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Chapter 10: Problem 10

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? $$\begin{aligned}&H_{0}: p=0.2\\\&H_{1}: p<0.2\end{aligned}$$

Short Answer

Expert verified
The test is left-tailed, and the parameter being tested is the population proportion \(p\).

Step by step solution

01

Identify the Null Hypothesis

The null hypothesis (denoted as \(H_0\)) is given as \(p = 0.2\). This represents the initial assumption that the population proportion is 0.2.
02

Identify the Alternative Hypothesis

The alternative hypothesis (denoted as \(H_1\)) is given as \(p < 0.2\). This represents the scenario being tested against the null hypothesis, suggesting that the population proportion is less than 0.2.
03

Determine the Direction of the Test

Compare the alternative hypothesis to the null hypothesis. Given that \(H_1: p < 0.2\), the test is left-tailed because the alternative hypothesis is concerned with values of \(p\) that are less than 0.2.
04

Identify the Parameter Being Tested

The parameter being tested in both the null and alternative hypotheses is the population proportion, denoted as \(p\).

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Key Concepts

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

null hypothesis
In hypothesis testing, the null hypothesis (denoted as \(H_0\)) is a statement that represents the default or initial assumption about a population parameter. It usually states that there is no effect or no difference, and it is what we assume to be true before collecting any evidence. For example, in the given exercise, the null hypothesis is \(p = 0.2\). This means the initial assumption is that the population proportion \(p\) is 0.2. The null hypothesis is crucial because it serves as a baseline for comparison with the alternative hypothesis.
alternative hypothesis
The alternative hypothesis (denoted as \(H_1\)) is the statement that we are trying to find evidence for. It represents an alternative assumption to the null hypothesis and suggests that there is an effect or a difference. In the same exercise, the alternative hypothesis is \(p < 0.2\). This hypothesis indicates that we are testing if the population proportion is actually less than 0.2, challenging the initial assumption of the null hypothesis. The alternative hypothesis can be one-sided or two-sided, depending on the nature of the test.
left-tailed test
A left-tailed test is a type of one-tailed test in hypothesis testing. It is used when the alternative hypothesis states that the parameter is less than the null hypothesis value. In other words, we are only interested in values that are smaller than a specific number. From our example, since the alternative hypothesis is \(p < 0.2\), it means we are conducting a left-tailed test. Here, we focus on the left side of the distribution curve. It is crucial to visualize this, as it helps to understand where to look for evidence against the null hypothesis.
population proportion
The population proportion (denoted as \(p\)) is a parameter that represents the fraction of individuals in a population who have a particular attribute or characteristic. It is essential to note that this is not the same as the sample proportion, which is calculated from observed data. In hypothesis testing, we often test claims about this population proportion. In the exercise given, both the null and alternative hypotheses deal with the population proportion, with the null hypothesis stating \(p = 0.2\) and the alternative hypothesis stating \(p < 0.2\). Thus, understanding population proportion is key to interpreting the results of the test.

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Most popular questions from this chapter

To test \(H_{0}: \mu=4.5\) versus \(H_{1}: \mu>4.5,\) a simple random sample of size \(n=13\) is obtained from a population that is known to be normally distributed. (a) If \(\bar{x}=4.9\) and \(s=1.3,\) compute the test statistic. (b) Draw a \(t\) -distribution with the area that represents the \(P\) -value shaded. (c) Approximate and interpret the \(P\) -value. (d) If the researcher decides to test this hypothesis at the \(\alpha=0.1\) level of significance, will the researcher reject the null hypothesis? Why?

Load the hypothesis tests for a mean applet. (a) Set the shape to normal, the mean to \(100,\) and the standard deviation to \(15 .\) These parameters describe the distribution of IQ scores. Obtain 1000 simple random samples of size \(n=10\) from this population, and test whether the mean is different from 100. How many samples led to a rejection of the null hypothesis if \(\alpha=0.05 ?\) How many would we expect to lead to a rejection of the null hypothesis? For this level of significance, what is the probability of a Type I error? (b) Set the shape to normal, the mean to 100 , and the standard deviation to \(15 .\) These parameters describe the distribution of IQ scores. Obtain 1000 simple random samples of size \(n=30\) from this population, and test whether the mean is different from \(100 .\) How many samples led to a rejection of the null hypothesis if \(\alpha=0.05 ?\) How many would we expect to lead to a rejection of the null hypothesis? For this level of significance, what is the probability of a Type I error? (c) Compare the results of parts (a) and (b). Did the sample size have any impact on the number of samples that incorrectly rejected the null hypothesis?

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? $$\begin{aligned}&H_{0}: \sigma=4.2\\\&H_{1}: \sigma \neq 4.2\end{aligned}$$

SAT Exam Scores A school administrator believes that students whose first language learned is not English score worse on the verbal portion of the SAT exam than students whose first language is English. The mean SAT verbal score of students whose first language is English is 515 on the basis of data obtained from the College Board. Suppose a simple random sample of 20 students whose first language learned was not English results in a sample mean SAT verbal score of \(458 .\) SAT verbal scores are normally distributed with a population standard deviation of \(112 .\) (a) Why is it necessary for SAT verbal scores to be normally distributed to test the hypotheses using the methods of this section? (b) Use the classical approach or the \(P\) -value approach at the \(\alpha=0.10\) level of significance to determine if there is evidence to support the administrator's belief.

Conduct the appropriate test. A simple random sample of size \(n=14\) is drawn from a population that is normally distributed with \(\sigma=20 .\) The sample mean is found to be \(\bar{x}=60 .\) Test whether the population mean is less than 70 at the \(\alpha=0.1\) level of significance.
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