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MINITAB EXPRESS TIP SHEET FOR BUSINESS STATISTICS – CENGAGE LEARNING

INTRODUCTION

Minitab Express software provides introductory statistics in a package designed to let students focus on the concepts, not the

software. Students can quickly access the statistics they need from menus and toolbars organized to complement leading textbooks.

Minitab Express is available for both Mac and PC, allowing you to teach and share files seamlessly across platforms. Data can be

easily typed, imported, or copied into familiar worksheets and simple dialogs completed to allow creation of complementary graphs

within a single analysis.

The Minitab Express workspace includes three main components:

The Navigator shows all the analyses in your project. Select an analysis to display it in the output pane. Right-click an

analysis to copy, export, print, or delete it.

The Output pane displays the results of your most recent analysis, or the analysis that you select in the Navigator.

The Data pane displays the data for your project. You can enter and edit data in the data pane.

More detail about the statistical processes described in this card can be found in the Help menu. This tip sheet will provide tips for

analyses completed in introductory and intermediate business statistics courses.

BASIC STATISTICS

This statistics section includes information on probability distributions, summary statistics, inference, tables, and graphs.

PROBABILITY DISTRIBUTIONS

DISTRIBUTION PLOT

Use a probability distribution plot to view distribution curves that you specify. When you create a probability distribution plot, you

specify one or more distributions and parameters so that you can visualize and compare distributions. You can also choose to shade

the area under the distribution curve that represents the value that you specify.

WHERE TO FIND THIS ANALYSIS

Mac: Statistics > Probability Distributions > Distribution Plots

PC: STATISTICS > Distribution Plot

DIALOG BOX ITEMS

Choose a probability distribution plot based on what you want the plot to display. Choose Single Distribution to create a plot that

shows a single distribution with the parameters that you specify; Vary Parameters to create a plot that shows the specified

distribution with different parameter values; Two Distributions to create a plot that shows two distributions with the parameters

that you specify; or Display Probability to create a plot that shows x-values and probabilities in a shaded area.

In the Distribution Plot of a Single Distribution dialog box, specify the distribution in the drop down list box as Binomial, Chi-square,

Discrete, Exponential, F, Integer, Lognormal, Normal, Poisson, t, Uniform, or Weibull. Each distribution will have its own set of

parameters to enter.

For the Normal distribution, the default Mean: of 0 and Standard deviation: of 1 result in a display (as shown in the Sample Output)

of a Standard Normal distribution.

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SAMPLE OUTPUT

This plot shows a normal distribution that has a mean of 0 and a standard deviation of 1.

CDF/PDF

The cumulative distribution function (CDF) calculates the cumulative probability for a given x-value. Use the CDF to determine the

probability that a random observation that is taken from the population will be less than or equal to a certain value. You can also use

this information to determine the probability that an observation will be greater than a certain value, or between two values.

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The probability density function (PDF) is an equation that represents the probability distribution of a continuous random variable. The PDF curve indicates regions of higher and lower probabilities for values of the random variable. For example, for a normal distribution, the highest PDF value is at the mean, while lower PDF values are in the tails of the distribution.

For a discrete distribution, such as a binomial distribution, you can use the PDF to determine the probability of exact data values.

WHERE TO FIND THIS ANALYSIS

Mac: Statistics > Probability Distributions > Cumulative Distribution Function

PC: STATISTICS > CDF/PDF > Cumulative Distribution Function

Mac: Statistics > Probability Distributions > Probability Density Function

PC: STATISTICS > CDF/PDF > Probability Density Function

DIALOG BOX ITEMS

CUMULATIVE PROBABILITY DISTRIBUTION

Minitab calculates the cumulative probabilities for specific data values from the distribution that you specify. For example, consider a Grear Tire Company problem where tire mileage was described by a normal distribution with μ = 36500 and σ = 5000. What is the probability that the tire mileage will exceed 40000 miles?

In the cumulative distribution function dialog box, select the Form of input as A single value to calculate the cumulative probability of a single x-value or A column of values to calculate cumulative probabilities for multiple x-values. The x-values must be in a column in the worksheet. In the Value: textbox enter a single x-value (40000 in this example) or a column that contains multiple x-values.

In the cumulative distribution function dialog box, select the appropriate distribution (normal) and enter the parameters associated with the problem (Mean: 36500; Standard deviation: 5000).

In Output, select Store cumulative probabilities in a column or Display a table of cumulative probabilities. If you save the probabilities to a worksheet column, you can use them for further analyses more easily.

