Showing posts with label statistics test. Show all posts

Sunday, August 22, 2010

Use the Excel t Test To Find Out What the Best Days To Sell Are

The t Test in Excel

Can Determine What Your

Best Sales Days Are

It's always great to know what day of the week you can expect to get peak sales. The t test in Excel can provide that information for you. It's quite a simple test to run, as you will see. This blog article will walk you step-by-step through a t Test in Excel. The t Test compares two groups of samples and determines whether the mean of one sample is different than the other. Most types of t Tests require that each sample group have the same number of samples and also have the same variance. This Excel t Test has neither of those requirements.

Here is the scenario we are going to test: Suppose that you have been monitoring your daily sales for about a year. Your two best sales during the week are normally Monday and Wednesday. You would like to know which of those two days really does produce the best sales. You've tracked Monday sales for the last 40 weeks and Wednesday sales for the last 42 weeks. Mean sales for Wednesday is a bit higher than mean sales for Monday, but you would like to know with 95% certainty whether the difference in means is not just by chance and that Wednesday really is a better day for selling.

You can run your data through an Excel t Test and know within a minute whether Wednesday really is the best sales day. Excel has several built-in t Tests. The specific Excel t Test we will use is called the "t Test: Two Sample Assuming Unequal Variances." This t Test allows for two samples that have unequal sizes and variances. The only requirement is that both samples are Normally distributed. This will be discussed shortly. In Excel 2003, this test can be accessed through this menu path: Tools / Data Analysis / t Test: Two Sample Assuming Unequal Variances. Before we perform this t Test, we need to have a discussion of what the t Test is.

t Test - General Description

This test will tell you whether the difference between the before and after numbers is genuine or whether this difference could merely have been the result of chance. Overall a t-test compares two means and determines within a specified degree of certainty whether the two means really are different, or whether the difference might have occurred by chance.

t Test for Two Samples Having
Unequal Sizes and Variances

The t Test that can be applied to two samples with unequal sizes and unequal variances determines whether the means of both samples are the same. In other words, this test evaluates within a specified degree of certainty whether the measured difference between the meaqns is real or could have occurred merely by chance.

Before we start discussing this specific test in detail, The t-test needs to be generally explained. The basic question to be answered is:

The t Test - What Is It?

The t test is a statistics test generally used to test whether means of populations are different. In the t test, a t value is calculated based upon the difference in the means and variances of the two populations. The greater the t value, the more certain it is that the means are different.

The t value can be generally described as follows:

t value = (Difference between the group means) / (Variability of the groups)

There are many variations of the t test. Each has its own specific formula for calculating a t value for the sampled data sets. All of the t value formulas can be described by the above formula.

The Higher the t Value - The More Likely the Groups Are Different

The higher the t value is, the more likely it is that the two means are different. If the two groups being compared have a high degree of variance (t value has a high denominator), it is much harder to tell them apart. On the other hand, if the two groups being compared have a low degree of variance (the t value has a low denominator), it is much easier to tell the two groups apart.

The Lower the Combined Variance, the Higher the t Value

The illustrations below should clarify how the degree of variance in the two groups determines how easy or difficult it is to state that the means of the two groups are really different. The t test quantifies this relationship and provides a way to determine whether the measured difference between two means can be considered real or not based upon the amount of variance in both groups. Here are illustrations that should clarify this relationship.

t tests in excel, t test, normal distribution, normally distributed, statistical tests in excel

We can see that pair of data sets on the right are much easier to differentiate because they have much less overlap than the pair of data sets on the right. The overlap represents the overall variability between the two data sets in each pair. The higher the total variablility within the pair of data sets, the higher will be the denominator in the t value formula. The higher the denominator, the lower the t value for the pair of data sets. The lower the t value, the less likely it is that the two data sets are separate data sets with different means.

T-Test for Two Samples Having Unequal Sizes and Variances

The t Test for comparing two samples with unequal sizes and variance is a variation of the t Test called Welch's t Test. It is not the classic Student's t Test, which does not allow for samples having unqual variances.

