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Chapter 11: Problem 1

When using the \(F\) distribution to test variances from two populations, should the random variables from each population be independent or dependent?

Short Answer

Expert verified
The random variables should be independent.

Step by step solution

01

Understand the Concept of Independence

In statistics, independence refers to the idea that the outcome or result from one random variable does not affect the outcome from another random variable. When comparing populations, it's important to determine whether the samples influence each other.
02

Recall the Requirements of the F-Test

The F-test is used to compare the variances of two populations. One of the fundamental requirements for using the F distribution is that the samples drawn from each population must be independent of each other. This ensures valid comparison of the variances.
03

Conclusion on Variable Dependency

Since an F-test requires the samples to be independent, the random variables from each population must also be independent. This assumption is critical to ensure the reliability and validity of the results obtained from the F-test.

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Key Concepts

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

Understanding Independent Samples
When working with statistical tests, it is crucial to understand the concept of independent samples. These are samples that are drawn from populations in such a way that the outcome or result of one sample does not influence or affect the other sample.
Why is this important? Well, dependency between samples can skew the results of a statistical test, making it unreliable. If samples are dependent, it means they are somehow linked or related, potentially leading to conclusions that aren't accurately reflective of the populations.
In the context of the F-test, which is used for variance comparison, using independent samples is a strict requirement. This ensures that the comparison between the two population variances is unbiased and valid. Therefore, before conducting an F-test, we must ensure that the samples are drawn independently.
Exploring Variance Comparison
Variance is a measure of how much values in a data set differ from the mean value. When comparing two populations, examining their variance helps us understand if there is more variability in one population compared to the other.
The F-test serves precisely this purpose: it helps us compare and determine if the variances are statistically different. This comparison is vital as it can inform decisions or lead to further statistical analysis.
To conduct a variance comparison using an F-test, follow these steps:
  • Calculate the variances for both populations.
  • Formulate a hypothesis where you state that the variances are equal as the null hypothesis.
  • Use the F-test to test the null hypothesis. A corresponding F statistic is calculated.
  • Compare this statistic to a critical value from the F distribution table, considering the confidence level and degree of freedom.
If the F statistic is larger than the critical value, we reject the null hypothesis, indicating a significant difference in variances.
Key F-test Requirements
The F-test is a popular statistical test to compare variances, but it comes with specific requirements that must be adhered to for reliable results. First and foremost, as previously mentioned, the samples from the two populations must be independent. This is non-negotiable.
Another critical requirement is that the populations should be normally distributed. If the data does not exhibit normal distribution, the validity of the F-test results is compromised.
Moreover, the ratio of variances should be considered. The F distribution tends to assume that both variances are non-negative, and the larger variance should be in the numerator when forming the F statistic.
Here are the essential requirements for conducting an F-test:
  • Independence of samples between the populations.
  • Approximately normally distributed populations.
  • Non-zero variances and preferably equal sample sizes for more power.
Meeting these requirements ensures that your F-test results are valid and that your conclusions are trustworthy.

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Most popular questions from this chapter

A set of solar batteries is used in a research satellite. The satellite can run on only one battery, but it runs best if more than one battery is used. The variance \(\sigma^{2}\) of lifetimes of these batteries affects the useful lifetime of the satellite before it goes dead. If the variance is too small, all the batteries will tend to die at once. Why? If the variance is too large, the batteries are simply not dependable. Why? Engineers have determined that a variance of \(\sigma^{2}=23\) months (squared) is most desirable for these batteries. A random sample of 22 batteries gave a sample variance of \(14.3\) months (squared). (i) Using a \(0.05\) level of significance, test the claim that \(\sigma^{2}=23\) against the claim that \(\sigma^{2}\) is different from 23 . (ii) Find a \(90 \%\) confidence interval for the population variance \(\sigma^{2}\). (iii) Find a \(90 \%\) confidence interval for the population standard deviation \(\sigma .\)

How are expected frequencies computed for goodness-of-fit tests?

Jim Mead is a veterinarian who visits a Vermont farm to examine prize bulls. In order to examine a bull, Jim first gives the animal a tranquilizer shot. The effect of the shot is supposed to last an average of 65 minutes, and it usually does. However, Jim sometimes gets chased out of the pasture by a bull that recovers too soon, and other times he becomes worried about prize bulls that take too long to recover. By reading journals, Jim has found that the tranquilizer should have a mean duration time of 65 minutes, with a standard deviation of 15 minutes. A random sample of 10 of Jim's bulls had a mean tranquilized duration time of close to 65 minutes but a standard deviation of 24 minutes. At the \(1 \%\) level of significance, is Jim justified in the claim that the variance is larger than that stated in his journal? Find a \(95 \%\) confidence interval for the population standard deviation.

For chi-square distributions, as the number of degrees of freedom increases, does any skewness increase or decrease? Do chi-square distributions become more symmetric (and normal) as the number of degrees of freedom becomes larger and larger?

A sociologist studying New York City ethnic groups wants to determine if there is a difference in income for immigrants from four different countries during their first year in the city. She obtained the data in the following table from a random sample of immigrants from these countries (incomes in thousands of dollars). Use a \(0.05\) level of significance to test the claim that there is no difference in the earnings of immigrants from the four different countries. \(\begin{array}{rrcr}\text { Country I } & \text { Country II } & \text { Country III } & \text { Country IV } \\ 12.7 & 8.3 & 20.3 & 17.2 \\\ 9.2 & 17.2 & 16.6 & 8.8 \\ 10.9 & 19.1 & 22.7 & 14.7 \\ 8.9 & 10.3 & 25.2 & 21.3 \\ 16.4 & & 19.9 & 19.8\end{array}\)
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