More On The Power For A Different Alternative: One-Sided Tes

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1. More on the power for a different alternative. A one-sided test of the null hypothesis μ = 20 versus the alternative μ = 35 has power equal to 0.60. Will the power for the alternative μ = 28 be higher or lower than 0.73? Explain your answer.

2. Where do you buy? Consumers can purchase nonprescription medications at food stores, mass merchandise stores such as Target and Walmart, or pharmacies. About 45% of consumers make such purchases at pharmacies. What accounts for the popularity of pharmacies, which often charge higher prices? A study examined consumers’ perceptions of overall performance of the three types of stores, using a long questionnaire that asked about such things as “neat and attractive store,” “knowledgeable staff,” and “assistance in choosing among various types of nonprescription medication.” A performance score was based on 27 such questions. The subjects were 201 people chosen at random from the Indianapolis telephone directory. What population do you think the authors of the study want to draw conclusions about? What population are you certain they can draw conclusions about?

Paper For Above instruction

The first question involves understanding the statistical concept of power in hypothesis testing, particularly in the context of one-sided tests examining different alternatives. In the scenario, a one-sided hypothesis test compares a null hypothesis μ = 20 against an alternative μ = 35, with a calculated power of 0.60. Power, in statistical testing, refers to the probability of correctly rejecting the null hypothesis when the alternative is true—in this case, when μ actually equals 35.

When considering a different alternative, such as μ = 28, the probability of rejecting the null hypothesis depends on the distance of this alternative from the null value (μ = 20). Since μ = 28 is closer to the null value than μ = 35, the test's power to detect μ = 28 as different from 20 would be lower than when testing against μ = 35. This is because the statistical power increases as the true mean moves further away from the null hypothesis value, assuming all other factors remain constant.

Mathematically, power is a function of effect size, sample size, significance level, and variability within the data. Given that the power for μ = 35 is 0.60, we expect the power for μ = 28 to be less than 0.60, hence lower than 0.73, due to the smaller deviation from the null hypothesis. The value 0.73 reflects a higher likelihood of detecting a difference when the alternative is farther from the null value (35), compared to when the true mean is only 28. Therefore, the power for an alternative hypothesis of μ = 28 will be lower than 0.73 because the smaller effect size reduces the test's ability to discriminate from the null hypothesis.

The second scenario examines the reasons behind the popularity of pharmacies for purchasing nonprescription medications, despite potentially higher prices. Approximately 45% of consumers prefer buying from pharmacies. This preference may be explained by several factors, including perceived higher quality of service, the availability of knowledgeable staff, and the trustworthiness of pharmacies in providing reliable advice, which influences consumer perceptions and decision-making.

The study’s methodology involved surveying 201 individuals randomly selected from the Indianapolis telephone directory. The questionnaire assessed perceptions about store appearance, staff knowledge, and assistance in selecting medications, culminating in a performance score based on 27 questions. The authors likely intend to generalize their findings to the broader population of consumers in Indianapolis or a similar demographic region. The population they aim to draw conclusions about is probably the entire population of consumers who purchase nonprescription medications in Indianapolis or comparable urban areas.

However, based on the sampling method—randomly choosing individuals from the telephone directory—certainty about the generalizability is limited to the population of telephone users in Indianapolis with listed numbers. The researchers can confidently draw conclusions about this specific subset but may face limitations in extending results to the entire consumer population without accounting for unlisted numbers, non-telephone users, or different geographic regions. Therefore, their conclusions are most valid for the population of telephone directory subscribers in Indianapolis.

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