We are going to use this t test to determine within 95% certainty whether the means sales from Wednesdays is different than the mean sales from Monday. We have measured from the last 42 Wednesdays and the last 40 Mondays and we will apply this Excel t test to determine whether the measured difference between the means is real or not.

A Little Bit More About This t Test

The t Test in general is a special case of one-way (sometimes called “single factor”) ANOVA. This paired two-sample student’s t test is applied when there is a natural pairing of samples. It is most often used to determine whether “before” and “after” means of a sample of the same objects have changed during an experiment. One really great thing about this t test is that the paired two-sample t test does not require that the variances of both populations to be the same.

Here is the formula to calculate the t value for a two-sample t test of unequal variances if you are testing to determine whether there difference between the two samples:

t value = [ X1 - X2 ] / [ SQRT( (s1^2 / n1) + (s2^2/n2) ) ]

Degree of Freedom = df =

[ [ (s1^2 / n1) + (s2^2 / n2) ]^2 ] / [ ( [ (s1^2 / n1)^2 ] / [n1 -1 ] ) + ( [ (s2^2 / n2)^2 ] / [n2 -1 ] ) ]

X1 and X2 are the sample means. s1 and s2 are the sample standard deviations, and n1 and n2 are sample sizes.

You can see that this follows the general formula for calculating the t value in a t test, which is:

t value = (Difference between the group means) / (Variability of the groups)

The t value is a specific point on the x-axis in the t distribution (student’s t distribution). If this t value falls outside the region of required certainty, it can be stated that the two means are probably different. If this t value falls within the region of required certainty, it cannot be stated that the two means are probably different.

The required region of certainty depends upon the degree of certainty required in the test. If 95% certainty is required, then the required region of certainty consists of 95% of the area under the student’s t distribution. The outer 5% is the region of uncertainty. This is also referred to as α (alpha) or the degree of significance. If the t value is large enough to be located all the way out on the x-axis in the 5% region of uncertainty, it can be stated within 95% certainty that the two means are different.

A t test can be a one-tailed test or a two-tailed test. A one-tailed test determines whether the means are different in one specific direction. For example, a one-tailed test could be used to determine only if the mean of the “after” measurements is greater than the mean of the “before” measurements. A two-tailed test determines whether the two means are merely different.

Two-Tailed t Test Is More Stringent

The two-tailed test is more stringent because the area in the outer tails outside of the region of required degree of certainty is split into two tails. For example, if the required degree of certainty is 95% on a two-tailed test, the calculated t value must be all the way out in the outer 2.5% of either tail for the t test to conclude within 95% certainty that the means are different.

One-Tailed t Test Is Less Stringent

A one-tailed test is less stringent. If the required degree of certainty is 95% on a one-tailed test, the calculated t value only has to be within the outer 5% of whatever tail is being tested to be able to state the two means are probably different.

Doing The Two-Sample t Test for Unequal Variances in Excel

We are testing to determine whether there really is a difference between mean sales on Monday and mean sales on Wednesday.

The data need to be arranged in Excel as follows:

Click on Image To See Enlarged View

The t Test we are about to use allows for different sample sizes and different variances, but that standard requirement for all t Tests is that both samples being compared are Normally distributed. There are a number of different ways of doing this. For brevity, we are going to do it the simplest possible way. We will make an Excel histogram of each sample's data and simple eyeball the shape of the histogram. If the shape of each histogram resembles the Normal curve, we will go with it.

There are a number of better ways of checking for Normality and here is a link to an article in this blog which describes how to do a simple but more accurate Excel Normality test called the Normal Probability Plot.

The Excel histogram is a simple thing to construct. If you haven't ever done one, here is a link to an article in this blog which shows how to create a histogram in Excel from sample data.
Completed histograms for each of the two samples are as follows:

histogram, t tests in excel, t test, normal distribution, normally distributed, statistical tests in excel

Both histograms appear to be Normally distributed so we can use t Test to compare the two samples. If either sample is not Normally distributed, the t test cannot be used because the output is likely to be totally incorrect. If either sample is not Normally distributed, we must use a nonparametric test such as the Mann-Whitney U Test to compare the samples.