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CUMULATIVE PROBABILITY DISTRIBUTION - SAMPLE OUTPUT

Minitab shows that this probability is .7580. For the Grear Tire Company, we are interested in the probability that the tire mileage

will be greater than 40000, thus the desired probability is 1 - .7580 = .2420.

PROBABILITY DENSITY FUNCTION

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Consider the probability that a particular baseball player hits one home run during a game where the player goes up to bat 4 times.

Based on the player's past games, the coach assumes that the player has a 0.10 probability of hitting a home run in the current

game. Because the player will either hit a home run or he will not for each time at bat, the coach uses the binomial distribution.

In the Probability Density dialog box, select the Form of input as A single value to calculate the cumulative probability of a single x-value or A column of values to calculate cumulative probabilities for multiple x-values. The x-values must be in a column in the worksheet. In the Value: textbox enter a single x-value or a column that contains multiple x-values. Select the appropriate distribution (binomial) and enter the parameters associated with the problem (Number of trials: 4; Event probability: 0.1).

In Output, select Store cumulative probabilities in a column or Display a table of cumulative probabilities. If you save the probabilities to a worksheet column, you can use them for further analyses more easily.

PROBABILITY DENSITY FUNCTION - SAMPLE OUTPUT

In these results, the probability density values are given for a binomial distribution with 4 trials and event probability of 0.10. The

probability that one event occurs in four trials is 0.291600, and the probability that four events occur in four trials is 0.000100.

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SUMMARY STATISTICS

DESCRIPTIVE STATISTICS

To ensure that your results are valid, consider the following guidelines when you collect data, perform the analysis, and interpret your results: data must be continuous (weights) or discrete (number) numeric data; sample data should be collected randomly; collect a medium to large sample of data to ensure data represent the distribution of your data.

WHERE TO FIND THIS ANALYSIS

Mac: Statistics > Summary Statistics > Descriptive Statistics

PC: STATISTICS > Descriptive Statistics

DIALOG BOX ITEMS

From the drop-down list in the Data tab, select All data are in one column or Data are in more than one column. In the Variable: text box, enter the column(s) of numeric data to analyze. In Group variable: (an optional entry), enter the column that contains the group identifier to explore the differences between the groups. Minitab calculates the descriptive statistics separately for each group.

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On the Statistics tab of the descriptive statistics dialog box, select the statistics to include in your output. The screen shot below

indicates the default settings. Click on a checkbox to select/deselect the desired statistics.

On the Display tab of the descriptive statistics dialog box, select the graphs to include in your output. Use a histogram to assess the shape and spread of the data. Histograms are best when the sample size is greater than 20. Use an individual value plot to examine the spread of the data and to identify any potential outliers. Individual value plots are best when the sample size is less than 50. Use a boxplot to examine the spread of the data and to identify any potential outliers. Boxplots are best when the sample size is greater than 20.

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SAMPLE OUTPUT

TALLY

Use this analysis to list the unique values in a column and to count the number of times each value occurs. You can display counts,

cumulative counts, percentages, and cumulative percentages.

WHERE TO FIND THIS ANALYSIS

Mac: Statistics > Summary Statistics > Tally

PC: STATISTICS > Tally

DIALOG BOX ITEMS

In the Tally dialog box, specify the column of data that you want to analyze. In the Variables: textbox, enter a column of data to analyze. Minitab displays the count of each unique value in the column of data that you enter. Click the appropriate checkboxes, to select the Statistics: to include in your output.

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SAMPLE OUTPUT

NORMALITY TEST

Use this analysis to determine whether data do not follow a normal distribution. For example, a food scientist for a company that

manufactures processed food wants to assess the percentage of fat in the company's bottled sauce to ensure the percentage is not

different from the advertised value of 15%. The scientist wants to determine whether the data follow a normal distribution before

performing other analyses with this data.

DIALOG BOX ITEMS

In the Normality Test dialog box, specify the column of data to analyze. In the Variables: textbox, enter the numeric column that you want to test.

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SAMPLE OUTPUT

A probability plot creates an estimated cumulative distribution function (CDF) from your sample by plotting the value of each observation against the observation's estimated cumulative probability. Use a probability plot to visualize how well your data fit the normal distribution. To visualize the fit of the normal distribution, examine the probability plot and assess how closely the data points follow the fitted distribution line. If your data are perfectly normal, the data points on the probability plot form a straight line. Skewed data form a curved line.

The Anderson-Darling goodness-of-fit statistic (AD-Value) measures the area between the fitted line (based on the normal distribution) and the empirical distribution function (which is based on the data points). The Anderson-Darling statistic is a squared distance that is weighted more heavily in the tails of the distribution.

