Bootstrapping Analysis Excel

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Spreadsheets in Education (eJSiE)Volume 4|Issue 3Article 46-21-2010Bootstrapping Analysis, Inferential Statistics andEXCELJohn A. Rochowicz JrAlvernia University, icz@low this and additional works at:/ejsieRecommended CitationRochowicz, John A. Jr (2011) "Bootstrapping Analysis, Inferential Statistics and EXCEL,"Spreadsheets in Education (eJSiE): Vol. 4: Iss.3, Article ble at:/ejsie/vol4/iss3/4This Regular Article is brought to you by the Faculty of Business atePublications@bond. It has been accepted for inclusion in Spreadsheets inEducation (eJSiE) by an authorized administrator of ePublications@bond. For more information, please contactBond University's RepositoryCoordinator.

Bootstrapping Analysis, Inferential Statistics and EXCELAbstractPerforming a parametric statistical analysis requires the justification of a number of necessary assumptions. Ifassumptions are not justified research findings are inaccurate and in question. What happens whenassumptions are not or cannot be addressed? When a certain statistic has no known sampling distributionwhat can a researcher do for statistical inference? Options are available for answering these questions andconducting valid research. This paper provides various numerical approximation techniques that can be usedto analyze data and make inferences about populations from samples. The application of confidence intervalsto inferential statistics is addressed. The analysis of data that is parametric as well as nonparametric isdiscussed. Bootstrapping analysis for inferential statistics is shown with the application of the Index Functionand the use of macros and the Data Analysis Toolpak on the EXCEL spreadsheet. A variety of interestingobservations are dsBootstrapping, Resampling, Statistics, Inferences, ApproximationThis regular article is available in Spreadsheets in Education (eJSiE):/ejsie/vol4/iss3/4

Rochowicz: Bootstrapping AnalysisBootstrapping Analysis, Inferential Statistics and EXCEL

John A. Rochowicz Jr

Alvernia University

icz@

Abstract

Performing a parametric statistical analysis requires the justification of a number

of necessary assumptions. If assumptions are not justified research findings are

inaccurate and in question. What happens when assumptions are not or cannot

be addressed? When a certain statistic has no known sampling distribution what

can a researcher do for statistical inference? Options are available for answering

these questions and conducting valid research. This paper provides various

numerical approximation techniques that can be used to analyze data and make

inferences about populations from samples. The application of confidence

intervals to inferential statistics is addressed. The analysis of data that is

parametric as well as nonparametric is discussed. Bootstrapping analysis for

inferential statistics is shown with the application of the Index Function and the

use of macros and the Data Analysis Toolpak on the EXCEL spreadsheet. A

variety of interesting observations are described.

Keywords: EXCEL, Spreadsheets, Inferential Statistics, Bootstrapping,

Resampling Data, Numerical Approximations.

Published by ePublications@bond, 20111

Spreadsheets in Education (eJSiE), Vol. 4, Iss. 3 [2011], Art. 41. Introduction

EXCEL is prevalent, easy to learn and can be applied for numerous statistical

projects. EXCEL, part of the Microsoft Office software package, has uses in a

variety of educational and research settings including many for statistics and

mathematics.

From my various classroom experiences, students doing quantitative research for

undergraduate degrees in social work, psychology, and science to graduate

degrees such as the MBA and PhD program in leadership and social sciences,

apply most parametric tests of hypotheses without checking assumptions. Many

examples of this lack of thoroughness on the part of the researcher have been

observed from my personal experiences of working with faculty colleagues and

mentoring research students. When parametric research is conducted

assumptions must be met. Many students and faculty colleagues doing research

do not address the assumptions required to conduct any of the typical

parametric inferential statistical methods including t-tests, Analysis of Variances

tests of hypothesis (ANOVA’s) and regression analysis. As a result, testing

hypotheses for inferential statistics are never completed and the reported results

are open to debate.

2. Statistics: Parametric or Nonparametric

Parametric statistics are taught at all levels of post secondary education.

Undergraduate to graduate levels and for all types of majors from the social to

the physical sciences, statistics are studied and applied in all kinds of research.