Before we run the t Test, we would like to take a look at a description of of each sample. In Exzcel 2003, this can be quickly done by the following tool: Tools / Data Analysis / Descriptive Statistics. The Descriptive Statistics for each sample are as follows:

Now, access this Excel t Test as follows (this is Excel 2003):

Tools / Data Analysis / t-Test: Two Sample Assuming Unequal Variances

This following dialogue box will appear:

Click on Image To See Enlarged View

Input the data as followings:

Variable 1 Range: Select everything that is highlighted yellow, including the label “Monday Sales.”

Variable 2 Range: Select everything that is highlighted tan, including the label “Wednesday Sales.”

Hypothesized Mean Difference: 0

Labels: Check the box because you included the labels for Variables 1 and 2.

Alpha: This depends on your desired degree of certainty. 0.05, if you desired 95% certainty. 0.20 if you desire 80% certainty.

Output Range: Select the cell that you want the upper left corner of the output to appear in.

Hit “OK” to run the analysis and the following Excel output appears:

Click on Image To See Enlarged View

This output can be interpreted as follows:

The t value is -6.088.

α = 0.05 = 1 - Required Degree of Certainty = 1 - 95%

p Value (1-Tailed) = 1.88E-08

p Value (2-Tailed) = 3.77E-08

One-tailed Test

This t value has a greater absolute value (6.088) than the critical t value for a one-tailed test (1.664). We can therefore state with 95% certainty that there really is a difference between Wednesday sales and Monday sales.

The above conclusion can also be reached because the p Value for the one-tailed test (highlighted in light red on the Excel output) is 1.88E-08. This is much less than alpha (0.05). The p Value being less than alpha is an equivalent result to the t value being greater than the t critical value.

Two-Tailed Test

The same result is arrived at for the two-tailed test. The two-tailed test is more stringent because the alpha region of uncertainty (5% of the area under the student’s t distribution curve) is now divided between both outer tails. The t value needs to be larger for the two-tailed test to wind up in the outer 2.5% area of either outer tail.

In this case, the t value was large enough to be positioned in the outer 2.5% of either outer tail. The absolute value of t value (6.088) is much larger than the critical t value for the two-tailed test (1.990). This indicates that it can be stated with 95% certainty that there really is a difference between Wednesday sales and Monday sales in this case as well.The p value calculated for the two-tailed test (3.77E-08) is much smaller than alpha (0.05). This is an equivalent result to the above.

Hand Calculation of the t Value and p Value

Let’s calculate the t value and p values for the one and two-tailed tests by hand to make sure that Excel has done a correct job. The t value is stated as the t statistic.

Here is the original test data Excel Descriptive Statistics:

Click on Images To See Enlarged View

Here is the hand calculation of the t value and p values for the one and two-tailed tests for this Two-Sample t Test Assuming Unequal Variance. The hand calculations below of the t Value and p Values agree with the Excel outputs. There are very slight differences due to rounding differences:

t value = [ X1 - X2 ] / [ SQRT( (s1^2 / n1) + (s2^2/n2) ) ]

Degree of Freedom = df =

[ [ (s1^2 / n1) + (s2^2 / n2) ]^2 ] / [ ( [ (s1^2 / n1)^2 ] / [n1 -1 ] ) + ( [ (s2^2 / n2)^2 ] / [n2 -1 ] ) ]

The Degrees of Freedom calucation must be rounded to the nearest whole number, which in this case is 80.

X1 and X2 are the sample means. s1 and s2 are the sample standard deviations, and n1 and n2 are sample sizes.

p Value = TDIST ( T Statistic, df, Number of Tails )

Here are the actual calculations done by hand in Excel:

Click on Image To See Enlarged View

The Two Sample t Test Assuming Unequal Variances.is a very simple test to run in Excel and can be applied to nearly any aspect of your marketing program to see if one group of samples is different from another group of samples. One note: both sample groups must be continuous and measured using the using the same scale.

Please post any comments you have on this article. Your opinion is highly valued!

If You Like This, Then Share It...