Minitab uses the Anderson-Darling statistic to calculate the p-value. The p-value is a probability that measures the evidence against the null hypothesis. Smaller p-values provide stronger evidence against the null hypothesis. Larger values for the Anderson-Darling statistic indicate that the data do not follow the normal distribution.

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INFERENCE

1-SAMPLE T

Use a 1-sample t-test to estimate the mean of a population and to compare it to a target value or a reference value when you do not

know the standard deviation of the population. For example, the ratings that 60 business travelers gave for Heathrow Airport are

entered in column C1 of a Minitab worksheet. The level of significance for the test is α = .05, and the population standard deviation

will be estimated by the sample.

WHERE TO FIND THIS ANALYSIS

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Mac: Statistics > 1-Sample Inference > t

PC: STATISTICS > One Sample > t

DIALOG BOX ITEMS

In the Data tab of the 1-Sample t dialog box, select the option that best describes your data as Sample data in a column or

Summarized data. In the Sample: textbox, enter the column of numeric data that you want to analyze. To perform a hypothesis test

select the Perform hypothesis test checkbox. Enter a value in the hypothesized mean as a target or reference value for the

population (usually given).

On the Options tab of the 1-sample t dialog box, select the alternative hypothesis or specify the confidence level for the confidence

interval. From the drop-down list box, choose Mean < hypothesized value to determine whether the population mean is less than

the hypothesized mean, and to get an upper bound. Choose Mean ≠ hypothesized value to determine whether the population mean

differs from the hypothesized mean, and to get a two-sided confidence interval. Choose Mean > hypothesized value to determine

whether the population mean is greater than the hypothesized mean, and to get a lower bound. In the Confidence level: drop-down

list, choose 90, 95, or 99 as the level of confidence for the interval.

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On the Display tab of the 1-Sample t dialog box, select the graphs to include in your output. Use a histogram to assess the shape and

spread of the data. Histograms are best when the sample size is greater than 20. Use an individual value plot to examine the spread

of the data and to identify any potential outliers. Individual value plots are best when the sample size is less than 50. Use a boxplot

to examine the spread of the data and to identify any potential outliers. Boxplots are best when the sample size is greater than 20.

SAMPLE OUTPUT

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In these results, the p-value (.0353) is less than the significance level (.05) indicating that the Null hypothesis is rejected in favor of

the alternative hypothesis ( μ > 7) indicating that the ratings that 60 business travelers gave for Heathrow Airport are greater than 7.

1-SAMPLE Z

Use a 1-sample Z-test to estimate the mean of a population and to compare it to a target value or a reference value when you know

the standard deviation of the population. For example, the data in column C1 of a Minitab worksheet represent the flight distance in

yards of the MaxFlight golf ball. The population standard deviation σ = 12 is assumed known and the level of significance is α = .05.

WHERE TO FIND THIS ANALYSIS

Mac: Statistics > 1-Sample Inference > Z

PC: STATISTICS > One Sample > Z

DIALOG BOX ITEMS

In the Data tab of the 1-Sample Z dialog box, select the option that best describes your data as Sample data in a column or

Summarized data. In the Sample: textbox, enter the column of numeric data that you want to analyze. To perform a hypothesis test

select the Perform hypothesis test checkbox. Enter a value in the hypothesized mean as a target or reference value for the

population (usually given).

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On the Options tab of the 1-sample Z dialog box, select the alternative hypothesis or specify the confidence level for the confidence

interval. From the drop-down list box, choose Mean < hypothesized value to determine whether the population mean is less than

the hypothesized mean, and to get an upper bound. Choose Mean ≠ hypothesized value to determine whether the population mean

differs from the hypothesized mean, and to get a two-sided confidence interval. Choose Mean > hypothesized value to determine

whether the population mean is greater than the hypothesized mean, and to get a lower bound. In the Confidence level: drop-down

list, choose 90, 95, or 99 as the level of confidence for the interval.

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On the Display tab of the 1-Sample Z dialog box, select the graphs to include in your output. Use a histogram to assess the shape and

spread of the data. Histograms are best when the sample size is greater than 20. Use an individual value plot to examine the spread

of the data and to identify any potential outliers. Individual value plots are best when the sample size is less than 50. Use a boxplot

to examine the spread of the data and to identify any potential outliers. Boxplots are best when the sample size is greater than 20.

