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Ch. 8 - Hypothesis Testing
Triola - Elementary Statistics 14th Edition
Triola14th EditionElementary StatisticsISBN: 9780137366446Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoTriola 14th EditionCh. 8 - Hypothesis TestingProblema 8.5.5
Capitolo 8, Problema 8.5.5

Randomization: Testing a Claim About a Proportion
In Exercises 5–8, use the randomization procedure for the indicated exercise.
Section 8-2, Exercise 9 “Cursed Movie”

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Step 1: Understand the problem. The goal is to test a claim about a proportion using a randomization procedure. Specifically, we are analyzing data related to the 'Cursed Movie' scenario from Section 8-2, Exercise 9. Review the claim and the data provided in the exercise to identify the null hypothesis (H₀) and the alternative hypothesis (H₁).
Step 2: Define the hypotheses. Typically, the null hypothesis (H₀) states that the proportion is equal to a specified value (e.g., p = p₀), while the alternative hypothesis (H₁) states that the proportion is different (e.g., p ≠ p₀, p > p₀, or p < p₀). Write these hypotheses clearly.
Step 3: Simulate randomization. To perform the randomization procedure, shuffle or resample the data under the assumption that the null hypothesis is true. This involves generating a large number of simulated samples where the proportion matches the null hypothesis value. Use statistical software or manual methods to perform this step.
Step 4: Calculate the test statistic for each simulated sample. The test statistic could be the sample proportion, the difference between proportions, or another relevant measure. Record the test statistic for each simulation.
Step 5: Compare the observed test statistic to the distribution of simulated test statistics. Determine the p-value by finding the proportion of simulated test statistics that are as extreme or more extreme than the observed test statistic. Use this p-value to decide whether to reject or fail to reject the null hypothesis based on the significance level (e.g., α = 0.05).

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Randomization

Randomization is a statistical technique used to assign subjects to different groups in a way that eliminates bias. It ensures that each participant has an equal chance of being placed in any group, which helps to create comparable groups and allows for valid inferences about the population. In hypothesis testing, randomization can be used to simulate the distribution of a test statistic under the null hypothesis.
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Intro to Random Variables & Probability Distributions

Proportion

A proportion is a statistical measure that represents the fraction of a whole, often expressed as a percentage. In the context of hypothesis testing, it refers to the ratio of a specific outcome to the total number of observations. Understanding proportions is crucial when testing claims about population characteristics, as it helps to quantify the likelihood of observing certain results under different conditions.
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Difference in Proportions: Hypothesis Tests

Hypothesis Testing

Hypothesis testing is a method used to determine whether there is enough statistical evidence in a sample to infer that a certain condition holds true for the entire population. It involves formulating a null hypothesis (no effect or no difference) and an alternative hypothesis (indicating an effect or difference), followed by calculating a test statistic and comparing it to a critical value to make a decision. This process is essential for validating claims about proportions in statistical studies.
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Step 1: Write Hypotheses
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Finding P-values

In Exercises 5–8, either use technology to find the P-value or use Table A-3 to find a range of values for the P-value. Based on the result, what is the final conclusion?


Weights of Quarters The claim is that weights (grams) of quarters made after 1964 have a mean equal to 5.670 g as required by mint specifications. The sample size is and the test statistic is t = -3.135.

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Using Technology

In Exercises 5–8, identify the indicated values or interpret the given display. Use the normal distribution as an approximation to the binomial distribution, as described in Part 1 of this section. Use a 0.05 significance level and answer the following:


a. Is the test two-tailed, left-tailed, or right-tailed?

b. What is the test statistic?

c. What is the P-value?

d. What is the null hypothesis, and what do you conclude about it?

e. What is the final conclusion?


Adverse Reactions to Drug The drug Lipitor (atorvastatin) is used to treat high cholesterol. In a clinical trial of Lipitor, 47 of 863 treated subjects experienced headaches (based on data from Pfizer). The accompanying TI-83/84 Plus calculator display shows results from a test of the claim that fewer than 10% of treated subjects experience headaches.

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Finding P-values

In Exercises 5–8, either use technology to find the P-value or use Table A-3 to find a range of values for the P-value. Based on the result, what is the final conclusion?


Cotinine in Smokers The claim is that smokers have a mean cotinine level greater than the level of 2.84 ng/mL found for nonsmokers. (Cotinine is used as a biomarker for exposure to nicotine.) The sample size is n = 902 and the test statistic is t = 56.319.

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