The assumptions associated with parametric tests of hypothesis [4] include: 1)

Data are collected randomly and independently (tests of hypothesis for means,

Analysis of Variances (ANOVA’s) for means and regression analysis). 2) Data are

collected from normally distributed populations (tests of hypothesis for means,

ANOVA’S for means and regression analysis). 3) Variances are equal when

comparing two or more groups (tests of hypothesis for means, ANOVA’S for

means and regression analysis).

Many ways exist for checking assumptions. In order to check that sampled data

is from a normally distributed population, histograms or stem-leaf plots are

usually displayed. If they appear mound-shaped the assumption of normality is

accepted as verified. Another way to verify normality is applying a

nonparametric test of hypothesis, the Kolgomorov-Smirnov goodness of fit test

of hypothesis for fitting data to a specific distribution. In this test of hypothesis

the researcher needs to accept the null hypothesis that the data are normal. In

this way the assumption of normality is justified. Levene’s Test of Hypothesis of

the Equality of Variances determines whether variances for populations being

studied are equal. The need to accept the null hypothesis of equal variances

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Rochowicz: Bootstrapping Analysisverifies that the assumption that variances are equal. Usually these tests are used

in conjunction with conducting parametric statistical inferences. The manner in

which the research experiment is defined and how data are collected would

justify the requirements that data are collected randomly and independently.

Examples of where parametric statistics are used include:

a) The researcher is interested in studying if there is a difference in

the means for scores achieved by students taking a mathematics

course compared to students taking a social science course. This

is an example of where a parametric independent samples t-test

would be conducted

b) A researcher is looking for a linear relationship between the time

a student puts into study and the final grade for a course. In this

situation linear regression and correlation analysis would be

performed.

If the data collected are not normal or the homogeneity or equality of variance

assumption is violated, there are alternative ways to make inferences. When

assumptions are not validated, published results become invalid and

questionable. A researcher that does not justify assumptions or cannot determine

a specific sampling distribution for a statistic must apply nonparametric

distribution-free statistical techniques.

Nonparametric techniques exist where no assumptions are needed or no

sampling distributions are found. For example the Kruskal-Wallis test of

hypothesis is a nonparametric alternative to the one way ANOVA This

nonparametric technique tests a hypothesis about the nature of three or more

populations and whether they have identical distributions and if they differ with

respect to location [7]. There are many others [4], [5], and [7].

Another concern is the inability of determining the sampling distribution for

certain statistics. For example there is no sampling distribution that can be used

for population medians [5]. Computing technology and bootstrapping can be

used for inferential statistics where the sampling distribution of the statistic is

unknown. Bootstrapping is a nonparametric, numerical application that can be

applied in EXCEL.

3. Bootstrapping Analysis: Nonparametric Analysis

The analysis of data without checking assumptions can be done with

bootstrapping techniques. Bradley Efron [2] described a computer intensive

technique that resamples collected data in order to study the behavior of a

distribution of a specific statistic. As a result inferences are made on populations

that are parametric as well as nonparametric.

Published by ePublications@bond, 20113

Spreadsheets in Education (eJSiE), Vol. 4, Iss. 3 [2011], Art. 4Bootstrapping is a numerical sampling technique where the data sampled are

resampled with replacement [2]. This means that you acquire a sample. Place

sample values back and then select another sample. In this way you get sample

data from which you can generate summary statistics for each resample. Various

descriptive statistics such as mean, median, mode, variance and correlation can

be bootstrapped.

With the use of a computer the student or researcher can create many resamples.

Bootstrapping statistics allows the student or researcher to analyze any

distribution and make inferences. The sampled data becomes the population and

the resampled data are the samples. There are advantages as well as

disadvantages when bootstrapping.

3.1 Advantages and Disadvantages:

Advantages include:

1) Verifying assumptions of normality and equality of variances for the

population is unnecessary. Inferences are valid even when assumptions

are not verified.

2) There is no need to determine the underlying sampling distribution for

any population quantity.

3) Interpretations and results are based upon many observations.

Disadvantages include:

1) Powerful computers are necessary

2) Randomness must be understood

3) Computers have built-in error.

4) Large sample sizes must be generated.

4. Inferential Statistics: Confidence Intervals

In any parametric or nonparametric statistics the researcher’s goal, is to infer

something about the population. In order to make inferences about the

population, constructing confidence intervals is acceptable. A confidence interval

includes a sample statistic or point estimate plus or minus a constant error term.