SAMPLE OUTPUT

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In these results, the p-value (.1255) is not less than the significance level (.05) indicating that the Null hypothesis is not rejected in

favor of the alternative hypothesis ( μ ≠295) indicating that the population average driving distance of the MaxFlight golf ball may be

295, that is, there is insufficient evidence to conclude that the population average driving distance is not 295. An examination of the

confidence interval [294.274, 300.926] confirms this conclusion since the hypothesized value (295) is in the confidence interval.

1 PROPORTION

Use a 1 proportion test to estimate a binomial population proportion and to compare the proportion to a target value or a reference value. For example, consider the proportion of female golfers at Pine Creek golf course. Pine Creek golf course is interested in increasing the proportion of women players and has implemented a special promotion designed to attract women golfers. One month after the promotion was implemented; the course manager would like to determine if the proportion of women players has increased.

WHERE TO FIND THIS ANALYSIS

Mac: Statistics > 1-Sample Inference > Proportion

PC: STATISTICS > One Sample > Proportion

DIALOG BOX ITEMS

In the Data tab of the 1-Sample Proportion dialog box, select the option that best describes your data as Sample data in a column or

Summarized data. In the Sample: textbox, enter the column of numeric data that you want to analyze. In the Event: drop-down list,

select the outcome that Minitab uses as the event (also called success). To perform a hypothesis test select the Perform hypothesis

test checkbox. Enter a value in the hypothesized proportion: textbox as a target or reference value for the population (usually given).

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On the Options tab of the 1 proportion dialog box, specify the confidence level for the confidence interval, define the alternative

hypothesis, or specify the method for the test and confidence interval. From the drop-down list box, choose Population <

hypothesized value to determine whether the population proportion is less than the hypothesized proportion, and to get an upper

bound. Choose Proportion ≠ hypothesized value to determine whether the population proportion differs from the hypothesized

proportion, and to get a two-sided confidence interval. Choose Proportion > hypothesized value to determine whether the

population proportion is greater than the hypothesized proportion, and to get a lower bound. In the Confidence level: drop-down

list, choose 90, 95, or 99 as the level of confidence for the interval. In the Method: drop-down list, choose Exact or Normal

approximation. Minitab uses the exact method because it is more accurate and powerful. However, many statistics textbooks use

the normal approximation method because it is easier for students to calculate manually.

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SAMPLE OUTPUT

In these results, the p-value (.0062) is less than the significance level (.05) indicating that the Null hypothesis is rejected in favor of

the alternative hypothesis (H1: p > 0.2). The proportion of women players at Pine Creek golf course has increased.

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1 VARIANCE

Use a 1 variance test to estimate either the variance or the standard deviation of a population and to compare that value to a target value or a reference value. Use this test to determine whether the population variance or the standard deviation differs from the hypothesized value that you specify or to calculate a range of values that is likely to include the population variance or the population standard deviation. For example, consider the arrival times at bus stops for the St. Louis Metro Bus Company. The company standard specifies an arrival time variance of 4 or less when arrival times are measured in minutes.

WHERE TO FIND THIS ANALYSIS

Mac: Statistics > 1-Sample Inference > Variance

PC: STATISTICS > One Sample > Variance

DIALOG BOX ITEMS

On the Data tab of the 1-Sample Variance dialog box, specify the data and the null hypothesis for your analysis. Select the option

that best describes your data as Sample data in a column, Sample standard deviation, or Sample variance. In the Sample: textbox,

enter the column of numeric data that you want to analyze. To perform a hypothesis test select the Perform hypothesis test

checkbox. Enter a value in the hypothesized standard deviation: textbox as a target or reference value for the population (usually

given). In this example, the population variance was given as 4 so a value of 2 is entered as the hypothesized standard deviation.

On the Options tab of the 1-Sample Variance choose the Alternative hypothesis: from the drop down list box as Standard deviation ≠

hypothesized value, Standard deviation > hypothesized value, or Standard deviation < hypothesized value. In this example, the

alternative hypothesis is Standard deviation > hypothesized value. In the Confidence level: drop down list box, select the level of

confidence for the confidence interval.

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SAMPLE OUTPUT

In these results, the p-value (.2091) is not less than the significance level (.05) indicating that there is insufficient evidence to reject

the Null hypothesis. Bus arrival time may be less than or equal to 4 minutes.

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2-SAMPLE T

Use a 2-sample t-test to determine whether the population means of two independent groups differ and calculate a range of values that is likely to include the difference between the population means. For example, checking account balances for the Cherry Grove and the Beechmont branches of a local bank are found in columns C1 and C1. A hypothesis test will be conducted to determine if the account balances differ and a 95% confidence interval provided.