Excel Master Series Blog Directory

Statistical Topics and Articles In Each Topic

Wednesday, July 28, 2010

How To Solve ALL Hypothesis Tests in Only 4 Steps

Every Hypothesis Test

Done in Excel in 4 Steps

As an Internet marketing manager I use hypothesis testing all the time. There are quite a few great marketing uses of the hypothesis test with Excel that I will explain in detail in future articles of this blog. If you would like to see one very useful application of the hypothesis test in an article in this blog, check out this blog article on how to construct a split-tester in Excel that is better than the Google Website Optimizer. The basic test of this split-tester (and the Google Website Optimizer) is a hypothesis test.

Hypothesis Test Determines if Something Changed

In a nutshell, a hypothesis test is used to determine if something really has changed. For example, maybe you changed your Intenet marketing program slightly and you want to determine within 95% certainty whether the sales results that you've noticed are caused by your changes or are they just the result of random chance. The hypothesis test is the perfect tool to quickly answer that question. I will go so far as to say that the hypothesis test is my favorite Internet marketing statistical tool.

Hypothesis Test - Solved With 4-Step Framework

Right now I would like to present a 4-step framework that can be used to solve ALL hypothesis tests. To my knowledge, I have not seen this framework presented anywhere else, but it definitely works for every type of hypothesis test.

Hypothesis Test Must 1st Be Classified

Before you can begin the 4-step procedure, you must classify the hypothesis test you are about to perform. There are 4 separate categories in which the hypothesis test must be classified before applying the 4-step method. Each classification must be solved a slightly different way while applying the 4-step method. You therefore must determine upfront the type of hypothesis test so you will know exactly how to apply the 4-step method. The 4 categories of hypothesis tests are as follows:

Problem Classification:

Select the proper choice of each of the four ways that a Hypothesis problem is classified as follows:

1) Mean Testing vs. Proportion Testing

• Proportion test samples have only two possible outcomes.
• Mean test samples have multiple possible outcomes.

2) One-Tailed vs. Two-Tailed Testing

    • Two-tailed tests determine whether two means are merely different.
    • One-tailed tests determine whether one mean is different in one
         direction.

3) One-Sample vs. Two Sample Testing

    • One sample is taken if original or "Before" comparison data is
           available.
    • Two samples are taken if no comparison data is available.

4) Unpaired Data Testing vs. Paired Data Testing

    • Paired data testing can be performed if "Before" and "After" data
         are collected from the same objects. Mean testing can be
         performed on paired data - Proportion testing cannot.
    • Unpaired data testing is performed on data collected in groups.

Here below is a more detailed explanation of the above classifications:

1) Mean testing vs. Proportion testing -

This is the most important distinction that must be made. Mean testing and proportion are both solved using the same 4-step method but use different formulas.

Mean testing – Hypothesis tests of mean use samples that can taken a range of values. For example, you are testing to determine if sales have gone up over the course of a month. The sampled daily sales can have a wide range of values.

Proportion testing – Hypothesis test of proportion use samples that can have only 2 values. For example, you are testing to determine whether new keywords in a Google AdWords ad group have increased conversion rate. You are sampling whether or not a click converted. Your sample has only 2 possible values. The click either converted or it didn’t.

2) One-tailed vs. Two-tailed testing

– This depends upon whether you are using the hypothesis test to determine whether the mean or proportion of one sampled group is merely different that the mean or proportion of another sampled group, or whether it is specifically different in one direction – whether it is larger or smaller.

One-tailed test – You are testing to determine if the mean or proportion of one sampled group is different in one specific direction than the mean or proportion of the other sampled group.

Two-tailed test – You only want to know if the mean or proportion of one group is different than that of the other group, but aren’t testing for direction.

3) One-sample vs. two-sample testing

– Whether you need to take one sample or two samples depends on whether you need have original or “before” sample data available. Two-sample testing is performed if no “before” data is available, or if no data is available on either side.

Paired data testing – An example of this would be “before” and “after” testing of the same object. For example, you are measuring whether sales really went up. Paired data testing can only be performed for a hypothesis test of mean, not proportion.

Unpaired data samples – Groups of unpaired data testing are treated independently of each other.

4) Unpaired data testing vs. paired data testing

– Most hypothesis tests use unpaired data. Whether data is paired or unpaired depends on whether both samples were collected from the same objects or not.

