not sufficient evidence that (Question asked)
Multi (c) Sample test of Proportions with categorical data
HypothesisHo: p1 = p2 = p3… pcc = the number of samples
Ha: not all p’s are equal
Procedure PHStat | Multiple-Sample Tests | Chi-Square Test
Test Statistic2
Decision Rule If the p-value is less than alpha Reject the Hypothesis
If the p-value is greater than or equal to alpha Fail to Reject the Hypothesis
ConclusionIf rejected – There is sufficient evidence that (Question asked)
If not rejected – There is not sufficient evidence that (Question asked)
Be sure to check the box for the Marascuilo Procedure to determine which proportions are different.
2 Test of Independence
HypothesisHo: Two categorical variables are independent
Ha: Two categorical variables are related
Procedure PHStat | Multiple-Sample Tests | Chi-Square Test
Test Statistic2
Decision Rule If the p-value is less than alpha Reject the Hypothesis
If the p-value is greater than or equal to alpha Fail to Reject the Hypothesis
ConclusionIf rejected – There is sufficient evidence that the variables are related
If not rejected – There is not sufficient evidence that the variables are related.
Two Sample test of Medians with numerical data
HypothesisHo: M1 = M2 a two tail testHo: M1 M2 Ha: M1 M2 upper tail test
Ha: M1 M2Ho: M1 M2 Ha: M1 M2 lower tail test
ProcedureRaw Data PHStat | Two-Sample Tests | Wilcoxon Rank Sum Test
Summary DataNo Tests available.
Test StatisticZ
Decision Rule If the p-value is less than alpha Reject the Hypothesis
If the p-value is greater than or equal to alpha Fail to Reject the Hypothesis
ConclusionIf rejected – There is sufficient evidence that (Question asked)
If not rejected – There is not sufficient evidence that (Question asked)
Kruskal-Wallis Rank Test for Differences Between c Medians
HypothesisHo: M1 = M2 = M3 = MC
Ha: Not all Mj are equal ( j=1,2,…C)
ProcedureRaw Data PHStat | Multiple-Sample Tests | Kruskal-Wallis Rank Test
Summary DataNo PHStat or Excel calculation available
Test StatisticH
Decision Rule If the p-value is less than alpha Reject the Hypothesis
If the p-value is greater than or equal to alpha Fail to Reject the Hypothesis
ConclusionIf rejected – There is sufficient evidence that (Question asked)
If not rejected – There is not sufficient evidence that (Question asked)
Chapter 12 Simple Linear Regression
Linear Regression Model: relationship represented as