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Chapter 10: Problem 68

What assumptions are made about the populations from which independent random samples are obtained when the \(t\) distribution is used to make small-sample inferences concerning the differences in population means?

Short Answer

Expert verified
The populations are assumed to be normally distributed with equal variances and the samples should be independent.

Step by step solution

01

Understanding the Context

We begin by understanding that the question is asking about the assumptions needed when using the t-distribution for small-sample inferences about the difference in means from two independent samples.
02

Identifying Assumptions of Independence

One key assumption when using the t-distribution for two independent samples is that each sample must be an independent random sample. This means that the selection of one sample does not influence or depend on the selection of the other sample.
03

Homogeneity of Variances

Another assumption is that the populations from which the samples are drawn should have equal variances. This is often referred to as the assumption of homogeneity of variances.
04

Normality Assumption

Finally, it is assumed that the populations from which the samples are drawn follow a normal distribution. This assumption is particularly crucial when dealing with small sample sizes, as the Central Limit Theorem does not apply as strongly as it does with larger samples.

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Key Concepts

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

Small-sample inference
When we deal with small-sample inference, we're exploring how to draw conclusions about populations from small samples. This is important for situations where collecting large samples may not be feasible. In many real-world scenarios, we only have access to a limited amount of data. Here, employing the right statistical methods to make inferences about the population is crucial.

Using the t-distribution is common in small-sample inference for the following reasons:
  • It is specifically designed for situations where sample sizes are small (typically below 30).
  • It accounts for the added variability that occurs in smaller samples.
To make legitimate conclusions using small samples, certain assumptions鈥攕uch as independence and normality鈥攎ust be satisfied.
Independent random samples
For statistical tests, like those using the t-distribution, a core assumption is having independent random samples. These are samples that are chosen without influence from each other. This means:
  • Each sample should be selected randomly from its respective population.
  • The outcome of one sample does not affect the outcome of another.
When samples are independent and random, the results of statistical tests, like identifying differences in population means, are more reliable. It helps to prevent bias and ensures that the results are truly reflective of the broader populations being studied.
Homogeneity of variances
For accurate results using t-tests in small-sample inferences, the assumption of homogeneity of variances must hold. But what does this mean?
  • It asserts that the variance (or spread) of data points in each population is equal.
  • This is essential for calculating the pooled variance, which is used in many forms of t-tests.
If the variances are not equal, the calculated test statistic may not follow the assumed t-distribution, potentially leading to inaccurate results. In cases where this assumption does not hold, alternative methods like Welch's t-test might be used, which adjusts for differences in variance.
Normal distribution assumption
The normal distribution assumption is vital in making small-sample inferences. This assumption stipulates that the populations from which our samples are drawn should follow a normal distribution. Here鈥檚 why:
  • It allows us to apply the t-distribution to calculate probabilities and make inferences.
  • It becomes particularly crucial when the sample size is small, as under these conditions, the Central Limit Theorem (which aids in achieving a normal distribution as sample size increases) isn't fully robust.
If the data are not normally distributed, especially in small samples, the validity of the test's conclusions might be jeopardized. In such cases, transformations or non-parametric methods may be considered to meet the normality assumption.

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Most popular questions from this chapter

Let \(Y_{1}, Y_{2}, \ldots, Y_{n}\) denote a random sample from a uniform distribution over the interval \((0, \theta)\) a. Find the most powerful \(\alpha\) -level test for testing \(H_{0}: \theta=\theta_{0}\) against \(H_{a}: \theta=\theta_{a},\) where \(\theta_{a}<\theta_{0}\) b. Is the test in part (a) uniformly most powerful for testing \(H_{0}: \theta=\theta_{0}\) against \(H_{a}: \theta<\theta_{0} ?\)

Under what assumptions may the \(F\) distribution be used in making inferences about the ratio of population variances?

High airline occupancy rates on scheduled flights are essential for profitability. Suppose that a scheduled flight must average at least \(60 \%\) occupancy to be profitable and that an examination of the occupancy rates for 12010: 00 A.M. flights from Atlanta to Dallas showed mean occupancy rate per flight of \(58 \%\) and standard deviation \(11 \% .\) Test to see if sufficient evidence exists to support a claim that the flight is unprofitable. Find the \(p\) -value associated with the test. What would you conclude if you wished to implement the test at the \(\alpha=.10\) level?

A political researcher believes that the fraction \(p_{1}\) of Republicans strongly in favor of the death penalty is greater than the fraction \(p_{2}\) of Democrats strongly in favor of the death penalty. He acquired independent random samples of 200 Republicans and 200 Democrats and found 46 Republicans and 34 Democrats strongly favoring the death penalty. Does this evidence provide statistical support for the researcher's belief? Use \(\alpha=.05\)

Operators of gasoline-fueled vehicles complain about the price of gasoline in gas stations. According to the American Petroleum Institute, the federal gas tax per gallon is constant \((18.4 c\) as of January 13,2005 ), but state and local taxes vary from \(7.5 c\) to \(32.10 c\) for \(n=18\) key metropolitan areas around the country. \(^{\star}\) The total tax per gallon for gasoline at each of these 18 locations is given next. Suppose that these measurements constitute a random sample of size 18 $$\begin{aligned} &\begin{array}{llllll} 42.89 & 53.91 & 48.55 & 47.90 & 47.73 & 46.61 \end{array}\\\ &\begin{array}{llllll} 40.45 & 39.65 & 38.65 & 37.95 & 36.80 & 35.95 \end{array}\\\ &\begin{array}{llllll} 35.09 & 35.04 & 34.95 & 33.45 & 28.99 & 27.45 \end{array} \end{aligned}$$ a. Is there sufficient evidence to claim that the average per gallon gas tax is less than \(45 \%\) ? Use the \(t\) table in the appendix to bound the \(p\) -value associated with the test. b. What is the exact \(p\) -value?: c. Construct a \(95 \%\) confidence interval for the average per gallon gas tax in the United States.
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