These ranges of values are found by setting the significance level (usually 0.05)

for making a decision for the hypothesis about the population; determining the

correct sampling distribution; and finding whether the theorized population

value is in the confidence interval or not. The confidence interval takes on the

form point estimate ± a critical value (zα/2) times the standard error (SE) of the

statistic. For the mean, the confidence interval looks like

xbar ± z

α/2 SE(sample mean), where xbar is the sample mean.

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Rochowicz: Bootstrapping AnalysisTesting hypothesis using confidence intervals is accomplished by checking

whether the theorized population quantity is or is not in a certain confidence

interval [1] and [4]. If the theorized population value is in the interval the null

hypothesis is accepted and if not the null hypothesis is rejected. This means that

the value obtained in the sample is not significantly different from the theorized

population value if in the interval and the sample and population are

significantly different when the population value is not in the interval. Suppose

a researcher wants to find a 95% confidence interval for the mean. The error term

is comprised of the standard error of the mean, that is the standard deviation

over the square root of the sample size and a critical z for large samples (where

population standard deviation is known) and t for small (where population

standard deviation is unknown) samples. These z or t values are critical values

and are based of the significance level set by a researcher. Alpha is the type I

error or the probability of rejecting a true null hypothesis [4]. For a 95%

confidence interval alpha is 1-.95 or 0.05. The values for zα/2 or tα/2 are found from

z or t tables or in EXCEL with the application of the function “=TINV(

probability, degrees of freedom)”.

Consider finding a 95% confidence interval for the mean. Calculating such an

interval indicates that upon repeated sampling 95% of the samples will contain

the population mean. Suppose a researcher wishes to test the hypothesis that the

population mean is 80 from a set of grades for 20 students in a certain statistics

course. The data were 81 70 79 86 89 65 76 69 71 88 78 79 81 79 73 84 73 81 89 88.

The sample mean was 79.02 and the standard deviation was 7.26. In order to test

the hypothesis that sample mean is not significantly different from the theorized

mean of 80, a 95% confidence interval is constructed. The researcher never knows

what the population mean is, only an approximation. The researcher is 95%

confident it’s between the numbers that define the interval upon repeated

sampling. That is the interval is in agreement with the sample the researcher

obtained. If the population mean is outside that interval, then the sample mean is

significantly different from the population mean.

The results of the calculations of the confidence interval for the mean in this

example are as follows: The sample mean is 79.02. The confidence interval is

comprised of the sample mean plus or minus tα/2 times the standard error of the

mean. In this case, the t value based on an alpha of 0.05 and 20-1 degrees of

freedom is 2.09 from a statistical table [4] and the standard error of the mean or

the standard deviation divided by the square root of the sample size is 1.62.

Combining these numbers provides the 95% confidence interval for the mean as

79.02 ± (2.09)(1.62). The confidence interval is 75.62 < μ < 82.42, where μ is the

population mean.

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Spreadsheets in Education (eJSiE), Vol. 4, Iss. 3 [2011], Art. 4At the 5% or 0.05 significance level, the theorized mean of 80 is contained in the

interval found and so the null hypothesis is not rejected. There is no significant

difference between the sample mean of 79.02 and the population mean of 80.

Example 1 shows how to use EXCEL to test this hypothesis about the mean.

If there is no known sampling distribution for a particular theorized population

quantity such as median or log means [6] an alternative nonparametric

confidence interval is found by using the resampled data and applying the 2.5

percentile and 97.5 percentile for the generated distribution In this way a 95%

confidence interval is obtained from the lower 2.5 percentile and upper 97.5

percentile. These bootstrapped confidence intervals are used as any other

confidence intervals to make inferences about the population [6].

The examples that follow show the application of EXCEL to inferential statistics

and bootstrapping analysis. Classical confidence intervals and bootstrapped

percentile confidence intervals are presented. Similar conclusions and results are

reached. The determination of bootstrapped percentile confidence intervals is

necessary for distributions of medians since there is no known sampling

distribution .

5. EXCEL Applications

EXCEL is useful for doing parametric as well as nonparametric statistics. The

simulation of normal data, the determination of t-values, means and medians for

sets of data are described in the following examples. Also percentiles for sets of

data can be found in EXCEL.

