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Chapter 10: Problem 49

For an \(F\) distribution, find the following: (a) \(f_{0.25,5,10}\) (b) \(f_{0.10,24,9}\) (c) \(f_{0.05,8,15}\) (d) \(f_{0.75,5,10}\) (e) \(f_{0.90,24,9}\) (f) \(f_{0.95,8,15}\)

Short Answer

Expert verified
Use F-distribution tables or software for each critical value.

Step by step solution

01

Understanding the F-distribution

The F-distribution is a continuous probability distribution that arises frequently as the null distribution of a test statistic, especially in ANOVA. It's characterized by two degrees of freedom parameters, usually denoted as \( d_1 \) and \( d_2 \), each connected to a different dataset. The critical value \( f_{\alpha, d_1, d_2} \) indicates the value of the F-distribution at which a specified proportion \( \alpha \) of the data falls below.
02

Find critical value for (a)

To find \( f_{0.25,5,10} \), we need to determine the value of the F-distribution for an area of 0.25 in the right tail, with 5 and 10 degrees of freedom respectively. This value can be found using statistical tables of the F-distribution or a statistical software. It is essential to look for the entry corresponding to \( \alpha = 0.25 \) with \( d_1 = 5 \) and \( d_2 = 10 \).
03

Find critical value for (b)

To find \( f_{0.10,24,9} \), determine the F-distribution value with an area of 0.10 in the right tail. Utilize statistical tables or software, and look for the entry with \( \alpha = 0.10 \), \( d_1 = 24 \), and \( d_2 = 9 \).
04

Find critical value for (c)

To calculate \( f_{0.05,8,15} \), identify the critical value for an F-distribution with 0.05 in the right tail. Search tables or software for \( \alpha = 0.05 \), \( d_1 = 8 \), and \( d_2 = 15 \).
05

Find critical value for (d)

Finding \( f_{0.75,5,10} \) involves searching for an F-value where 75% of the distribution is to the right (since F-tables are for right tails). Convert this by using symmetry or software. Find the entry for \( 1-0.75 = 0.25 \) with the same degrees of freedom.
06

Find critical value for (e)

For \( f_{0.90,24,9} \), you seek the point where 90% is to the right. Use the tables for \( \alpha = 0.10 \) as found in step (c), because of symmetry properties of F-distribution.
07

Find critical value for (f)

To find \( f_{0.95,8,15} \): Locate the value in tables or use software for \( \alpha = 0.05 \), \( d_1 = 8 \), and \( d_2 = 15 \). This uses the symmetric property, converting the original 95% to its complement to find the typical value.

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Key Concepts

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

Degrees of Freedom
Degrees of freedom are essential numbers used in statistical models that quantify the number of independent values in a data set which are free to vary. In the context of the F-distribution, there are typically two degrees of freedom: \(d_1\) and \(d_2\). These represent:
  • \(d_1\): The degrees of freedom for the numerator, often the variation within groups being compared.
  • \(d_2\): The degrees of freedom for the denominator, usually representing the variation within all the groups combined.
Understanding degrees of freedom is crucial because they determine the shape of the distribution. Higher degrees of freedom result in an F-distribution that is more symmetrical and less spread out, while lower degrees result in a more skewed distribution. Degrees of freedom help us to locate the appropriate critical value for our test, which indicates whether our results are statistically significant.
Critical Value
The critical value in statistics is a threshold at which we decide whether the null hypothesis can be rejected. For an F-distribution, finding the critical value involves evaluating how a specified proportion \(\alpha\) relates to the distribution's tail.
  • A common approach is to find \(f_{\alpha, d_1, d_2}\), where \(\alpha\) is the significance level, indicating the area in the right tail of the F-distribution.
  • This value is typically located using statistical tables or tools designed to calculate F-distribution probabilities.The critical value helps define the rejection region of a hypothesis test, where if the calculated F-test statistic exceeds this critical value, we reject the null hypothesis.
Knowing how to find and interpret the critical value is key to correctly conducting statistical tests like ANOVA.It informs us whether the observed data is extreme enough to suggest a significant effect.
ANOVA
ANOVA, short for Analysis of Variance, is a statistical test used to determine if there are significant differences between the means of three or more independent groups. It helps to analyze variances within and among the groups, identifying any significant difference in sample means.
  • The basic principle of ANOVA involves comparing the variance within groups to the variance between groups using the F-test.
    • The calculated F-statistic is then compared to the critical value from the F-distribution to determine the significance of the results.
  • If the F-statistic exceeds the critical value, it suggests that at least one group mean is significantly different from the others, leading to a rejection of the null hypothesis that all group means are equal.
ANOVA is an essential tool in experimental research, helping to assess multiple groups simultaneously and offering insights into multiple factors and their interactions.
Statistical Tables
Statistical tables are crucial tools for statisticians and researchers. They provide pre-calculated values that allow users to find critical values needed for hypothesis testing.
  • For the F-distribution, tables list critical values needed to assess if findings are statistically significant.
  • The user-friendly nature of these tables enables quick determination of values without complex calculations or software.
Using statistical tables involves looking up values based on degrees of freedom and the desired significance level \(\alpha\). They are invaluable in educational settings, providing a tangible way to grasp statistical concepts essential for hypothesis testing such as ANOVA.Although modern statistical software can compute these values, understanding how to use tables solidifies one's comprehension of statistical testing and is an important skill for any statistics student.