WHERE TO FIND THIS ANALYSIS

Mac: Statistics > 2-Sample Inference > t

PC: STATISTICS > Two Samples > t

DIALOG BOX ITEMS

On the Data tab of the 2-sample t dialog box, select the option that best describes your data from the drop down list box as Both

samples are in one column, Each sample is in its own column, or Summarized data. Choose Cherry Grove for the Sample 1: textbox

and Beechmont for the Sample 2: textbox.

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In the Options tab of the 2-Sample t, specify the alternative hypothesis from the drop down list box as Difference ≠ 0, Difference < 0,

or Difference > 0. In the Confidence level: drop down list box, select the level of confidence for the confidence interval. Select the

Assume equal variances: checkbox when the variances of the two populations are equal or when the samples are about the same

size. If the two population variances are truly equal, then the 2-sample t-test with the assumption of equal variances is slightly more

powerful than the 2-sample t-test without the assumption of equal variances. If you assume equal variances when the variances are

not equal or when the sample sizes are very different, serious error can occur.

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On the Display tab of the 2-Sample t dialog box, select the graphs to include in your output. Use an individual value plot to examine the spread of the data and to identify any potential outliers. Individual value plots are best when the sample size is less than 50. Use a boxplot to examine the spread of the data and to identify any potential outliers. Boxplots are best when the sample size is greater than 20.

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SAMPLE OUTPUT

In these results, the p-value (.0058) is less than the significance level (.05). The null hypothesis is rejected and the conclusion is that

there is a difference in mean account balances between the Cherry Grove and the Beechmont bank branches. A 95% confidence

interval indicates that the population mean difference may be between 35 and 194.9 dollars.

PAIRED T

Use a paired t-test to determine whether the mean of the differences between two paired samples differs from 0 and/or calculate a range of values that is likely to include the population mean of the differences. The paired t-test is useful for analyzing the same set of items that were measured under two different conditions, differences in measurements made on the same subject before and after a treatment, or differences between two treatments given to the same subject. For example, suppose employees at a manufacturing company can use two different methods to perform a production task. To maximize production the company wants to identify the population method with the smaller mean production time. The tentative assumption is that the two production methods have the same population mean completion time.

WHERE TO FIND THIS ANALYSIS

Mac: Statistics > 2-Sample Inference > Paired t

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PC: STATISTICS > Two Samples > Paired t

DIALOG BOX ITEMS

On the Data tab of the paired t-test dialog box, select the option that best describes your data as Each sample is in a column or

Summarized data (differences). In the Sample 1: textbox, indicate the sample associated with the population μ1. In the Sample 2:

textbox, indicate the sample associated with the population μ2.

On the Options tab of the Paired t dialog box, specify the confidence level for the confidence interval and/or define the alternative

hypothesis. From the drop down list box, choose Mean difference ≠ 0, Mean difference < 0, or Mean difference > 0. In the

Confidence level: drop down list box, select the level of confidence for the confidence interval.

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On the Display tab of the Paired t dialog box, select the graphs to include in your output. Choose Histogram of differences, Individual value plot of differences, and/or Boxplot of differences.

SAMPLE OUTPUT

In these results, the p-value (.0795) is NOT than the significance level (.05). There is insufficient evidence to reject the null

hypothesis. There may be no difference in the two mean production methods to perform a production task.

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2 PROPORTIONS

Use a 2 proportions test to determine whether the population proportions of two groups differ and to calculate a range of values

that is likely to include the difference between the population proportions. For example, a tax preparation firm is interested in

comparing the quality of work at two of its regional offices. Samples of tax returns are examined to estimate the proportion of

erroneous returns prepared at each office. The alternative hypothesis indicates that the proportion of erroneous returns in Office 1

is not the same in Office 2.

WHERE TO FIND THIS ANALYSIS

Mac: Statistics > 2-Sample Inference > Proportions

PC: STATISTICS > Two Samples > Proportions

DIALOG BOX ITEMS

On the Data tab of the 2 proportions dialog box, select the option that best describes your data as Both samples are in one column,

Each sample is in its own column, and Summarized data. In the Sample 1: textbox, enter the column of data that you want to analyze

for Office 1. In the Sample 2: textbox, enter the column of data to analyze for Office 2. In the Event: drop-down list, select the

outcome that Minitab uses as the event (also called success). Choose Yes for the event for this example.

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On the Options tab of the 2-Sample Proportions dialog box, choose the Alternative hypothesis: from the drop down list box as

Difference ≠ 0, Difference > 0, or Difference < 0. Specify the confidence level from the Confidence level: drop down list. In the drop

down list for the Test method: specify Estimate the proportions separately or Use the pooled estimate of the proportion.