5.1 EXCEL: Classical Example

Consider the dataset of 20 statistics grades analyzed above. The classical way to

make an inference concerning the mean is to: a) Identify the null and alternative

hypothesis. b) Construct a confidence interval and c) Decide to reject or fail to

reject the null hypothesis. The assumption that data are normal can be checked

by checking a histogram. Using the fact that a histogram appears normal and

data were generated using the function

“=(NORMSINV(RAND())*standard

deviation)+mean”, the t-test of hypothesis for means can be applied. The rejection

or failure to reject a null hypothesis is accomplished by using confidence

intervals [4].

Figure 1 displays the classical method, the traditional textbook method for

determining confidence intervals.

A class of 20 students took a statistics test and the class mean was 79.02 with a

standard deviation of 7.26. A researcher wants to test whether the sample mean

is significantly different from a theorized population mean of 80, an average

grade necessary for meeting a statewide assessment mandate. In figure 1 the

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Rochowicz: Bootstrapping Analysisresults for a 95% confidence interval for the mean are shown. The confidence

interval does contain the population mean of 80 and so the hypothesis of no

difference between the population mean and the sample men is not rejected.

Findings indicate there is no significant difference. And the tα/2 is found using

“=tinv(probability, degrees of freedom)”. Notice the value 2.09 agrees with the

value from a textbook [4].

Figure 1: Classical t-test of Hypothesis with Confidence Intervals

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Figure 2: The Data are Normal.

The histogram appears mound shaped and the application of the t-distribution is

appropriate. For any dataset the distribution would be normal since the function

“=(NORMSINV(RAND())*standard deviation)+mean” was applied.

5.2 EXCEL EXAMPLE 2: Large Sample Statistics

The data for this example are based on IQ scores with a mean of 100 and a

standard deviation of 15. A distribution of these IQ scores of 40 samples of size

30 is shown in figure 3. The distribution of means of the 40 samples with a

histogram and frequency distribution are displayed in figure 4. The distribution

of means appears to be mound shaped. For parametric statistics the shape of the

sampling distribution is supposed to be approximately mound-shaped or

normal. The data were simulated using the function

“=NORMINV(RAND())*$C$2)+$C$1” where cell C2 contains the standard

deviation and cell C1 contains the mean. This function is copy and pasted into as

many cells as desired. In figure 5, the 95% confidence intervals for each sample

and the distribution of means are illustrated. As expected about 95% of the

classical confidence intervals for the mean do contain 100.

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Rochowicz: Bootstrapping Analysis

Figure 3: Section of 40 Samples

Figure 4: The Distribution of Means Appears Mound-shaped.

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Figure 5: Classical Confidence Intervals for the Sample Means

Tests of hypothesis using confidence intervals can be conducted using EXCEL. If

a theorized value is in a certain confidence interval accept the null hypothesis of

no difference and if the theorized value is outside the interval reject the null

hypothesis. The findings suggest that the population quantity is significantly

different from the sample quantity acquired for a specific experiment. And the

difference has not occurred by chance.

A hypothesis test can de performed on means by applying a 95% confidence

interval for the mean using the classical t-distribution and the percentile method.

In the case where no sampling distribution of a statistics is known constructing a

95% confidence interval using percentiles is applied. See examples 7 and 8 for

percentile confidence intervals.

5.3 EXCEL Example 3: Bootstrapping a small sample

EXCEL does not have any built-in commands or programs to perform

bootstrapping. But there are ways to do bootstrapping in EXCEL without the

purchase and learning of other software such as SPSS. They include:

1) Applying the INDEX Function

2) The application of Data Analysis Toolpak and Macros. A detailed

description of using the Data Analysis Toolpak and Macros for

bootstrapping data is supplied in the Appendix

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Rochowicz: Bootstrapping AnalysisApplying the EXCEL INDEX Function is a way to conduct bootstrapping

analysis without using an add-in or any macros. If you wish to do a large

number of resamples, use the INDEX command and the random number

generator. The primary use of the Index Function is to return a value from a table

or range of data. The structure of the Index Function is: INDEX(table or a range

or an array, row location, column location). The use of rand()+1 will generate a

random row or column location. The syntax for the command that generates

random rows and columns of data from a sample is the following:

“=INDEX(range of cells, ROWS(range of cells)*RAND()+1,COLUMNS(range of

cells)*RAND()+1)”. Next copy and paste this command for as many resamples as

desired. Pressing F9 key on computer keyboard recalculates data.

