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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

Historically (from about 2001 to 2014 ), \(57 \%\) of Americans believed that global warming is caused by human activities. A March 2017 Gallup poll of a random sample of 1018 Americans found that 692 believed that global warming is caused by human activities. a. What percentage of the sample believed global warming was caused by human activities? b. Test the hypothesis that the proportion of Americans who believe global warming is caused by human activities has changed from the historical value of \(57 \%\). Use a significance level of \(0.01\). c. Choose the correct interpretation: i. In 2017 , the percentage of Americans who believe global warming is caused by human activities is not significantly different from \(57 \%\). ii. In 2017 , the percentage of Americans who believe global warming is caused by human activities has changed from the historical level of \(57 \%\).

A magazine advertisement claims that wearing a magnetized bracelet will reduce arthritis pain in those who suffer from arthritis. A medical researcher tests this claim with 233 arthritis sufferers randomly assigned either to wear a magnetized bracelet or to wear a placebo bracelet. The researcher records the proportion of each group who report relief from arthritis pain after 6 weeks. After analyzing the data, he fails to reject the null hypothesis. Which of the following are valid interpretations of his findings? There may be more than one correct answer. a. The magnetized bracelets are not effective at reducing arthritis pain. b. There's insufficient evidence that the magnetized bracelets are effective at reducing arthritis pain. c. The magnetized bracelets had exactly the same effect as the placebo in reducing arthritis pain. d. There were no statistically significant differences between the magnetized bracelets and the placebos in reducing arthritis pain.

The label on a can of mixed nuts says that the mixture contains \(40 \%\) peanuts. After opening a can of nuts and finding 22 peanuts in a can of 50 nuts, a consumer thinks the proportion of peanuts in the mixture differs from \(40 \%\). The consumer writes these hypotheses: \(\mathrm{H}_{0}: \mathrm{p} \neq 0.40\) and \(\mathrm{H}_{\mathrm{a}}: \mathrm{p}=0.44\) where \(p\) represents the proportion of peanuts in all cans of mixed nuts from this company. Are these hypotheses written correctly? Correct any mistakes as needed.

No-Carb Diet A weight-loss diet claims that it causes weight loss by eliminating carbohydrates (breads and starches) from the diet. To test this claim, researchers randomly assign overweight subjects to two groups. Both groups eat the same amount of calories, but one group eats almost no carbs, and the other group includes carbs in their meals. After 2 months, the researchers test the claim that the no-carb diet is better than the usual diet. They record the proportion of each group that lost more than \(5 \%\) of their initial weight. They then announce that they failed to reject the null hypothesis. Which of the following are valid interpretations of the researchers' findings? a. There were no significant differences in effectiveness between the no-carb diet and the carb diet. b. The no-carb diet and the carb diet were equally effective. c. The researchers did not see enough evidence to conclude that the no-carb diet was more effective. d. The no-carb diet was less effective than the carb diet.

If we reject the null hypothesis, can we claim to have proved that the null hypothesis is false? Why or why not?
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