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Chapter 8: Problem 7

We give sample sizes for the groups in a dataset and an outline of an analysis of variance table with some information on the sums of squares. Fill in the missing parts of the table. What is the value of the F-test statistic? Three groups with \(n_{1}=5, n_{2}=5,\) and \(n_{3}=5\). ANOVA table includes:$$ \begin{array}{|l|l|c|l|l|} \hline \text { Source } & \text { df } & \text { SS } & \text { MS } & \text { F-statistic } \\ \hline \text { Groups } & & 120 & & \\ \hline \text { Error } & & 282 & & \\ \hline \text { Total } & & 402 & & \\ \hline \end{array} $$

Short Answer

Expert verified
The filled ANOVA table will be as follows, where df for Groups = 2, MS for Groups = 60, df for Error = 12, MS for Error = 23.5, and the F-statistic = 2.553.

Step by step solution

01

Calculate degrees of freedom (df)

Firstly, calculate the degrees of freedom. For the groups this will be: df for Groups = number of groups - 1 = 3 - 1 = 2. For the error, this will be: df for Error = total number of data points - number of groups = 15 (since \(n_{1} = n_{2} = n_{3} = 5\)) - 3 = 12.
02

Calculate mean square (MS)

Next, calculate the mean square. This is done by dividing the sum of squares (SS) by the degrees of freedom (df). For the groups this will be: MS for Groups = SS for Groups / df for Groups = 120 / 2 = 60. For the error, this will be: MS for Error = SS for Error / df for Error = 282 / 12 = 23.5.
03

Calculate F-statistic

Lastly, calculate the F-test statistic. The formula for the F-statistic is the ratio of the mean square for Groups to the mean square for Error. So: F-statistic = MS for Groups / MS for Error = 60 / 23.5 = 2.553.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Degrees of Freedom
In the context of Analysis of Variance (ANOVA), degrees of freedom (often abbreviated as df) are crucial in understanding how data is partitioned and how variability is measured.
Essentially, degrees of freedom refer to the number of values that are free to vary when making statistical calculations.
When calculating the degrees of freedom for the groups, it represents the number of groups minus one.
  • For example, with three groups, the degrees of freedom would be 3 minus 1, which equals 2.
Similarly, when determining the degrees of freedom for the error, it accounts for the variability within each group. To find this, subtract the number of groups from the total number of observations in the dataset.
  • In this exercise, each group has 5 data points, so with three groups, that results in 15 total data points.
  • The degrees of freedom for the error would thus be 15 minus 3, giving us 12.
Understanding and calculating degrees of freedom correctly is vital because it impacts the next stages of statistical analysis, including the sum of squares and the F-test statistic calculations.
Sum of Squares
Sum of Squares (SS) is a measure of the total variability within a dataset. In ANOVA, we separate the total sum of squares into components to understand different sources of variation.
The two main components are:
  • **Between-Groups Sum of Squares:** This reflects the variation due to differences between group means. In simple terms, it measures how much the group means differ from the overall mean.
  • **Within-Groups (or Error) Sum of Squares:** This is the variation within each group. It captures how much individual observations differ from their respective group means.
In the given problem, the sum of squares for groups is provided as 120, while the error sum of squares is 282. The total sum of squares is the sum of these two components, equating to 402.
Calculating these components allows us to further assess the statistical significance of the differences between group means, laying the groundwork for the F-test statistic.
F-test Statistic
The F-test Statistic is a critical value calculated in ANOVA used to determine if the means of several groups are equal.
It is formed by assessing the ratio of the variance between the groups to the variance within the groups.
To find the F-test statistic:
  • Firstly, calculate the Mean Square (MS) for Groups by dividing the sum of squares for groups by its respective degrees of freedom. For our example, it is 120 divided by 2, resulting in 60.
  • Next, calculate the Mean Square for Error, which involves dividing the error sum of squares by its degrees of freedom. Here, 282 divided by 12 results in 23.5.
  • The F-test statistic is then derived by dividing the MS for Groups by the MS for Error. In this problem, it is 60 divided by 23.5, giving an F-statistic of approximately 2.553.
A higher F-statistic suggests a greater degree of difference between group means compared to variations within the groups. Understandably, this plays a vital role in hypothesis testing to decide whether the differences observed are statistically significant.