With the INDEX function, bootstrapping can be performed on any statistic

including means, medians, modes, analyses of variance and regression analysis

such as correlation and beta weights.

Consider the following set of data: 1 5 8 9 12 15 18. Using the function

“=INDEX(($c$4:$c$10),ROWS($c$4:$c$10*RAND()+1,COLUMNS($c$4:$c$10*RAND()+1)” will generate random resamples. As expected some of the values are

shown and others are not. Also some of the numbers occur more than once. The

results are 18 9 5 8 1 8 8. Pressing F9 will show many resamples. Figure 6 shows

one resample using “INDEX”

Figure 6: Resampling One Sample

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5.4 EXCEL Example 4: Bootstrapping Analysis

Resample the data for 30 IQ scores from example 2 using the INDEX function.

Forty resamples were obtained and displayed in figure 7. Notice that the

distribution of means on the resampled data appears normal (figure 8). Since the

data appears normal, the t-test of hypothesis can be applied and the classical

confidence intervals can be determined.

Suppose you with to test whether the mean is not equal to 100, the mean for IQ

scores for the population of all persons that take the IQ test.

With the use of confidence intervals a student or researcher can test this

hypothesis.

Figure 7: Section of Resampled Means

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Rochowicz: Bootstrapping Analysis

Figure 8: The Distribution of Resampled Means Appears Mound-Shaped

Figure 9: Confidence Intervals for Resampled Means

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In figure 9 the classical confidence intervals for resampled means are presented.

Using the distribution of means and the percentile functions “=PERCENTILE

(range of cells, .025)” and “=PERCENTILE (range of cells, .975)” a 95% percentile

confidence interval for the mean is found.

For figure 8 apply “=PERCENTILE (C3:C42, .025)” and “=PERCENTILE (C3:C42,

.975)” to obtain the percentile confidence intervals. Using the distribution of

resampled means of figure 8, the percentile confidence interval is 92.403 to

103.096. In the distribution of means with percentile confidence intervals, notice

that results are very similar. That is about 95% of the confidence intervals for

sample means contain the hypothesized mean of 100. And upon repeated

resampling about 95% of the resamples contain 100.

When testing whether the population mean is 100, note that since 100 is in the

interval the null hypothesis of no difference between the sample mean and the

theorized mean of 100 is not rejected.

From figure 4 the confidence interval for means based on percentiles is 94.99 to

104.34.

5.5 EXCEL Example 5: Bootstrapping with the Data Analysis Toolpak

Check the appendix for installing Data Analysis Toolpak, applying the Data

Analysis Toolpak and applying a macro in Data Analysis Toolpak. The following

set of data 1 5 8 9 12 15 18 was resampled using the Data Analysis Toolpak. The

results after one application are 18 18 12 15 9 15 8. Some data are the same and

some are missing. Bootstrapping has occurred. Figure 10 shows the results of

using the Toolpak.

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Rochowicz: Bootstrapping Analysis

Figure 10: Resampling with Data Analysis Toolpak

5.6 EXCEL Example 6: Bootstrapping Analysis with Data Analysis Toolpak:

Many Resamples

Since bootstrapping analysis requires many “resampled” samples the process

can be repeated as many times as desired by changing the output range in the

Sampling Dialog Box but this tends to be tedious In order to automate this

process the development of a macro will help. The macro is in the appendix. The

macro automates the calculation of many reseamples. The example above for

bootstrapped means is analyzed by using the Toolpak. Figure 11 displays the

results. The distribution appears mound-shaped and t-distribution confidence

intervals can be constructed as in example 4.

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Figure 11: Distribution of Resampled Means with Data Analysis Toolpak

5.7 EXCEL Example 7: Inferential Statistics on Medians Using Bootstrapping

Techniques

Consider a dataset of size 30. The data are 20 25 33 42 48 51 60 72 75 74 81 87 102

105 110 123 142 151 159 200 214 234 244 300 500 602 603 604 609 651.