SAMPLE OUTPUT

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In these results, the p-value (.0471) is less than the significance level (.10). There is sufficient evidence to reject the null hypothesis

and conclude that the proportion of erroneous returns in Office 1 and Office 2 are not the same.

2 VARIANCES

Use a 2 variances test to determine whether the variances or the standard deviations of two groups differ and to calculate a range of

values that is likely to include the population ratio of the variances or the standard deviations of the two groups. For example, Dullus

County Schools is considering the renewal of its school bus service contract. Two bus companies’ variances are compared. Low

variance values indicate more consistent and higher-quality service.

WHERE TO FIND THIS ANALYSIS

Mac: Statistics > 2-Sample Inference > Variances

PC: STATISTICS > Two Samples > Variances

DIALOG BOX ITEMS

On the Data tab of the 2-Sample Variance dialog box, select the option that best describes your data from the drop down list box as

Both samples are in one column, Each sample is in its own column, Sample standard deviations, or Sample variances. In the Sample

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1: textbox, Milbank is chosen as sample 1 since it provides the larger sample variance. In the Sample 2: textbox, Gulf Park is chosen

as sample 2.

On the Options tab of the 2-Sample Variance dialog box, specify the hypothesized ratio, alternative hypothesis, confidence level confidence level and indicate whether or not test and confidence intervals are based on normal distribution.

In the Hypothesized ratio textbox: enter a value that defines your null hypothesis. Think of this value as a target value or a reference

value. When considering differences in population variances, a value of 1 is typically used. In the Alternative hypothesis: drop down

list box, indicate the alternative hypothesis as Ratio ≠ hypothesized ratio, Ratio < hypothesized ratio, or Ratio > hypothesized ratio.

Specify the confidence level from the Confidence level: drop down list. If test or confidence intervals are based on the normal

distribution, check the indicated textbox.

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On the Display tab of the 2-Sample Variance dialog box, select the graphs to include in your output. Choose the Summary plots checkbox to display a summary of the results. Choose the histogram checkbox to assess the shape and spread of the

data. Choose the Individual value plot to examine the spread of the data and to identify any potential outliers.

SAMPLE OUTPUT

In these results, the p-value (.0811) is NOT less than the significance level (.05). There is insufficient evidence to reject the null

hypothesis. The arrival times of the two bus companies may be the same.

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TABLES

Use a chi-square goodness-of-fit test to determine whether the proportions of categories from sample data are consistent with a

hypothesized distribution.

Use a cross tabulation and chi-square analysis when you have data that are categorized by one or more categorical variables. With a cross tabulation and chi-square analysis, you can (1) determine the counts or percentages for each category, and (2) investigate the relationship between variables.

CHI-SQUARE GOODNESS OF FIT TEST

Chi-square goodness-of-fit tests use the following hypotheses:

H0: The data are consistent with the specified distribution.

H1: The data are not consistent with the specified distribution.

You can test equal proportions of observations, specified proportions of observations, or proportions based on historical data.

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Consider a market share study being conducted by Scott Marketing Research. Over the past year, market shares have stabilized at 30% for company A, 50% for company B, and 20% for company C. Company C has retained Scott Marketing Research to determine whether a new product introduced will alter or change the market shares for the 3 companies.

H0: pA = .50; pB = .30, and pC = .20 Ha: The population proportions are not pA = .50; pB = .30, and pC = .20 Let us assume that the market research firm has used a consumer panel of 200 customers and each customer was asked to specify a purchase preference. Company preference responses are summarized as follows

Observed Frequency

Company A’s Product Company B’s Product Company C’s Product

48 98 54

WHERE TO FIND THIS ANALYSIS

Mac: Statistics > Tables > Chi-Square Goodness-of-Fit

PC: STATISTICS > Chi-Square Goodness-of-Fit

DIALOG BOX ITEMS

On the Data tab of the Chi-Square Goodness-of-Fit Test dialog box, specify the data for your analysis and set up the null hypothesis.

In the drop down list box, indicate the format of the data as Categorical data in a column or Summarized data in a column. Indicate

the appropriate data location for the Observed counts and Category names. In the Test: drop down list box select Equal proportions

to determine whether the proportions in each category are the same. Select Specified proportions to enter the proportion to test for

each category of your worksheet. Select Proportions based on historical counts to enter the counts for each category that you want

to test.

