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Chapter 10: Problem 12

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? \(H_{0}: p=0.76\) \(H_{1}: p>0.76\)

Short Answer

Expert verified
Right-tailed test, testing the population proportion \(p\).

Step by step solution

01

- Identify the Null Hypothesis

The null hypothesis is given as \(H_{0}: p=0.76\). This represents the assumption that the population proportion is equal to 0.76.
02

- Identify the Alternative Hypothesis

The alternative hypothesis is given as \(H_{1}: p>0.76\). This represents the assumption that the population proportion is greater than 0.76.
03

- Determine the Type of Test

The alternative hypothesis \(H_{1}: p>0.76\) indicates that the test is looking for values greater than 0.76. Therefore, this is a right-tailed test.
04

- Identify the Parameter Being Tested

Both hypotheses involve the population proportion \(p\). Thus, the parameter being tested is \(p\).

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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, often represented as \( H_0 \), is a fundamental concept in hypothesis testing. It represents a statement of no effect or no difference. In many cases, it's a statement that the population parameter is equal to a specific value. For example, the null hypothesis \( H_0: p=0.76 \) suggests that the population proportion \( p \) is equal to 0.76. The null hypothesis is assumed to be true until evidence suggests otherwise. It forms the baseline that the test will challenge. Rejecting the null hypothesis means that there is enough statistical evidence to support an alternative claim.
alternative hypothesis
The alternative hypothesis, denoted as \( H_1 \) or \( H_a \), is what you want to prove to be true. It is a statement that indicates the presence of an effect, a difference, or a relationship. In this case, \( H_1: p>0.76 \) means that the researcher believes the population proportion \( p \) is greater than 0.76. Unlike the null hypothesis, the alternative hypothesis represents change or a new perspective. It's what researchers aim to provide evidence for through their testing.
right-tailed test
A right-tailed test is a type of hypothesis test where the region of rejection is on the right side of the sampling distribution. This type of test is used when the alternative hypothesis states that a parameter is greater than the null hypothesis value. For example, with \( H_1: p>0.76 \), we are interested in finding whether the proportion \( p \) is significantly greater than 0.76. The test statistic will be calculated, and if it falls in the right tail of the distribution beyond a critical value, we will reject the null hypothesis. This implies that our observation is extreme enough to support the greater than claim.
population proportion
The population proportion \( p \) is a key parameter in statistics, representing the ratio of members of a population that have a particular attribute. It is a measure often tested in hypothesis testing scenarios. For example, if we say that \( H_0: p=0.76 \) then \( p \) represents the proportion of the population possessing a specific characteristic being 76%. Hypothesis tests on population proportion are common in surveys and quality control, where understanding the fraction of a group that exhibits a certain trait is crucial. The test evaluates whether sample data provides strong enough evidence to infer something about the population proportion.

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Most popular questions from this chapter

In \(1994,52 \%\) of parents with children in high school felt it was a serious problem that high school students were not being taught enough math and science. A recent survey found that 256 of 800 parents with children in high school felt it was a serious problem that high school students were not being taught enough math and science. Do parents feel differently today than they did in 1994 ? Use the \(\alpha=0.05\) level of significance? Source: Based on "Reality Check: Are Parents and Students Ready for More Math and Science?" Public Agenda, \(2006 .\)

A simple random sample of size \(n=40\) is drawn from a population. The sample mean is found to be \(108.5,\) and the sample standard deviation is found to be \(17.9 .\) Is the population mean greater than 100 at the \(\alpha=0.05\) level of significance?

The website pundittracker.com keeps track of predictions made by individuals in finance, politics, sports, and entertainment. Jim Cramer is a famous TV financial personality and author. Pundittracker monitored 678 of his stock predictions (such as a recommendation to buy the stock) and found that 320 were correct predictions. Treat these 678 predictions as a random sample of all of Cramer's predictions. (a) Determine the sample proportion of predictions Cramer got correct. (b) Suppose that we want to know whether the evidence suggests Cramer is correct less than 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 area representing the \(P\) -value 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 \(\alpha=0.05\) level of significance.

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.

Explain the difference between statistical significance and practical significance.
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