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Most popular questions from this chapter

Some computer output for an analysis of variance test to compare means is given. (a) How many groups are there? (b) State the null and alternative hypotheses. (c) What is the p-value? (d) Give the conclusion of the test, using a \(5 \%\) significance level. \(\begin{array}{lrrrr}\text { Source } & \text { DF } & \text { SS } & \text { MS } & \text { F } \\ \text { Groups } & 4 & 1200.0 & 300.0 & 5.71 \\ \text { Error } & 35 & 1837.5 & 52.5 & \\ \text { Total } & 39 & 3037.5 & & \end{array}\)

Two sets of sample data, \(\mathrm{A}\) and \(\mathrm{B}\), are given. Without doing any calculations, indicate in which set of sample data, \(\mathrm{A}\) or \(\mathrm{B}\), there is likely to be stronger evidence of a difference in the two population means. Give a brief reason, comparing means and variability, for your answer. $$ \begin{array}{cc|cc} \hline {\text { Dataset A }} & {\text { Dataset B }} \\ \hline \text { Group 1 } & \text { Group 2 } & \text { Group 1 } & \text { Group 2 } \\ \hline 13 & 18 & 13 & 48 \\ 14 & 19 & 14 & 49 \\ 15 & 20 & 15 & 50 \\ 16 & 21 & 16 & 51 \\ 17 & 22 & 17 & 52 \\ \bar{x}_{1}=15 & \bar{x}_{2}=20 & \bar{x}_{1}=15 & \bar{x}_{2}=50 \end{array} $$

Color affects us in many ways. For example, Exercise C.92 on page 498 describes an experiment showing that the color red appears to enhance men's attraction to women. Previous studies have also shown that athletes competing against an opponent wearing red perform worse, and students exposed to red before a test perform worse. \(^{3}\) Another study \(^{4}\) states that "red is hypothesized to impair performance on achievement tasks, because red is associated with the danger of failure." In the study, US college students were asked to solve 15 moderately difficult, five-letter, single-solution anagrams during a 5-minute period. Information about the study was given to participants in either red, green, or black ink just before they were given the anagrams. Participants were randomly assigned to a color group and did not know the purpose of the experiment, and all those coming in contact with the participants were blind to color group. The red group contained 19 participants and they correctly solved an average of 4.4 anagrams. The 27 participants in the green group correctly solved an average of 5.7 anagrams and the 25 participants in the black group correctly solved an average of 5.9 anagrams. Work through the details below to test if performance is different based on prior exposure to different colors. (a) State the hypotheses. (b) Use the fact that sum of squares for color groups is 27.7 and the total sum of squares is 84.7 to complete an ANOVA table and find the F-statistic. (c) Use the F-distribution to find the p-value. (d) Clearly state the conclusion of the test.

A recent study \(^{2}\) examines the impact of a mother's voice on stress levels in young girls. The study included 68 girls ages 7 to 12 who reported good relationships with their mothers. Each girl gave a speech and then solved mental arithmetic problems in front of strangers. Cortisol levels in saliva were measured for all girls and were high, indicating that the girls felt a high level of stress from these activities. (Cortisol is a stress hormone and higher levels indicate greater stress.) After the stress-inducing activities, the girls were randomly divided into four equal-sized groups: one group talked to their mothers in person, one group talked to their mothers on the phone, one group sent and received text messages with their mothers, and one group had no contact with their mothers. Cortisol levels were measured before and after the interaction with mothers and the change in the cortisol level was recorded for each girl. (a) What are the two main variables in this study? Identify each as categorical or quantitative. (b) Is this an experiment or an observational study? (c) The researchers are testing to see if there is a difference in the change in cortisol level depending on the type of interaction with mom. What are the null and alternative hypotheses? Define any parameters used. (d) What are the total degrees of freedom? The \(d f\) for groups? The \(d f\) for error? (e) The results of the study show that hearing mother's voice was important in reducing stress levels. Girls who talk to their mother in person or on the phone show decreases in cortisol significantly greater, at the \(5 \%\) level, than girls who text with their mothers or have no contact with their mothers. There was not a difference between in person and on the phone and there was not a difference between texting and no contact. Was the p-value of the original ANOVA test above or below \(0.05 ?\)

Exercises 8.46 to 8.52 refer to the data with analysis shown in the following computer output: \(\begin{array}{lrrrr}\text { Level } & \text { N } & \text { Mean } & \text { StDev } & \\ \text { A } & 6 & 86.833 & 5.231 & \\ \text { B } & 6 & 76.167 & 6.555 & \\ \text { C } & 6 & 80.000 & 9.230 & \\ \text { D } & 6 & 69.333 & 6.154 & \\ \text { Source } & \text { DF } & \text { SS } & \text { MS } & \text { F } & \text { P } \\ \text { Groups } & 3 & 962.8 & 320.9 & 6.64 & 0.003 \\ \text { Error } & 20 & 967.0 & 48.3 & & \\ \text { Total } & 23 & 1929.8 & & & \end{array}\) Is there evidence for a difference in the population means of the four groups? Justify your answer using specific value(s) from the output.
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