The 4-Step Method To Solve ALL Hypothesis Tests

After having classified the hypothesis test according to the 4 categories, you are now ready to perform the 4-step method. In summary, the steps are as follows:

1) Create the Null and Alternate Hypotheses

2) Map the Normal Curve

- Showing the Distribution of the Variable Used by the Null Hypothesis.

3) Map the Region of Certainty

– The Area Under the Normal Curve That Corresponds With the Degree of Certainty You Require For Your Hypothesis Test.

4) Perform Either the Critical Value Test or the P Value Test

– to Determine Whether To Reject or Fail To Reject the Null Hypothesis

Without going into too much detail, we will take a brief look at solving a hypothesis test using the 4-step method.

Problem - One-Tailed, One-Sample, Unpaired Hypothesis Test of Mean

Testing whether a delivery time has gotten worse

Problem: A furniture company states that its average delivery time is 15 days with a (population) standard deviation of 4 days. A random sample of 50 deliveries showed an average delivery time of 17 days. Determine within 98% certainty (0.02 significance level) whether delivery time has increased.

SOLUTION:

We know that this is a test of mean and not proportion because each individual sample taken can have a wide range of values: Any delivery time sample measurement from 12 to 18 days is probably reasonable.

We know that this is a one-tailed test because we are trying to determine if the "After Data" mean delivery time is larger than the "Before Data" mean delivery time, not whether the mean delivery times are merely different.

We know that only one sample needs to be taken because the population data being tested is already available.

This is unpaired data because groups are sampled independently. Below is the Before and After sample data:

hypothesis test, hypothesis tests, ttest, normal distribution, normally distributed

Click On Image To See Enlarged View

Step 1 - Create the Null and Alternate Hypotheses

The Null Hypothesis normally states that both means are the same.

If the "Before Data" population mean, µ, equals the "After Data" sample mean, xavg, then

xavg = µ = 15

The Null Hypothesis states that both means are the same, which is equivalent to:

The Null Hypothesis, which states that xavg is the same as µ (which is 15), is as follows:

Null Hypothesis, H0, is that xavg = 15

*****************************************************************************

The Alternate Hypothesis states that the After Data mean is larger, which is equivalent to:

The Alternate Hypothesis, which states that xavg is larger than µ (which is 15), is as follows:

Alternate Hypothesis, H1, is that xavg is greater than 15

Step 2 - Map the Normal Curve

We now create a Normal curve showing a distribution of the same variable that is used by the Null Hypothesis, which is xavg.

The mean of this Normal curve will occur at the same value of the distributed variable as stated in the Null Hypothesis.

Since the Null Hypothesis states that xavg = 15, the Normal curve will map the distribution of the variable xavg with a mean of xavg = 15.

This Normal curve will have a standard error that is calculated as the standard error of a sample taken from a population is normally calculated, as follows:

Sample Standard Error = sxavg = σ / SQRT(n) = 4 / SQRT(50) = 0.566

Here is the Normal Curve mapped with the mean of xavg = 15

and Sample Standard Error = 0.566

Click On Image To See Enlarged View

Step 3 - Map the Region of Certainty

The problem requires a 98% Level of Certainty so the Region of Certainty will contain 98% of the area under the Normal curve.

We know that this problem uses a one-tailed test with the Region of Uncertainty entirely contained in the outer right tail. The Region of Uncertainty contains 2% of the total area under the Normal curve. The entire 98% Region of Certainty lies to the left of the 2% Region of Uncertainty, which is entirely contained in the outer right tail.

*****************************************************************************
We need to find out how far the boundary of the Region of Certainty is from the Normal curve mean. Calculating the number of standard errors from the Normal curve mean to the outer boundary of the Region of Certainty in the right tail for a one-tailed test is done in Excel as follows:

z 98%,1-tailed = NORMSINV(1 - α) = NORMSINV(0.98) = 2.05

Excel Note - NORMSINV(x) = The number of standard errors from the Normal curve mean to a point right of the Normal curve mean at which x percent of the area under the Normal curve will be to the left of that point.

Additional note - For a one-tailed test, NORMSINV(x) can be used to calculate the number of standard errors from the Normal curve mean to the boundary of the Region of Certainty whether it is in the left or the right tail.