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Most popular questions from this chapter

An electrical engineer must design a circuit to deliver the maximum amount of current to a display tube to achieve sufficient image brightness. Within her allowable design constraints, she has developed two candidate circuits and tests prototypes of each. The resulting data (in microamperes) are as follows: $$\begin{array}{l|l}\text { Circuit } 1: & 251,255,258,257,250,251,254,250,248 \\\\\hline \text { Circuit } 2: & 250.253 .249 .256 .259 .252 .260 .251\end{array}$$ (a) Use the Wilcoxon rank-sum test to test \(H_{0}: \mu_{1}=\mu_{2}\) against the alternative \(H_{1}: \mu_{1}>\mu_{2} .\) Use \(\alpha=0.025 .\) (b) Use the normal approximation for the Wilcoxon rank-sum test. Assume that \(\alpha=0.05 .\) Find the approximate \(P\) -value for this test statistic.

Two machines are used to fill plastic bottles with dishwashing detergent. The standard deviations of fill volume are known to be \(\sigma_{1}=0.10\) fluid ounces and \(\sigma_{2}=0.15\) fluid ounces for the two machines, respectively. Two random samples of \(n_{1}=12\) bottles from machine 1 and \(n_{2}=10\) bottles from machine 2 are selected, and the sample mean fill volumes are \(\bar{x}_{1}=30.87\) fluid ounces and \(\bar{x}_{2}=30.68\) fluid ounces. Assume normality. (a) Construct a \(90 \%\) two-sided confidence interval on the mean difference in fill volume. Interpret this interval. (b) Construct a \(95 \%\) two-sided confidence interval on the mean difference in fill volume. Compare and comment on the width of this interval to the width of the interval in part (a). (c) Construct a \(95 \%\) upper-confidence interval on the mean difference in fill volume. Interpret this interval. (d) Test the hypothesis that both machines fill to the same mean volume. Use \(\alpha=0.05 .\) What is the \(P\) -value? (e) If the \(\beta\) -error of the test when the true difference in fill volume is 0.2 fluid ounces should not exceed \(0.1,\) what sample sizes must be used? Use \(\alpha=0.05 .\)

Consider the hypothesis test \(H_{0}: \mu_{1}=\mu_{2}\) against \(H_{1}: \mu_{1}>\mu_{2}\) with known variances \(\sigma_{1}=10\) and \(\sigma_{2}=5\) Suppose that sample sizes \(n_{1}=10\) and \(n_{2}=15\) and that \(\bar{x}_{1}=24.5\) and \(\bar{x}_{2}=21.3 .\) Use \(\alpha=0.01\). (a) Test the hypothesis and find the \(P\) -value. (b) Explain how the test could be conducted with a confidence interval. (c) What is the power of the test in part (a) if \(\mu_{1}\) is 2 units greater than \( \mu_{2}\)? (d) Assuming equal sample sizes, what sample size should be used to obtain \(\beta=0.05\) if \(\mu_{1}\) is 2 units greater than \(\mu_{2} ?\) Assume that \(\alpha=0.05 .\)

Consider the hypothesis test \(H_{0}: \mu_{1}=\mu_{2}\) against \(H_{1}: \mu_{1}>\mu_{2} .\) Suppose that sample sizes \(n_{1}=10\) and \(n_{2}=10\) that \(\bar{x}_{1}=7.8\) and \(\bar{x}_{2}=5.6,\) and that \(s_{1}^{2}=4\) and \(s_{2}^{2}=9\). Assume that \(\sigma_{1}^{2}=\sigma_{2}^{2}\) and that the data are drawn from normal distributions. Use \(\alpha=0.05 .\) (a) Test the hypothesis and find the \(P\) -value. (b) Explain how the test could be conducted with a confidence interval. (c) What is the power of the test in part (a) if \(\mu_{1}\) is 3 units greater than \(\mu_{2}\)? (d) Assuming equal sample sizes, what sample size should be used to obtain \(\beta=0.05\) if \(\mu_{1}\) is 3 units greater than \(\mu_{2}\) ? Assume that \(\alpha=0.05\).

The overall distance traveled by a golf ball is tested by hitting the ball with Iron Byron, a mechanical golfer with a swing that is said to emulate the legendary champion, Byron Nelson. Ten randomly selected balls of two different brands are tested and the overall distance measured. The data follow: $$\begin{array}{l}\text { Brand } 1: 275,286,287,271,283,271,279,275,263,267 \\\\\text { Brand } 2: 258,244,260,265,273,281,271,270,263,268 \end{array}$$ (a) Is there evidence that overall distance is approximately normally distributed? Is an assumption of equal variances justified? (b) Test the hypothesis that both brands of ball have equal mean overall distance. Use \(\alpha=0.05 .\) What is the \(P\) -value? (c) Construct a \(95 \%\) two-sided \(\mathrm{CI}\) on the mean difference in overall distance between the two brands of golf balls. (d) What is the power of the statistical test in part (b) to detect a true difference in mean overall distance of 5 yards? (e) What sample size would be required to detect a true difference in mean overall distance of 3 yards with power of approximately \(0.75 ?\)
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