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Chapter 10: Problem 9

A null and alternative hypothesis is given. Determine whether the hypothesis test is left-tailed, right-tailed, or two tailed. What parameter is being tested? $$\begin{aligned}&H_{0}: \mu=5\\\&H_{1}: \mu>5\end{aligned}$$

Short Answer

Expert verified
Right-tailed test; population mean (\mu\) is being tested.

Step by step solution

01

Identify the null hypothesis

The null hypothesis is usually denoted as \(H_0\). In this case, \(H_0: \mu = 5\). This means we assume the population mean \(\mu\) is equal to 5.
02

Identify the alternative hypothesis

The alternative hypothesis is usually denoted as \(H_1\). In this case, \(H_1: \mu > 5\). This suggests that we are testing whether the population mean \(\mu\) is greater than 5.
03

Determine the type of test

Since the alternative hypothesis \(H_1\) states \(\mu > 5\), we are testing for whether the mean is greater than a certain value. This makes it a right-tailed test.
04

Identify the parameter being tested

The parameter being tested is the population mean, denoted by \(\mu\).

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Key Concepts

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

null hypothesis
The null hypothesis, denoted as \( H_0 \), is a fundamental concept in hypothesis testing. It represents a default or initial assumption about a population parameter. In this context, it's the assumption that the population mean \( \mu \) is equal to 5: \( H_0: \mu = 5 \). The null hypothesis is what we test against and aim to find evidence to disprove. Without any evidence to the contrary, we stick to this assumption.
alternative hypothesis
The alternative hypothesis, represented as \( H_1 \), proposes an alternative to what is stated in the null hypothesis. In the given exercise, the alternative hypothesis is \( H_1: \mu > 5 \). This states that the population mean \( \mu \) is greater than 5. The goal of hypothesis testing is to determine whether there is enough evidence to support this alternative hypothesis. It's crucial because it reflects the claim we're trying to test.
right-tailed test
A right-tailed test is one type of statistical hypothesis test it focuses on whether there is evidence that the population parameter (such as the mean) is greater than a certain value. In our exercise, since the alternative hypothesis \( H_1 \) claims that \( \mu > 5 \), we're performing a right-tailed test. The 'right-tail' refers to the end of the distribution in the positive direction. If our test statistic falls significantly into this tail, we may reject the null hypothesis in favor of the alternative.
population mean
A population mean, denoted by \( \mu \), is a measure of central tendency that represents the average value of a population. In the hypothesis test at hand, we're examining whether this population mean \( \mu \) is exactly 5 or greater. Population mean is crucial because it provides insights about the overall characteristics of the population from which samples are drawn. Hypothesis testing involving the population mean helps us make inferences about this average value based on sample data.

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Most popular questions from this chapter

To test \(H_{0}: \mu=4.5\) versus \(H_{1}: \mu>4.5,\) a random sample of size \(n=13\) is obtained from a population that is known to be normally distributed with \(\sigma=1.2\) (a) If the sample mean is determined to be \(\bar{x}=4.9,\) compute and interpret the \(P\) -value. (b) If the researcher decides to test this hypothesis at the \(\alpha=0.1\) level of significance, will the researcher reject the null hypothesis? Why?

To test \(H_{0}: \mu=45\) versus \(H_{1}: \mu \neq 45,\) a random sample of size \(n=40\) is obtained from a population whose standard deviation is known to be \(\sigma=8\) (a) Does the population need to be normally distributed to compute the \(P\) -value? (b) If the sample mean is determined to be \(\bar{x}=48.3\) compute and interpret the \(P\) -value. (c) If the researcher decides to test this hypothesis at the \(\alpha=0.05\) level of significance, will the researcher reject the null hypothesis? Why?

Para-nonylphenol is found in polyvinyl chloride (PVC) used in the food processing and packaging industries. Researchers wanted to determine the effect this substance had on the organ weight of firstgeneration mice when both parents were exposed to \(50 \mu \mathrm{g} / \mathrm{L}\) of para- nonylphenol in drinking water for 4 weeks. After 4 weeks, the mice were bred. After 100 days, the offspring of the exposed parents were sacrificed and the kidney weights were determined. The mean weight of the 12 offspring was found to be \(396.9 \mathrm{mg}\) with a standard deviation of \(45.4 \mathrm{mg} .\) Is there significant evidence to conclude that the kidney weight of the offspring whose parents were exposed to \(50 \mu \mathrm{g} / \mathrm{L}\) of para- nonylphenol in drinking water for 4 weeks is greater than \(355.7 \mathrm{mg},\) the mean weight of kidneys in normal 100 -day old mice at the \(\alpha=0.05\) level of significance? Source: Vendula Kyselova et al., Effects of \(p\) -nonylphenol and resveratrol on body and organ weight and in vivo fertility of outbred CD-1 mice, Reproductive Biology and Endocrinology, 2003 )

To test \(H_{0}: \mu=20\) versus \(H_{1}: \mu<20,\) a random sample of size \(n=18\) is obtained from a population that is known to be normally distributed with \(\sigma=3\) (a) If the sample mean is determined to be \(\bar{x}=18.3\) compute and interpret the \(P\) -value. (b) If the researcher decides to test this hypothesis at the \(\alpha=0.05\) level of significance, will the researcher reject the null hypothesis? Why?

(a) determine the null and alternative hypotheses, (b) explain what it would mean to make a Type I error, and (c) explain what it would mean to make a Type II error. According to the Statistical Abstract of the United States, the mean monthly cell phone bill was \(\$ 49.91\) in \(2003 .\) A researcher suspects that the mean monthly cell phone bill is different today.
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