The Region of Certainty extends to the right of the Normal curve mean of xavg = 15 by 2.05 standard errors.

One standard error = sxavg = 0.566, so:

2.05 standard errors = (2.05) * (0.566) = 1.16

The outer boundary of the Region of Certainty has the value = µ + z 98%,1-tailed * sxavg

which equals 15 + (2.05) * (0.566) = 15 + 1.16 = 16.16

The point, 16.16, is 2.05 standard errors from the Normal curve mean of xavg = 15

This point, 16.16, is the right boundary of the 98% Region of Certainty on the Normal curve.

Here is the mapping of the Region of Certainty:

Click On Image To See Enlarged View

Step 4 - Perform Critical Value and p-Value Tests

a) Critical Value Test

The Critical Value Test is the final test to determine whether to reject or not reject the Null Hypothesis. The p Value Test, described later, is an equivalent alternative to the Critical Value Test.

The Critical Value test tells whether the value of the actual variable, xavg, falls inside or outside of the Critical Value, which is the boundary between the Region of Certainty and the Region of Uncertainty.

If the actual value of the distributed variable, xavg, falls within the Region of Certainty, the Null Hypothesis is not rejected.

If the actual value of the distributed variable, xavg, falls outside of the Region of Certainty and, therefore, into the Region of Uncertainty, the Null Hypothesis is rejected and the Alternate Hypothesis is accepted.

In this case, the actual value of the variable, xavg = 17

The actual value of the variable xavg = 17 and is therefore to the right of (outside of) the outer right Critical Value (16.16), which is the boundary between the Regions of Certainty and Uncertainty in the right tail.

The actual value of the variable xavg is outside the Region of Certainty and therefore outside the Critical Value.

We therefore reject the Null Hypothesis and accept the Alternate Hypothesis which states that delivery time has increased, with a maximum possible error of 2%. This is shown in the following Excel graph:

Click On Image To See Enlarged View

b) p Value Test
The p Value Test is an equivalent alternative to the Critical Value Test and also tells whether to reject or not reject the Null Hypothesis.

The p Value equals the percentage of area under the Normal curve that is in the tail outside of the actual value of the variable xavg.

For a one-tailed test, if the p Value is larger than α, the Null Hypothesis is not rejected.

For a two-tailed test, if the p Value is larger than α/2, the Null Hypothesis is not rejected.

For a one-tailed test, the Region of Uncertainty is contained entirely in one tail. Therefore the curve area contained by the Region of Uncertainty in that tail equals α.

For a two-tailed test, the Region of Uncertainty is split between both tails. Therefore the curve area contained by the Region of Uncertainty in that tail equals α/2.

The p Value for the actual value of the distributed variable, which in this case is greater than the mean (falls to the right of the mean in the right tail), calculated in Excel is:

p Valuexavg = 1 - NORMSDIST( [ xavg - µ ] / sxavg )

Excel note - NORMSDIST(x) calculates the total area under the Normal curve to the left of the point that is x standard errors to the right of the Normal curve mean.

p Valuexavg = 1 - NORMSDIST((17 - 15 ) / 0.566)

= 1 - NORMSDIST(2/0.566)

= 0.0002

The p Value (0.0002) is less than α (0.02), so the Null Hypothesis is rejected and the Alternate Hypothesis is accepted..

For a one-tailed test---> When the p Value is less than α, the actual value of the distributed variable falls outside the Region of Certainty and the Null Hypothesis is rejected.

This is the case here as shown in the Excel graph:

Click On Image To See Enlarged View

In subsequent articles to this blog, I will show some very useful ways of using various types of hypothesis tests in Excel to improve your marketing. If you are interested in getting a deeper understanding of how to use Excel to perform hypothesis tests, Chapters 8 and 9 of the
Excel Statistical Master go into a lot of detail with many examples of doing every type of hypothesis test in Excel.

Feel free to comment on this blog article. Your opinion is very important.

