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Chapter 9: Problem 2

For what type of variable does it makes sense to construct a confidence interval about a population mean?

Short Answer

Expert verified
Constructing a confidence interval about a population mean makes sense for quantitative variables.

Step by step solution

01

Understanding Variables

Identify the different types of variables: qualitative and quantitative. Qualitative variables represent categories or groups, while quantitative variables represent numerical values that can be measured or counted.
02

Focus on Quantitative Variables

Determine whether the confidence interval about a population mean applies to quantitative variables. Confidence intervals are used to estimate population parameters, such as the mean, based on sample data.
03

Consistency with Numerical Data

Recognize that constructing a confidence interval about a population mean requires numerical data. This is because the mean is a central tendency measurement that summarizes numerical data, and constructing an interval means estimating this value for a larger population from a sample.
04

Conclusion

It makes sense to construct a confidence interval about a population mean for quantitative variables, as these variables allow for the calculation of mean and subsequent statistical analysis.

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Key Concepts

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

Quantitative Variables
Quantitative variables are types of variables that represent numerical values. These values can be counted or measured.
For example, heights, weights, and ages are all quantitative variables. They allow us to perform mathematical operations like addition, subtraction, and averaging.
When we deal with quantitative data, we can summarize it using measures like the mean (average). The mean can provide a good sense of the 'center' or typical value in our data set.
Quantitative variables are crucial when constructing a confidence interval about a population mean since confidence intervals give a range of values that likely include the population mean based on sample data.
Population Mean
The population mean is the average of a set of numerical values for an entire population.
It is a fundamental concept in statistics, as it represents a central point in the data distribution. In reality, it's often impractical to measure the whole population.
So, we collect a sample and use it to estimate the population mean. When we calculate the mean from a sample, it helps us understand what the population mean might be.
To account for the variability between different samples, we use confidence intervals. These intervals provide a range within which we expect the population mean to fall, given a certain level of confidence (e.g., 95%).
Numerical Data
Numerical data refers to data expressed in numbers, making it suitable for statistical analysis and mathematical operations. There are two main types of numerical data:
1. Discrete data: These are counts that take on distinct, separate values (e.g., the number of students in a class).
2. Continuous data: These are measurements that can take any value within a range (e.g., weight, height).
In order to construct a confidence interval about a population mean, numerical data is essential. Without numerical data, you cannot compute measures of central tendency (like the mean) or variability.
The strength of numerical data lies in its ability to provide accurate and meaningful insights through techniques like confidence intervals, which help us make informed conclusions about the larger population from which the sample is drawn.

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Most popular questions from this chapter

The 114 th Congress of the United States of America has 535 members, of which 108 are women. An alien lands near the U.S. Capitol and treats members of Congress as a random sample of the human race. He reports to his superiors that a \(95 \%\) confidence interval for the proportion of the human race that is female has a lower bound of 0.168 and an upper bound of 0.236. What is wrong with the alien's approach to estimating the proportion of the human race that is female?

A survey of 2306 adult Americans aged 18 and older conducted by Harris Interactive found that 417 have donated blood in the past two years. (a) Obtain a point estimate for the population proportion of adult Americans aged 18 and older who have donated blood in the past two years. (b) Verify that the requirements for constructing a confidence interval about \(p\) are satisfied. (c) Construct a \(90 \%\) confidence interval for the population proportion of adult Americans who have donated blood in the past two years. (d) Interpret the interval.

56\. Population A has standard deviation \(\sigma_{\mathrm{A}}=5,\) and population \(\mathrm{B}\) has standard deviation \(\sigma_{\mathrm{B}}=10 .\) How many times larger than Population A's sample size does Population B's need to be to estimate \(\mu\) with the same margin of error? [Hint: Compute \(\left.n_{\mathrm{B}} / n_{\mathrm{A}}\right]\).

You Explain It! Hours Worked In a survey conducted by the Gallup Organization, 1100 adult Americans were asked how many hours they worked in the previous week. Based on the results, a \(95 \%\) confidence interval for mean number of hours worked was lower bound: 42.7 and upper bound: \(44.5 .\) Which of the following represents a reasonable interpretation of the result? For those that are not reasonable, explain the flaw. (a) There is a \(95 \%\) probability the mean number of hours worked by adult Americans in the previous week was between 42.7 hours and 44.5 hours. (b) We are \(95 \%\) confident that the mean number of hours worked by adult Americans in the previous week was between 42.7 hours and 44.5 hours. (c) \(95 \%\) of adult Americans worked between 42.7 hours and 44.5 hours last week. (d) We are \(95 \%\) confident that the mean number of hours worked by adults in Idaho in the previous week was between 42.7 hours and 44.5 hours.

A \(90 \%\) confidence interval for the number of hours that full-time college students sleep during a weekday is lower bound: 7.8 hours and upper bound: 8.8 hours. Which of the following represents a reasonable interpretation of the result? For those that are not reasonable, explain the flaw. (a) \(90 \%\) of full-time college students sleep between 7.8 hours and 8.8 hours. (b) We are \(90 \%\) confident that the mean number of hours of sleep that full- time college students get any day of the week is between 7.8 hours and 8.8 hours. (c) There is a \(90 \%\) probability that the mean hours of sleep that full-time college students get during a weekday is between 7.8 hours and 8.8 hours. (d) We are \(90 \%\) confident that the mean hours of sleep that fulltime college students get during a weekday is between 7.8 hours and 8.8 hours.
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