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Chapter 10: Problem 119

Find the critical value for the hypothesis test with \(H_{a}: \sigma_{1}>\sigma_{2},\) with \(n_{1}=7, n_{2}=10,\) and \(\alpha=0.05\).

Short Answer

Expert verified
The critical value for the hypothesis test is approximately 3.68.

Step by step solution

01

Calculate degrees of freedom

The degrees of freedom for the two samples are \(df_{1}=n_{1}-1=7-1=6\) and \(df_{2}=n_{2}-1=10-1=9\). Thus, the F-distribution is determined by these two degrees of freedom.
02

Look up the critical value in the F-table

With \(\alpha=0.05\), \(df_{1}=6\) and \(df_{2}=9\), we look up the corresponding critical value in the F-table which is approximately \(3.68\). This is the cut-off point beyond which we would reject the null hypothesis that the two standard deviations are equal in favor of the alternative hypothesis that \(\sigma_{1} > \sigma_{2}\).
03

Interpret the result

The critical value \(3.68\) means that if the calculated F-statistic from the samples were to exceed this value, we would have enough evidence at the 0.05 significance level to reject the null hypothesis that \(\sigma_{1} \leq \sigma_{2}\) in favor of the alternative hypothesis that \(\sigma_{1} > \sigma_{2}\).

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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 is an important concept in hypothesis testing and plays a key role in ensuring accurate statistical results. It essentially refers to the number of values in a calculation that are free to vary. When you are performing tests involving two samples, like in our problem, understanding how to calculate and use degrees of freedom is crucial because it affects the shape of the F-distribution.
  • For each sample, the degrees of freedom is calculated as the sample size minus one.
  • In our exercise, for the first sample ( _1), with 7 observations, the degrees of freedom is 6.
  • For the second sample ( n_2), with 10 observations, the degrees of freedom is 9.
These degrees of freedom are utilized when determining the critical value, which helps ascertain whether the observed results are statistically significant when compared to the expected outcome.
F-distribution
The F-distribution is a type of probability distribution that arises frequently in statistics, particularly in the context of comparing variances. It is a continuous distribution that helps us ascertain whether two population variances are significantly different.
  • The F-distribution is skewed to the right, meaning it is not symmetrical.
  • This distribution requires two sets of degrees of freedom: one for the numerator and one for the denominator.
In the context of our problem, the numerator degrees of freedom is 6 and the denominator is 9. This shape of the distribution helps statisticians decide the critical value by observing how far the values extend into the tail of the distribution curve. A larger F-statistic indicates a higher likelihood that the observed variance difference is significant.
Critical Value
The critical value in hypothesis testing is a crucial point on the test distribution that determines the threshold for considering whether to reject the null hypothesis. It essentially serves as a benchmark.
  • If the test statistic exceeds this value, we conclude there is significant evidence to reject the null hypothesis.
  • Conversely, if the test statistic does not exceed this value, we do not reject the null hypothesis.
In the given exercise, the critical value, identified as approximately 3.68, serves as this threshold. If the calculated F-statistic for this test exceeds 3.68, it implies the variance of the first population likely exceeds that of the second at the specified significance level. This critique helps researchers make informed decisions about their hypothesis, promoting accuracy in the testing process.
Significance Level
The significance level, often denoted by \( \alpha \), is a predetermined threshold that guides the decision-making process in hypothesis testing. It represents the probability of rejecting the null hypothesis when it is actually true, commonly known as a Type I error.
  • A conventional choice for significance level is 0.05, which means there is a 5% risk of concluding a difference exists when there is no actual difference.
  • The choice of \( \alpha \) directly influences the critical value and the robustness of the test results.
In our exercise, an \( \alpha \) level of 0.05 implies that the critical value of 3.68 applies at this stringency. Using this level, researchers can confidently assess whether the observed differences are not due to random chance, thereby validating or refuting the hypothesis they have set out to test.

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Most popular questions from this chapter

Both parents and students have many concerns when considering colleges. One of the top three concerns, based on a College Partnership study, is "Choosing best major/career." Nineteen percent of the parents reported "Choosing best major/career" as a major concern whereas \(15 \%\) of students reported it as a major concern. If the study was conducted with a sample of 1750 students and their parents, test the hypothesis that "Choosing best major/career" was a bigger concern for the parents, at the 0.05 level of significance.

a. Two independent samples, each of size \(3,\) are drawn from a normally distributed population. Find the probability that one of the sample variances is at least 19 times larger than the other one. b. Two independent samples, each of size \(6,\) are drawn from a normally distributed population. Find the probability that one of the sample variances is no more than 11 times larger than the other one.

Lauren, a brunette, was tired of hearing, "Blondes have more fun." She set out to "prove" that "brunettes are more intelligent." Lauren randomly (as best she could) selected 40 blondes and 40 brunettes at her high school.$$\begin{array}{llll}\hline \text { Blondes } & n_{\beta}=40 & \bar{x}_{\beta i}=88.375 & s_{\beta i}=6.134\\\\\text { Brunettes } & n_{\mathrm{S}}=40 & \bar{x}_{A r}=87.600 & s_{B r}=6.640\\\\\hline\end{array}$$ Upon seeing the sample results, does Lauren have support for her claim that "brunettes are more intelligent than blondes"? Explain. What could Lauren say about blondes' and brunettes' intelligence?

An insurance company is concerned that garage A charges more for repair work than garage B charges. It plans to send 25 cars to each garage and obtain separate estimates for the repairs needed for each car. a. How can the company do this and obtain independent samples? Explain in detail. b. How can the company do this and obtain dependent samples? Explain in detail.

A test concerning some of the fundamental facts about acquired immunodeficiency syndrome (AIDS) was administered to two groups, one consisting of college graduates and the other consisting of high school graduates. A summary of the test results follows: College graduates: \(\quad n=75, \bar{x}=77.5, s=6.2\) High school graduates: \(n=75, \bar{x}=50.4, s=9.4\) Do these data show that the college graduates, on average, scored significantly higher on the test? Use \(\alpha=0.05\)
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