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Chapter 9: Problem 3

Consider a hypothesis test of difference of means for two independent populations \(x_{1}\) and \(x_{2}\). What are two ways of expressing the null hypothesis?

Short Answer

Expert verified
The null hypothesis can be expressed as either H_0: 渭_1 = 渭_2 or H_0: 渭_1 - 渭_2 = 0 .

Step by step solution

01

Understanding the Problem

To solve this problem, we need to identify how to express the null hypothesis for a hypothesis test that compares the means of two independent populations.
02

The Null Hypothesis Concept

The null hypothesis ( H_0 ) is a statement that there is no effect or no difference. In the context of comparing two means, it is typically formulated to indicate that the means of the two populations are equal.
03

Express Null Hypothesis with Population Means

One way to express the null hypothesis mathematically is to say the population means are equal: H_0: 渭_1 = 渭_2 , where 渭_1 and 渭_2 are the means of populations x_1 and x_2 respectively.
04

Express Null Hypothesis with Difference in Means

Another way to express the null hypothesis is by stating that the difference between the two population means is zero: H_0: 渭_1 - 渭_2 = 0 . This is essentially equivalent to saying the means are equal.

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Key Concepts

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

Null Hypothesis
In statistics, a null hypothesis is a fundamental concept used when testing relationships between data sets. It is essentially a statement that nothing of interest is happening鈥攎eaning there is no effect or difference to be found in the data being tested. When dealing with two population means, the null hypothesis (denoted as \(H_0\)) usually states there is no difference between these means.

This can be expressed in two equivalent ways:
  • By stating that both population means are equal: \(H_0: \mu_1 = \mu_2\).
  • By stating there is no difference between the population means: \(H_0: \mu_1 - \mu_2 = 0\).
Both expressions convey the same intent鈥攖hat any observed difference in sample means is purely the result of random sampling variability. The null hypothesis serves as a starting point for statistical hypothesis testing.
Difference of Means
The difference of means refers to the contrast in averages between two sets of data, often from two different populations. In hypothesis testing, assessing whether this difference is statistically significant鈥攐r if it could have occurred by chance alone鈥攊s a common goal.

To understand the difference of means, consider these points:
  • A positive difference indicates that one population has a higher mean than the other.
  • A negative difference suggests the opposite effect.
  • A difference of zero implies both population means are the same, which aligns with the null hypothesis in tests comparing two independent groups.
Statisticians often use t-tests or z-tests to determine if the observed difference is statistically significant. This involves comparing the calculated difference to a critical value from the distribution appropriate to the sample size and variance.
Independent Populations
Independent populations refer to two or more groups that do not influence each other. In statistical terms, this means the values in one group have no effect or correlation with the values in the other group. This independence is crucial when conducting hypothesis tests involving multiple groups.

For hypothesis testing:
  • Samples must be randomly assigned or sourced from populations that do not overlap.
  • The assumption of independence allows the use of methods like the two-sample t-test.
  • Independence ensures that any observed differences in means are not due to shared factors or influences between populations.
Understanding the principle of independent populations is key to correctly applying statistical methods and validating the results derived from such analyses.
Population Means
A population mean denotes the average value of all observations in a single population. It is often represented by the Greek letter \(\mu\). Calculating population means provides a central tendency measure that captures an entire population's distribution.

Key aspects of population means include:
  • It is a parameter that embodies the true mean value of the entire population.
  • When comparing populations, the mean helps determine which population, on average, exhibits a higher or lower trait of interest.
  • Population means are typically unknown and are estimated using sample means (\(\bar{x}\)), which are derived from data.
In hypothesis testing, the goal is often to draw inferences about population means based on sample data, thus making decisions about the null hypothesis.

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Most popular questions from this chapter

USA Today reported that about \(47 \%\) of the general consumer population in the United States is loyal to the automobile manufacturer of their choice. Suppose Chevrolet did a study of a random sample of 1006 Chevrolet owners and found that 490 said they would buy another Chevrolet. Does this indicate that the population proportion of consumers loyal to Chevrolet is more than \(47 \%\) ? Use \(\alpha=0.01\).

Consider a test for \(\mu\). If the \(P\) -value is such that you can reject \(H_{0}\) at the \(5 \%\) level of significance, can you always reject \(H_{0}\) at the \(1 \%\) level of significance? Explain.

In environmental studies, sex ratios are of great importance. Wolf society, packs, and ecology have been studied extensively at different locations in the U.S. and foreign countries. Sex ratios for eight study sites in northern Europe are shown below (based on The Wolf by L. D. Mech, University of Minnesota Press). \(\begin{array}{lcc} \hline \text { Location of Wolf Pack } & \text { \% Males (Winter) } & \text { \% Males (Summer) } \\ \hline \text { Finland } & 72 & 53 \\ \text { Finland } & 47 & 51 \\ \text { Finland } & 89 & 72 \\ \text { Lapland } & 55 & 48 \\ \text { Lapland } & 64 & 55 \\ \text { Russia } & 50 & 50 \\ \text { Russia } & 41 & 50 \\ \text { Russia } & 55 & 45 \\ \hline \end{array}\) It is hypothesized that in winter, "loner" males (not present in summer packs) join the pack to increase survival rate. Use a \(5 \%\) level of significance to test the claim that the average percentage of males in a wolf pack is higher in winter.

This problem is based on information taken from The Merck Manual (a reference manual used in most medical and nursing schools). Hypertension is defined as a blood pressure reading over \(140 \mathrm{~mm}\) Hg systolic and/or over \(90 \mathrm{~mm}\) Hg diastolic. Hypertension, if not corrected, can cause long- term health problems. In the college-age population \((18-24\) years), about \(9.2 \%\) have hypertension. Suppose that a blood donor program is taking place in a college dormitory this week (final exams week). Before each student gives blood, the nurse takes a blood pressure reading. Of 196 donors, it is found that 29 have hypertension. Do these data indicate that the population proportion of students with hypertension during final exams week is higher than \(9.2 \%\) ? Use a \(5 \%\) level of significance.

A random sample of \(n_{1}=16\) communities in western Kansas gave the following information for people under 25 years of age. \(x_{1}:\) Rate of hay fever per 1000 population for people under 25 \(\begin{array}{rrrrrrrr}98 & 90 & 120 & 128 & 92 & 123 & 112 & 93 \\ 125 & 95 & 125 & 117 & 97 & 122 & 127 & 88\end{array}\) A random sample of \(n_{2}=14\) regions in western Kansas gave the following information for people over 50 years old. \(x_{2}\) : Rate of hay fever per 1000 population for people over 50 \(\begin{array}{llllllr}95 & 110 & 101 & 97 & 112 & 88 & 110 \\ 79 & 115 & 100 & 89 & 114 & 85 & 96\end{array}\) (Reference: National Center for Health Statistics.) i. Use a calculator to verify that \(\bar{x}_{1} \approx 109.50, s_{1} \approx 15.41, \bar{x}_{2} \approx 99.36\), and \(s_{2} \approx 11.57\) ii. Assume that the hay fever rate in each age group has an approximately normal distribution. Do the data indicate that the age group over 50 has a lower rate of hay fever? Use \(\alpha=0.05\).
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