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Chapter 8: Problem 2

Choose one of the answers in each case. In statistical inference, measurements are made on a population), and generalizations are made to a (sample or population).

Short Answer

Expert verified
In statistical inference, measurements are made on a sample, and generalizations are made to a population.

Step by step solution

01

Clarify the Definitions

Firstly, let's define the two terms. A 'population' is all the individuals or items under consideration in a statistical study. A 'sample' on the other hand, is a subset of the population and is the group on which measurements are actually taken. \[ \]
02

Analyze the Question

On careful reading, we can infer the following from the question: 'In statistical inference, measurements are made on a _______, and generalizations are made to a ________'.
03

Conclusion

Based on the above discussion, it is clear that in statistical inference, measurements are made on a 'sample', and generalizations are made to the 'population'. Thus, the correct answer is: 'Measurements are made on a sample, and generalizations are made to a population'.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Population
In statistics, the term "population" refers to the entire group of individuals or instances that you are interested in studying. This could be people, animals, plants, products, or any other category you can think of. For instance, if you were studying the average height of adult humans in a country, the entire adult population of that country would be your population of interest.

Populations are usually too large to measure entirely. This is where the concept of sampling comes into play. Instead of measuring every single member of the population, statisticians select a more manageable group, or 'sample', to make their measurements.
Sample
A sample is a subset of a population that is used in a study to represent the whole group. Imagine picking out a small group from a larger crowd to get an idea of the crowd's overall taste or behavior. This is precisely what a sample does. It reflects the characteristics of the population as closely as possible, allowing researchers to make inferences about the population using data from the sample.

When choosing a sample, it is crucial to ensure that it is representative of the population. This means it should, ideally, include all the diversity and variation found in the population. This involves important considerations such as random sampling and sample size.
Generalizations
After collecting data from a sample, statisticians perform various analyses. The goal of these analyses is to draw conclusions or make inferences about the broader population. This process is known as making generalizations. For example, if surveys show that 70% of a sample likes a particular product, statisticians may infer that a similar percentage of the entire population feels the same way.

The validity of these generalizations hinges on the quality of the sample. If the sample accurately mirrors the population, then the generalizations made are more likely to be accurate. However, if the sample is biased, the generalizations may be flawed. Therefore, statisticians must take great care in how they design studies to ensure reliable outcomes.

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Most popular questions from this chapter

When comparing two sample proportions with a two-sided alternative hypothesis, all other factors being equal, will you get a smaller p-value if the sample proportions are close together or if they are far apart? Explain.

Samuel Morse determined that the percentage of \(a\) 's in the English language in the 1800 s was \(8 \%\). A random sample of 600 letters from a current newspaper contained 60 a's. Using the \(0.10\) level of significance, test the hypothesis that the proportion of \(a\) 's in this modern newspaper is \(0.09\).

A proponent of a new proposition on a ballot wants to know whether the proposition is likely to pass. Suppose a poll is taken, and 580 out of 1000 randomly selected people support the proposition. Should the proponent use a hypothesis test or a confidence interval to answer this question? Explain. If it is a hypothesis test, state the hypotheses and find the test statistic, p-value, and conclusion. If a confidence interval is appropriate, find the approximate \(95 \%\) confidence interval. In both cases, assume that the necessary conditions have been met.

Incentives A psychologist is interested in testing whether offering students a financial incentive improves their video-game-playing skills. She collects data and performs a hypothesis test to test whether the probability of getting to the highest level of a video game is greater with a financial incentive than without. Her null hypothesis is that the probability of getting to this level is the same with or without a financial incentive. The alternative is that this probability is greater. She gets a p-value from her hypothesis test of \(0.003 .\) Which of the following is the best interpretation of the p-value? i. The p-value is the probability that financial incentives are not effective in this context. ii. The p-value is the probability of getting exactly the result obtained, assuming that financial incentives are \(n o t\) effective in this context. iii. The p-value is the probability of getting a result as extreme as or more extreme than the one obtained, assuming that financial incentives are not effective in this context. iv. The p-value is the probability of getting exactly the result obtained, assuming that financial incentives are effective in this context. \(\mathrm{y}\). The p-value is the probability of getting a result as extreme as or more extreme than the one obtained, assuming that financial incentives are effective in this context.

The null hypothesis on true/false tests is that the student is guessing, and the proportion of right answers is \(0.50 .\) A student taking a five-question true/false quiz gets 4 right out of 5 . She says that this shows that she knows the material, because the one-tailed p-value from the one-proportion \(z\) -test is \(0.090\), and she is using a significance level of \(0.10 .\) What is wrong with her approach?
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