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Chapter 10: Problem 10

In Problems \(9-14,\) the null and alternative hypotheses are given. Determine whether the hypothesis test is lefi-tailed, right-tailed, or two-tailed. What parameter is being tested? \(H_{0}: p=0.2\) \(H_{1}: p<0.2\)

Short Answer

Expert verified
Left-tailed test; parameter \( p \).

Step by step solution

01

Identify the null hypothesis

The null hypothesis provided is: \( H_{0}: p=0.2 \)
02

Identify the alternative hypothesis

The alternative hypothesis provided is: \( H_{1}: p<0.2 \)
03

Determine the type of test

Compare the alternative hypothesis with the null hypothesis. Since the alternative hypothesis is \( p < 0.2 \), it shows that we are interested in values of \( p \) less than the null hypothesis value. This indicates a left-tailed test.
04

Identify the parameter being tested

The parameter being tested is \( p \), which is the proportion parameter in the hypotheses.

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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 there is no effect or no difference. It's a default assumption that any kind of difference or significance you see in a set of data is due to chance.
For example, if a company claims that 20% of its customers are satisfied, we can test this claim by setting the null hypothesis as \(p = 0.2\). The null hypothesis often serves as a starting point for statistical testing.
The null hypothesis is tested directly, and the goal is to determine if there is enough evidence to reject it in favor of the alternative hypothesis.
alternative hypothesis
The alternative hypothesis, denoted as \(H_{1}\) or \(H_{a}\), is what you want to prove. This hypothesis is contrary to the null hypothesis and reflects the presence of an effect or difference.
In the given exercise, the alternative hypothesis is \(p < 0.2\), which suggests that the proportion parameter \(p\) is less than the assumed value by the null hypothesis.
When formulating an alternative hypothesis, you're essentially questioning the status quo and proposing that there is a new effect or difference to be considered.
left-tailed test
A left-tailed test is a type of hypothesis test where the area of interest is in the left tail of the distribution.
In other words, it tests whether the sample proportion is significantly less than the assumed population proportion (null hypothesis value).
In our example, since the alternative hypothesis is \(p < 0.2\), we're conducting a left-tailed test. We are looking specifically for evidence that \(p\) is less than 0.2.
proportion parameter
The proportion parameter, denoted as \(p\), represents the probability of a certain event occurring in a population.
In hypothesis tests involving proportions, we often focus on this parameter to make inferences about the population based on sample data.
In the given problem, the parameter \(p\) is being tested to see if it is significantly different (specifically lower) than 0.2.

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Most popular questions from this chapter

Decide whether the problem requires a confidence interval or hypothesis test, and determine the variable of interest. For any problem requiring a confidence interval, state whether the confidence interval will be for a population proportion or population mean. For any problem requiring a hypothesis test, write the null and alternative hypothesis. A researcher wanted to estimate the average length of time mothers who gave birth via Caesarean section spent in a hospital after delivery of the baby.

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In his book, "The Signal and the Noise," Nate Silver analyzed 733 predictions made by experts regarding political events. Of the 733 predictions, 338 were mostly true. (a) Determine the sample proportion of political predictions that were mostly true. (b) Suppose that we want to know whether the evidence suggests the political predictions were mostly true less thar half the time. State the null and alternative hypotheses. (c) Verify the normal model may be used to determine the \(P\) -value for this hypothesis test. (d) Draw a normal model with the area representing the \(P\) -val shaded for this hypothesis test. (e) Determine the \(P\) -value based on the model from part (d). (f) Interpret the \(P\) -value. (g) Based on the \(P\) -value, what does the sample evidence suggest? That is, what is the conclusion of the hypothesis test? Assume an \(\alpha=0.1\) level of significance.

A simple random sample of size \(n=200\) individuals with a valid driver's license is asked if they drive an American-made automobile. Of the 200 individuals surveyed, 115 responded that they drive an American-made automobile. Determine if a majority of those with a valid driver's license drive an American made automobile at the \(\alpha=0.05\) level of significance.
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