Ch. 7 - Hypothesis Testing with One Sample

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 7 - Hypothesis Testing with One Sample

Problema 7.1.1

Capitolo 7, Problema 7.1.1

What are the two types of hypotheses used in a hypothesis test? How are they related?

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The two types of hypotheses used in a hypothesis test are the null hypothesis (denoted as H₀) and the alternative hypothesis (denoted as Hₐ).

The null hypothesis (H₀) represents the default or status quo assumption. It is a statement that there is no effect, no difference, or no relationship between variables. For example, H₀ might state that the mean of a population is equal to a specific value.

The alternative hypothesis (Hₐ) is the statement that contradicts the null hypothesis. It represents the claim or effect that the researcher is trying to support. For example, Hₐ might state that the mean of a population is not equal to a specific value, or it could specify a direction (greater than or less than).

The null and alternative hypotheses are mutually exclusive and collectively exhaustive. This means that only one of them can be true, and together they cover all possible outcomes of the test.

The goal of hypothesis testing is to use sample data to determine whether there is enough evidence to reject the null hypothesis (H₀) in favor of the alternative hypothesis (Hₐ), based on a chosen significance level (α).

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Null Hypothesis (H0)

The null hypothesis is a statement that there is no effect or no difference, serving as a default position in hypothesis testing. It posits that any observed effect in the data is due to sampling variability rather than a true effect. For example, in a drug efficacy test, the null hypothesis might state that the drug has no effect on patients compared to a placebo.

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Step 1: Write Hypotheses

Alternative Hypothesis (H1)

The alternative hypothesis is a statement that contradicts the null hypothesis, suggesting that there is an effect or a difference. It represents the researcher's claim or what they aim to prove through the hypothesis test. For instance, in the same drug efficacy test, the alternative hypothesis would assert that the drug does have a significant effect on patients.

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Step 1: Write Hypotheses

Relationship Between Hypotheses

The null and alternative hypotheses are mutually exclusive; if one is true, the other must be false. In hypothesis testing, the goal is to gather evidence to either reject the null hypothesis in favor of the alternative or fail to reject the null. This relationship is fundamental to statistical inference, guiding the decision-making process based on sample data.

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Step 1: Write Hypotheses

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Hypothesis Testing Using Rejection Region(s) In Exercises 39–44, (a) identify the claim and state H0 and Ha, (b) find the critical value(s) and identify the rejection region(s), (c) find the standardized test statistic z, (d) decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim.

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In Exercises 3–8, find the critical value(s) and rejection region(s) for the type of t-test with level of significance alpha and sample size n.

Left-tailed test, α=0.10, n=20

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In Exercises 3–6, determine whether a normal sampling distribution can be used. If it can be used, test the claim.

Claim: p ≥0.48, α=0.08. Sample statistics: p_hat = 0.40, n=90

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Hypothesis Testing Using Rejection Regions In Exercises 19–26, (a) identify the claim and state H0 and Ha, (b) find the critical value(s) and identify the rejection region(s), (c) find the standardized test statistic t, (d) decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. Assume the population is normally distributed.

Lead Levels As part of your work for an environmental awareness group, you want to test a claim that the mean amount of lead in the air in U.S. cities is less than 0.032 microgram per cubic meter. You find that the mean amount of lead in the air for a random sample of 56 U.S. cities is 0.021 microgram per cubic meter and the standard deviation is 0.034 microgram per cubic meter. At α=0.01, can you support the claim?

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A travel analyst claims the mean daily base price for renting a full-size or less expensive vehicle in Vancouver, British Columbia, is more than \$86. You want to test this claim. In a random sample of 40 full-size or less expensive vehicles available to rent in Vancouver, British Columbia, the mean daily base price is \$93.23. Assume the population standard deviation is \$28.90. At α=0.10, do you have enough evidence to support the analyst’s claim?

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[APPLET] A weight loss program claims that program participants have a mean weight loss of at least 10.5 pounds after 1 month. The weight losses after 1 month (in pounds) of a random sample of 40 program participants are listed below. At α=0.01, is there enough evidence to reject the program’s claim?

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