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On the Display tab of the Chi-Square Goodness-of-Fit Test dialog box, select the graphs to include in your output as Bar chart of the observed and the expected counts or Bar chart of each category's contribution to the chi-square value. The bar chart of the observed and the expected counts plots each category's observed and expected values to determine whether there is a difference in a particular category. The bar chart of each category’s contribution to the chi-square value plots each category's contribution to the chi-square value to determine the category that contributes the most to the chi-square value. Use the radio buttons to indicate whether to Display bars from the largest to the smallest or Display bars in category order.

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SAMPLE OUTPUT

In these results, the Chi-Square test indicates a p-value of (.0255) is less than the significance level (.05) thus the population proportions are not as indicated in the null hypothesis. Market shares for the three companies have changed.

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CROSS TABULATION AND CHI-SQUARE

Chi-square tests of association (here known as cross tabulation and chi-square) use the following hypotheses:

H0: The variables are independent; no association between variables exists.

H1: The variables are not independent; association between variables exist and variables are dependent.

Consider the preferences of beer drinkers for light, regular, and dark beers. A sample of 200 beer drinkers is taken with each person

asked to indicate a preference for one of the three types (light, regular, dark) and their gender (male, female). The hypotheses for

this test of independence (association) are:

H0: Beer preference is independent of gender. H1: Beer preference is not independent of gender.

WHERE TO FIND THIS ANALYSIS

Mac: Statistics > Tables > Cross Tabulation and Chi-Square

PC: STATISTICS > Cross Tabulation and Chi-Square

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DIALOG BOX ITEMS

On the Data tab of the Cross Tabulation and Chi-Square dialog box, select the option that best describes your data as Raw data

(categorical variables) or Summarized data in a two-way table. For summarized, observed counts for categorical data, in Column

containing the table: enter the columns that contain the summarized counts. In the Row labels (optional column), enter the column

that contains the category identifiers.

On the Display tab of the Cross Tabulation and Chi-Square dialog box, use check boxes to select the statistics to include in your output. Choose Percent of row total to display the percentage that each cell represents within a table row. Choose Percent of column total to display the percentage that each cell represents within a table column. Choose Percent of overall total to display the percentage that each cell represents of the total observations. Choose Chi-Square test of association to perform a chi-square test of association. The null hypothesis of this test is that no association exists between variables. Choose Expected cell counts to display the expected count for each cell. Choose Raw residuals to display the difference between the observed count and the expected count for each cell. Choose Standardized residuals to display the standardized residuals for each cell. Standardized residuals are the raw residuals, divided by the square root of the expected counts. Choose Adjusted residuals to display the adjusted residuals for each cell. Adjusted residuals are the raw residuals, divided by the estimated standard deviation of the observed count. Choose Each cell's contribution to chi-square to display the contribution to the chi-square statistic for each cell. The contribution to the chi-square statistic is the standardized residual squared for each cell.

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SAMPLE OUTPUT

In these results, the Chi-Square test indicates a p-value of (.0398) which is less than the significance level (.05). The null hypothesis is rejected indicating that the variables are not independent; an association between beer preference and gender exists.

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GRAPHS

HISTOGRAM

Use a histogram to examine the shape and spread of your data. A histogram works best when the sample size is at least 20. A

histogram divides sample values into many intervals and represents the frequency of data values in each interval with a bar. After

you create a histogram, you can add a normal distribution fit line, change the scale type, and more.

WHERE TO FIND THIS ANALYSIS

GRAPHS – Histogram

Select the option that matches your data as Single Y Variable: Simple, to plot a single column of numeric data; Single Y Variable: With Groups, to plot a single column of numeric data, with groups that are defined by the values in a column of categorical data; or Multiple Y Variables: Simple to plot multiple columns of numeric data.

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DIALOG BOX ITEMS

In the Histogram of a Single Y Variable dialog box, select the Y variable: as the column in the worksheet to be plotted.

SAMPLE OUTPUT

In the sample output below, a histogram of the frequency of Audit Times is provided with a table of Summary Statistics.

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CUSTOMIZE THE OUTPUT

Click the graph to select it, then click the plus sign beside the graph and select items to display on the graph. By default, the title,

axis labels, and tick labels are shown.

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DOT PLOT

Use a dotplot to assess and compare sample data distributions. A dotplot divides sample values into small intervals and represents

each value or small group of values with a dot along a number line. A dotplot is best when the sample size is less than approximately

50. The following dotplot shows the distribution of the amount of torque that is required to remove the caps from a sample of

shampoo bottles.