Resample this data set 40 times and obtain a distribution of medians. Since there

is no known sampling distribution for medians the application of bootstrapping

techniques should be used. Generating a distribution of medians is accomplished

by resampling with the INDEX function. Figure 12 displays a sample of size 30

resampled 40 times. The distribution of medians is not usually normal and since

there is no known sampling distribution for medians bootstrapping analysis is

applied. Calculating a 95% confidence interval for the distribution of medians is

shown in figure 13. This confidence interval for the population median is based

on percentiles and leads to decisions about significant differences in the 2

groups.

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Figure 12: Section of Resampled Medians

Figure 13: Distribution of Resampled Medians.

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Spreadsheets in Education (eJSiE), Vol. 4, Iss. 3 [2011], Art. 4The histogram is not normal and so percentile confidence intervals are used to

make an inference about the population median. The percentile confidence

interval for the distribution of medians in figure 13 is 76 to 200. Suppose a

researcher wants to test whether the null hypothesis for the population median is

150. Based on the percentile confidence interval the null hypothesis would not be

rejected, since 150 is in the interval 76 to 200. Therefore there is no significant

difference between 150 and the sample median of 126, the average of all

resampled medians.

5.8 EXCEL Example 8: Inferential Statistics on the Differences in Medians

Suppose the following data represent the prices on homes in 2 different locations.

The prices on homes at location A are: 99 96 93 92 87 81 82 89 77 74 76 66 71 82 69

71 80 66 42 45 33 46 32 22 25 19 15 17 14 12

The prices on homes at location B are: 300 333 321 345 333 245 324 222 321 119 117

115 111 100 96 65 62 61 69 71 45 42 55 69 78 88 67 42 66 68

Consider the difference in the median prices of the homes in the 2 locations. Is

there a significant difference in the median prices?

The first column is the sampled data for each dataset. The data is resampled 40

times for each set of data. Inferences are made on whether there is a significant

difference in the median prices at the 2 locations based on percentile confidence

intervals. That is the difference in the median prices at the 2 locations is 0.

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Rochowicz: Bootstrapping AnalysisFigure 14: Display of Two Sets of Data

Figure 15: A Distribution of Median Differences with the Percentile Confidence

Interval

The sampling distribution for the difference in medians is shown in figure 15.

The 95% percentile confidence interval for the median difference is 2.5 to 83.

Since 0 is not in the interval the null hypothesis of no differences in the median

prices for the 2 groups is rejected. That is there is a difference in medians for the

2 sets of data. Note that pressing F9 many times provides about 95% of the

intervals with 0 in them. This is in agreement with the concept of confidence

interval. The medians of all these resamples were obtained by applying the

function “=MEDIAN(range of cells)”. For example “=MEDIAN(C9:P9)”

calculates the median of the numbers in cells c9 through p9

6. Bootstrapping Analysis in EXCEL: Observations

Students as well as researchers learn in applying bootstrapping there is no need

to satisfy assumptions when conducting inferential statistics. Also bootstrapping

analysis can be performed on not only sampling distributions that are known but

also on sampling distributions that are not known. Doing research this way

enables the research community to recognize valid research findings even where

assumptions are not justified.

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Spreadsheets in Education (eJSiE), Vol. 4, Iss. 3 [2011], Art. 4Meaningful analyses can be made on populations that could not be analyzed

before bootstrapping was developed. For example inferences can be made on

population log means [5].

Results from the parametric and resampling approaches are comparable

especially when the number of resamples is very large. With EXCEL

spreadsheets, implementing a large number of calculations can be conducted.

EXCEL spreadsheets are valuable for bootstrapping analysis. Various capabilities

of EXCEL for doing bootstrapping include the application of: a) the INDEX

Function b) random numbers and c) the Data Analysis Toolpak and macros. As a

result of using EXCEL, constructing confidence intervals, determining

significance and graphing results or outcomes are easily done. There is almost no

limitation on conducting tests of hypothesis for any statistic with bootstrapping.

7. Conclusions

EXCEL is easy to learn and useful for statistical analysis. Using the INDEX

function, the Data Analysis Toolpak and automating calculations with macros

provide students and researchers with a variety of useful techniques valuable not

only for inferential statistics but for many other mathematical applications

There is more than one way to conduct valid statistics and all approaches lead to