Ch. 12 - Analysis of Variance

Triola - Elementary Statistics 14th Edition

Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook

Triola 14th Edition

Ch. 12 - Analysis of Variance

Problem 12.1.1a

Chapter 12, Problem 12.1.1a

In Exercises 1–4, use the following listed measured amounts of chest compression (mm) from car crash tests (from Data Set 35 “Car Data” in Appendix B). Also shown are the SPSS results from analysis of variance. Assume that we plan to use a 0.05 significance level to test the claim that the different car sizes have the same mean amount of chest compression.

Anova

a. What characteristic of the data above indicates that we should use one-way analysis of variance?

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Step 1: Identify the groups in the data. The table shows measured amounts of chest compression (mm) for four different car sizes: Small, Midsize, Large, and SUV. Each group represents a distinct category of car size.

Step 2: Recognize the dependent variable. The dependent variable in this case is the measured amount of chest compression (mm), which is a continuous numerical variable.

Step 3: Understand the purpose of one-way ANOVA. One-way analysis of variance (ANOVA) is used to compare the means of more than two groups to determine if there is a statistically significant difference among them. Here, we are testing the claim that the different car sizes have the same mean chest compression.

Step 4: Note the requirement for one-way ANOVA. One-way ANOVA is appropriate when there is one independent variable (car size) with multiple levels (Small, Midsize, Large, SUV) and one dependent variable (chest compression). This data meets the criteria because car size is the independent variable with four levels, and chest compression is the dependent variable.

Step 5: Confirm the assumption of independence. The data appears to be grouped by car size, and the measurements within each group are independent of each other, which is a key assumption for one-way ANOVA.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

One-Way Analysis of Variance (ANOVA)

One-way ANOVA is a statistical method used to compare the means of three or more independent groups to determine if at least one group mean is significantly different from the others. It assesses the impact of a single categorical independent variable on a continuous dependent variable, making it suitable for experiments with multiple groups, such as different car sizes in this case.

Assumptions of ANOVA

ANOVA relies on several key assumptions: the samples must be independent, the data should be normally distributed within each group, and the variances among the groups should be approximately equal (homogeneity of variance). Checking these assumptions is crucial before performing ANOVA to ensure valid results and interpretations.

Significance Level (α)

The significance level, often denoted as alpha (α), is the threshold for determining whether the observed results are statistically significant. In this context, a significance level of 0.05 indicates that there is a 5% risk of concluding that a difference exists when there is none. It is used to decide whether to reject the null hypothesis, which states that all group means are equal.

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04:46

Step 4: State Conclusion Example 4

Related Practice

Textbook Question

Bonferroni Test Shown below are weights (kg) of poplar trees obtained from trees planted in a rich and moist region. The trees were given different treatments identified in the table below. The data are from a study conducted by researchers at Pennsylvania State University and were provided by Minitab, Inc. Also shown are partial results from using the Bonferroni test with the sample data.

a. Use a 0.05 significance level to test the claim that the different treatments result in the same mean weight.

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Textbook Question

Interaction

a. What is an interaction between two factors?

Textbook Question

Transformations of Data Example 1 illustrated the use of two-way ANOVA to analyze the sample data in Table 12-3. How are the results affected in each of the following cases?

a. The same constant is added to each sample value.

Textbook Question

Birth Weights The table below lists some of the same data used in the preceding exercise, but the seven days of the week are combined into weekday (Monday, Tuesday, Wednesday, Thursday, Friday) and weekend days (Saturday, Sunday). Also, the birth weights are converted to kilograms. What do you conclude?

Textbook Question

Pulse Rates Shown below are pulse rates from Data Set 1 “Body Data” in Appendix B, and the StatCrunch display from two-way analysis of variance of these data. In analyzing these data, what important feature is addressed with two-way analysis of variance that is not addressed with two separate tests of (1) difference between mean pulse rates based on gender, or (2) differences among the mean pulse rates in the different age brackets?

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Textbook Question

Birth Weights Data Set 6 “Births” includes birth weights (g), hospitals, and the day of the week that mothers were admitted to the hospital. Using rows to represent the four hospitals (Albany Medical Center, Bellevue Hospital Center, Olean General Hospital, Strong Memorial Hospital), and using columns to represent the seven different days of the week, a two-way table has 28 individual cells. Using five birth weights for each of those 28 cells and using StatCrunch for two-way analysis of variance, we get the results displayed below. What do you conclude?