WHERE TO FIND THIS ANALYSIS

GRAPHS – Dotplot

DIALOG BOX ITEMS

Select the option that matches your data as Single Y Variable: Simple, to plot a single column of numeric data; Single Y Variable: With Groups, to plot a single column of numeric data, with groups that are defined by the values in a column of categorical data; or Multiple Y Variables: Simple to plot multiple columns of numeric data.

In the Dotplot of a Single Y Variable dialog box, select the Y variable: as the column in the worksheet to be plotted.

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SAMPLE OUTPUT

In the sample output below, a dotplot of the frequency of Audit Times is provided with a table of Summary Statistics.

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CUSTOMIZE THE OUTPUT

Click the graph to select it, then click the plus sign beside the graph and select items to display on the graph. By default, the title,

axis labels, and tick labels are shown.

STEM-AND-LEAF PLOT

Use a stem-and-leaf plot to examine the shape and spread of sample data. A stem-and-leaf plot works best when the sample size is

less than approximately 50. A stem-and-leaf plot is similar to a histogram that is turned on its side. However, instead of displaying

bars, a stem-and-leaf plot displays digits from the actual data values to denote the frequency of each bin (row). The following stem-

and-leaf plot shows the distribution of the results of a 150-question aptitude test given to 50 individuals recently interviewed for a

position.

WHERE TO FIND THIS ANALYSIS

GRAPHS – Stem-and-Leaf Plot

DIALOG BOX ITEMS

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In the Variables text box of the Stem-and-Leaf plot dialog box, enter the numeric column (or double-click on the column) that you want to graph. In the Group variable, you may (optionally) enter the column of categorical data that defines the groups.

SAMPLE OUTPUT

In the sample output below, a stem-and-leaf plot of the number of aptitude questions answered correctly is provided. The graph

shows the distribution of test scores.

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SCATTERPLOT

Use a scatterplot to investigate the relationship between a pair of continuous variables. A scatterplot displays ordered pairs of X and

Y variables in a coordinate plane. After you create a scatterplot, you can add a fitted regression line and choose a linear, quadratic,

or cubic regression model. In the example below, the scatterplot investigates the relationship between the number of commercials

shown and sales at the store the following week.

WHERE TO FIND THIS ANALYSIS

GRAPHS – Scatterplot

DIALOG BOX ITEMS

Select the option that matches your data as Single Y Variable: Simple, to create a scatterplot that displays the data values for two variables at their X-Y coordinates; Single Y Variable: With Groups, to create a scatterplot that displays the data values for two variables at their X-Y coordinates, grouped by a third variable; or Multiple Y Variables: Simple to create a scatterplot that displays the data values for multiple Y variables and a single X variable.

In the Y variable: of the Scatterplot of a Single X-Y Pair dialog box, enter (or double-click to select) the numeric column that you want to explain or (Sales Volume). In the X variable:, enter the numeric column that might explain changes in the Y variable (No. of Commercials).

SAMPLE OUTPUT

In the sample output below, a scatterplot of sales volume vs number of commercials is displayed with summary statistics. The

completed scatterplot indicates a positive relationship between the number of commercials and sales.

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CUSTOMIZE THE OUTPUT

Click the graph to select it, then click the plus sign beside the graph and select items to display on the graph. By default, the title

and axis labels are shown.

BOXPLOT

Use a boxplot to assess and compare the shape, central tendency, and variability of sample distributions and to look for outliers. A

boxplot works best when the sample size is at least 20. A boxplot shows the median, interquartile range, range, and outliers for each

group. In the following example, a box plot graphical summary of data is based on a five-number summary of starting salaries for 12

business school graduates.

WHERE TO FIND THIS ANALYSIS

GRAPHS – Boxplot

DIALOG BOX ITEMS

Select the option that matches your data as Single Y Variable: Simple, to create a boxplot that displays the distribution of values for one variable; Single Y Variable: With Groups, to boxplot that displays the distribution of values for different groups when the groups are specified with a group variable; or Multiple Y Variables: Simple to create a boxplot that displays the distribution of values for

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different groups when the groups are in separate columns; or Multiple Y Variables: With Groups to create a boxplot that displays the distribution of values for two-way groups when the groups are in separate columns and there is also a grouping variable. In the Y variable: of the Boxplot of a Single Y Variable dialog box, enter (or double-click on) the column of numeric data that you want to graph in Y variable (Starting Salary).

SAMPLE OUTPUT

In the sample output below, a boxplot of starting salaries is displayed with summary statistics. The completed scatterplot indicates a

positive relationship between the number of commercials and sales.

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CUSTOMIZE THE OUTPUT

Click the graph to select it, then click the plus sign beside the graph and select items to display on the graph. By default, the title

and axis labels are shown.