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Chapter 13: Problem 9

Determine the F-test statistic based on the given summary statistics. $$ \begin{array}{cccc} \text { Population } & \text { Sample Size } & \text { Sample Mean } & \text { Sample Variance } \\ \hline 1 & 10 & 40 & 48 \\ \hline 2 & 10 & 42 & 31 \\ \hline 3 & 10 & 44 & 25 \end{array} $$

Short Answer

Expert verified
The F-test statistic \( F \approx 1.548 \).

Step by step solution

01

- Identify the formula for the F-test statistic

The F-test statistic is calculated using the formula: \[ F = \frac{s_1^2}{s_2^2} \] where \( s_1^2 \) and \( s_2^2 \) are the sample variances of the populations being compared.
02

- Identify the populations for comparison

From the given data, choose two populations to compare. For this example, let's compare Population 1 and Population 2.
03

- Insert the sample variances into the formula

For Population 1, the sample variance \( s_1^2 = 48 \). For Population 2, the sample variance \( s_2^2 = 31 \). The formula for the F-test statistic becomes: \[ F = \frac{48}{31} \]
04

- Calculate the F-test statistic

Perform the division to get the F-test statistic: \[ F = \frac{48}{31} \approx 1.548 \]

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Key Concepts

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

Sample Variance
To understand the F-test statistic, it's important to know about sample variance. Sample variance measures how spread out the values in a sample are around the sample mean. It tells us how much the values differ from the average value. The formula for sample variance is: \[ s^2 = \frac{\sum (x_i - \bar{x})^2}{n-1} \] where \(x_i\) are the values in the sample, \(\bar{x}\) is the sample mean, and \(n\) is the sample size. Calculating sample variance helps in understanding the dispersion in your data.
Sample Size
Sample size is the number of observations in your sample. In the context of statistical tests, knowing the sample size is crucial since it impacts the reliability of your results. Larger sample sizes yield more reliable estimates because they better approximate the population parameters. In our example, each population has a sample size of 10. This consistency helps simplify the F-test calculation.
Statistical Comparison
Statistical comparison involves comparing datasets to determine if there are significant differences between them. In an F-test, we compare the variances of two populations to see if one is significantly larger than the other. This can help determine if two datasets come from populations with different variability. Efficient statistical comparison allows us to make informed decisions based on data.
F-distribution
The F-distribution is a probability distribution that arises frequently in the context of variance analysis. It's skewed to the right and is used in F-tests to compare variances. The shape of the F-distribution depends on two degrees of freedom parameters, \(d_1\) and \(d_2\). These parameters are determined by the sample sizes of the populations being compared. Understanding the F-distribution is key to interpreting the results of an F-test correctly.

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Most popular questions from this chapter

Given the following ANOVA output, answer the questions that follow. \(\begin{array}{lrrrrr}\text { Source } & \text { df } & \text { SS } & \text { MS } & F & P \\ \text { Factor A } & 1 & 531.2 & 531.2 & 11.73 & 0.003 \\\ \text { Factor B } & 2 & 3018.0 & 1509.0 & 33.33 & 0.000 \\ \text { Interaction } & 2 & 16.3 & 8.2 & 0.18 & 0.836 \\ \text { Error } & 18 & 814.9 & 45.3 & & \end{array}\) (a) Is there evidence of an interaction effect? Why or why not? (b) Based on the \(P\) -value, is there evidence of a difference in the means from factor A? Based on the \(P\) -value, is there evidence of a difference in the means from factor \(\mathrm{B} ?\) (c) What is the mean square error?

The comparisonwise error rate, denoted \(\alpha_{c}\), is the probability of making a Type I error when comparing two means. It is related to the familywise error rate, \(\alpha\), through the formula \(1-\alpha=\left(1-\alpha_{c}\right)^{k},\) where \(k\) is the number of means being compared. (a) If the familywise error rate is \(\alpha=0.05\) and \(k=3\) means are being compared, what is the comparisonwise error rate? (b) If the familywise error rate is \(\alpha=0.05\) and \(k=5\) means are being compared, what is the comparisonwise error rate? (c) Based on the results of parts (a) and (b), what happens to the comparisonwise error rate as the number of means compared increases?

Given the following ANOVA output, answer the questions that follow: $$ \begin{aligned} &\text { Analysis of Variance for Response }\\\ &\begin{array}{lrrrrr} \text { Source } & \text { df } & \text { SS } & \text { MS } & F & P \\ \text { Block } & 6 & 1712.37 & 285.39 & 134.20 & 0.000 \\ \text { Treatment } & 3 & 2.27 & 0.76 & 0.36 & 0.786 \\ \text { Error } & 18 & 38.28 & 2.13 & & \\ \text { Total } & 27 & 1752.91 & & & \end{array} \end{aligned} $$ (a) The researcher wants to test \(H_{0}: \mu_{1}=\mu_{2}=\mu_{3}=\mu_{4}\) against \(H_{1}:\) at least one of the means is different. Based on the ANOVA table, what should the researcher conclude? (b) What is the mean square due to error? (c) Explain why it is not necessary to use Tukey's test on these data.

The following data are taken from four different populations that are known to be normally distributed, with equal population variances based on independent simple random samples. $$ \begin{array}{cccc} \text { Sample 1 } & \text { Sample 2 } & \text { Sample 3 } & \text { Sample 4 } \\ \hline 110 & 138 & 98 & 130 \\ \hline 85 & 140 & 100 & 116 \\ \hline 83 & 130 & 94 & 157 \\ \hline 95 & 115 & 110 & 137 \\ \hline 103 & 101 & 104 & 144 \\ \hline 105 & 130 & 118 & 124 \\ \hline 107 & 123 & 102 & 139 \\ \hline \end{array} $$ (a) Test the hypothesis that each sample comes from a population with the same mean at the \(\alpha=0.05\) level of significance. That is, test \(H_{0}: \mu_{1}=\mu_{2}=\mu_{3}=\mu_{4}\). (b) If you rejected the null hypothesis in part (a), use Tukey's test to determine which pairwise means differ using a familywise error rate of \(\alpha=0.05\). (c) Draw boxplots of each set of sample data to support your results from parts (a) and (b).

The variability among the sample means is called _____ sample variability, and the variability of each sample is the _____ sample variability.
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