Excel Master Series Blog Directory

Statistical Topics and Articles In Each Topic

Subscribe to: Posts (Atom)

Become an Excel Statistical Master

Excel Master Series - MBA-level statistics - Over 1,100+ Pages of Easy-To-Follow Instructions in Excel

It's a Full
Easy-To-Follow
MBA Course in Business Statistics

ALL IN EXCEL

&

MUCH Clearer

Than Your Text

Book

Download the
1,100+ Page Excel Statistical Master now

More Easy-To-

Follow eManuals

That You Will

Master Quickly

Step-By-Step Optimization With Excel Solver

What's In It?

For anyone who wants to be operating at a high level with the Excel Solver quickly, this is the book for you. Step-By-Step Optimization With Excel Solver is a 200+ page .pdf e-manual of simple yet thorough explanations on how to use the Excel Solver to solve today’s most widely known optimization problems. Loaded with screen shots that are coupled with easy-to-follow instructions, this book will simplify many difficult optimization problems and make you a master of the Excel Solver almost immediately.

Here are just some of the Solver optimization problems that are solved completely with simple-to-understand instructions and screen shots in this e-manual:

• The famous “Traveling Salesman” problem using Solver’s Alldifferent constraint and the Solver’s Evolutionary method to find the shortest path to reach all customers. This also provides an advanced use of the Excel INDEX function.

• The well-known “Knapsack Problem” which shows how optimize the use of limited space while satisfying numerous other criteria.

• How to perform nonlinear regression and curve-fitting on the Solver using the Solver’s GRG Nonlinear solving method.

• How to solve the “Cutting Stock Problem” faced by many manufacturing companies who are trying to determine the optimal way to cut sheets of material to minimize waste while satisfying customer orders.

• Portfolio optimization to maximize return or minimize risk.

• Venture capital investment selection using the Solver’s Binary constraint to maximize Net Present Value of selected cash flows at year 0. Clever use of the If-Then-Else statements makes this a simple problem.

• How use Solver to minimize the total cost of purchasing and shipping goods from multiple suppliers to multiple locations.

• How to optimize the selection of different production machine to minimize cost while fulfilling an order.

• How to optimally allocate a marketing budget to generate the greatest reach and frequency or number of inbound leads at the lowest cost.

Step-By-Step Optimization With Excel Solver has complete instructions and numerous tips on every aspect of operating the Excel Solver. You’ll fully understand the reports and know exactly how to tweek all of the Solver’s settings for total custom use. This e-manual also provides lots of inside advice and guidance on setting up the model in Excel so that it will be as simple and intuitive as possible to work with.

All of the optimization problems in this book are solved step-by-step using a 6-step process that works every time. In addition to detailed screen shots and easy-to-follow explanations on how to solve every optimization problem in the book, a link is provided to download an Excel workbook that has all problems completed exactly as they are in this e-manual.

Step-By-Step Optimization With Excel Solver is exactly the e-manual you need if you want to be optimizing at an advanced level with the Excel Solver quickly.

*******************

Become an Excel Statistical Master

It's a Full
Easy-To-Follow
MBA Course in Business Statistics

ALL IN EXCEL

&

MUCH Clearer

Than Your Text

Book

Download the
1,100+ Page Excel Statistical Master now

More Easy-To-

Follow eManuals

That You Will

Master Quickly

*******************

Become an Excel Statistical Master

It's a Full
Easy-To-Follow
MBA Course in Business Statistics

ALL IN EXCEL

&

MUCH Clearer

Than Your Text

Book

Download the
1,100+ Page Excel Statistical Master now

More Easy-To-

Follow eManuals

That You Will

Master Quickly

*******************

Become an Excel Statistical Master

It's a Full
Easy-To-Follow
MBA Course in Business Statistics

ALL IN EXCEL

&

MUCH Clearer

Than Your Text

Book

Download the
1,100+ Page Excel Statistical Master now

Immediate, Absolute, No-Questions-Asked, Money-Back Guarantee If Not TOTALLY, 100% Satisfied. In Other Words, If Any Excel Master Series eManual That You've Purchased Here Does Not Provide Instructions That Are CRYSTAL CLEAR and EASY TO UNDERSTAND, You Get All Of Your Money Back Immediately and Keep the eManual. Guaranteed!

Meet The Author