Ch. 8 - Hypothesis Testing

Triola - Elementary Statistics 14th Edition

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Triola 14th Edition

Ch. 8 - Hypothesis Testing

Problema 8.CQQ.1

Capitolo 8, Problema 8.CQQ.1

Discarded Plastic Data Set 42 “Garbage Weight” includes weights (pounds) of discarded plastic from 62 different households. Those 62 weights have a mean of 1.911 pounds and a standard deviation of 1.065 pounds. We want to use a 0.05 level of significance to test the claim that this sample is from a population with a mean less than 2.000 pounds. Identify the null hypothesis and alternative hypothesis.

Guida verificata passo dopo passo

Step 1: Understand the problem. We are testing a claim about the population mean (μ) of discarded plastic weights. Specifically, we want to test if the population mean is less than 2.000 pounds, using a sample mean of 1.911 pounds, a standard deviation of 1.065 pounds, and a significance level of 0.05.

Step 2: Define the null hypothesis (H₀) and the alternative hypothesis (Hₐ). The null hypothesis represents the status quo or no effect, while the alternative hypothesis represents the claim we are testing. In this case: H₀: μ ≥ 2.000 (the population mean is greater than or equal to 2.000 pounds), and Hₐ: μ < 2.000 (the population mean is less than 2.000 pounds).

Step 3: Identify the type of test. Since we are testing whether the population mean is less than a specific value, this is a one-tailed test. Additionally, because the population standard deviation is not provided and we are using the sample standard deviation, we will use a t-test.

Step 4: Calculate the test statistic. The formula for the t-test statistic is:

t = x̄ - μ

₀sn, where

x̄

is the sample mean,

μ

₀ is the hypothesized population mean,

s

is the sample standard deviation, and

n

is the sample size.

Step 5: Compare the test statistic to the critical value or use the p-value approach. Determine the critical t-value for a one-tailed test at a 0.05 significance level with degrees of freedom

df = n - 1

. If the test statistic is less than the critical value, or if the p-value is less than 0.05, reject the null hypothesis. Otherwise, fail to reject the null hypothesis.

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Null Hypothesis

The null hypothesis (H0) is a statement that indicates no effect or no difference, serving as a default position in hypothesis testing. In this context, it posits that the population mean of discarded plastic weights is equal to or greater than 2.000 pounds. It is the hypothesis that researchers aim to test against, and it is typically assumed true until evidence suggests otherwise.

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Percorso guidato

06:21

Step 1: Write Hypotheses

Alternative Hypothesis

The alternative hypothesis (H1) represents the statement that contradicts the null hypothesis, suggesting that there is an effect or a difference. In this scenario, it claims that the population mean of discarded plastic weights is less than 2.000 pounds. This hypothesis is what researchers hope to support through statistical testing, indicating a significant finding if the null hypothesis is rejected.

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Percorso guidato

06:21

Step 1: Write Hypotheses

Level of Significance

The level of significance, often denoted as alpha (α), is the threshold for determining whether to reject the null hypothesis. In this case, a significance level of 0.05 means there is a 5% risk of concluding that a difference exists when there is none (Type I error). It helps researchers decide how strong the evidence must be to reject the null hypothesis in favor of the alternative hypothesis.

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04:46

Step 4: State Conclusion Example 4

Pratica correlata

Domanda del libro di testo

Robust Explain what is meant by the statements that the t test for a claim about μ is robust, but the (chi)^2 test for a claim about σ is not robust.

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Estimates vs. Hypothesis Tests Labels on cans of Dr. Pepper soda indicate that they contain 12 oz of the drink. We could collect samples of those cans and accurately measure the actual contents, then we could use methods of Section 7-2 for making an estimate of the mean amount of Dr. Pepper in cans, or we could use those measured amounts to test the claim that the cans contain a mean of 12 oz. What is the difference between estimating the mean and testing a hypothesis about the mean?

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Discarded Plastic Find the test statistic used for the hypothesis test described in Exercise 1.

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Discarded Plastic

What distribution is used for the hypothesis test described in Exercise 1?

For the hypothesis test described in Exercise 1, is it necessary to determine whether the 62 weights appear to be from a population having a normal distribution? Why or why not?

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Lightning Deaths Listed below are the numbers of deaths from lightning strikes in the United States each year for a sequence of recent and consecutive years. Find the values of the indicated statistics.

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Discarded Plastic The P-value for the hypothesis test described in Exercise 1 is 0.2565.

What should be concluded about the null hypothesis?

What is the final conclusion that addresses the original claim?

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