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Chapter 7: Problem 13

Suppose that a study was carried out in which each person in a random sample of students at a particular college was asked how much money he or she spent on textbooks for the current semester. Would you use these data to estimate a population mean or to estimate a population proportion? How did you decide?

Short Answer

Expert verified
In this study, we would use the data collected from the random sample of students to estimate the population mean for the amount of money spent on textbooks for the current semester at the college. This decision is based on the fact that the variable of interest is a quantitative (numerical) variable representing the amount of money spent on textbooks, which is more suitable for estimating a population mean rather than a population proportion.

Step by step solution

01

Identify the Variable of Interest

The variable of interest in this study is the amount of money each student spent on textbooks for the current semester.
02

Determine the Type of Variable

The variable in question represents the amount of money spent on textbooks, which is a quantitative (numerical) variable. This is because spending on textbooks can take on a range of numerical values.
03

Assessing Population Mean or Population Proportion

A population mean is the average value of a quantitative variable in a population, whereas a population proportion represents the fraction of items in a specific category within a population, which is more suitable for a categorical variable. Since the variable in this study is quantitative (amount of money spent), we are estimating a population mean, not a population proportion.
04

Conclusion

In this study, we would use the data collected from the random sample of students to estimate the population mean for the amount of money spent on textbooks for the current semester at the college. We made this decision because the variable of interest is a quantitative variable (amount spent) rather than a categorical variable (proportion of students in a certain category).

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Key Concepts

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

Quantitative Variable
In the study, the variable of interest is how much money students spend on textbooks for the current semester. This type of variable is quantitative because it deals with numbers and amounts. Quantitative variables are capable of taking a wide range of numerical values and can be measured using units like dollars, meters, or pounds.
Unlike qualitative or categorical variables, which categorize or label groups, quantitative variables tell us about the magnitude or amount of something.
Common examples include height, weight, or test scores, where values can vary with each instance.
Sampling Methods
When conducting a study, choosing a sampling method is crucial to obtaining accurate and representative results. The original exercise describes the use of a random sample of students. Random sampling is a method where each member of the population has an equal chance of being selected.
This approach helps ensure that the sample represents the broader population, reducing bias in the results. Other sampling methods include stratified sampling, systematic sampling, and cluster sampling.
Each method has its own advantages and is chosen based on the study's goals and the population's characteristics. For studies aiming to estimate averages, like textbook costs, ensuring randomness in sampling is essential to obtaining reliable and valid estimates of population means.
Data Analysis
Data analysis involves examining, cleaning, transforming, and modeling data to extract useful information and support decision-making. In the context of estimating the population mean for money spent on textbooks, data analysis would involve calculating the average expenditure from the sample data.
We'd start by organizing our data and checking it for accuracy. Then, by computing the mean, measures of variability such as the standard deviation can also be calculated. These statistics give insights into the data's spread and the typical amount spent by students.
Additionally, data visualization techniques, such as histograms, can help us better understand spending patterns. All these steps ensure that the conclusions drawn are credible and informative.

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Most popular questions from this chapter

Refer to the instructions prior to Exercise 7.25. Can moving their hands help children learn math? This question was investigated by the authors of the paper "Gesturing Gives Children New Ideas about Math" (Psychological Science [2009]: 267-272). A study was conducted to compare two different methods for teaching children how to solve math problems of the form \(3+2+8=8\). One method involved having students point to the \(3+2\) on the left side of the equal sign with one hand and then point to the blank on the right side of the equal sign before filling in the blank to complete the equation. The other method did not involve using these hand gestures. To compare the two methods, 128 children were assigned at random to one of the methods. Each child then took a test with six problems, and the number correct was determined for each child. The researchers planned to see if the resulting data supported the theory that the mean number correct for children who use hand gestures is higher than the mean number correct for children who do not use hand gestures.

Refer to the instructions prior to this exercise. A study of fast-food intake is described in the paper "What People Buy From Fast-Food Restaurants" (Obesity [2009]: 1369-1374). Adult customers at three hamburger chains (McDonald's, Burger King, and Wendy's) at lunchtime in New York City were approached as they entered the restaurant and were asked to provide their receipt when exiting. The receipts were then used to determine what was purchased and the number of calories consumed. The sample mean number of calories consumed was \(857,\) and the sample standard deviation was 677 . This information was used to learn about the mean number of calories consumed in a New York fast-food lunch.

Consider the population that consists of all employees of a large computer manufacturer. a. Give an example of a question about this population that could be answered by collecting data and using it to estimate a population characteristic. b. Give an example of a question about this population that could be answered by collecting data and using it to test a claim about this population.

The article "More Communities Banning 'Television on a Stick"" (USA TODAY, March 23,2010 ) describes an ongoing controversy over the distraction caused by digital billboards along highways. One study mentioned in the newspaper article is described in "Effects of Advertising Billboards During Simulated Driving" (Applied Ergonomics [2010]: 1-8). In this study, 48 people made a \(9 \mathrm{~km}\) drive in a driving simulator. Drivers were instructed to change lanes according to roadside lane change signs. Some of the lane changes occurred near digital billboards. What was displayed on the digital billboard changed once during the time that the billboard was visible by the driver to simulate the changing digital billboards that appear along highways. Data from this study supported the theory that the time required to respond to road signs was greater when digital billboards were present. Is the inference made one that involves estimation or one that involves hypothesis testing? (Hint: See Example 7.1.)

Refer to the instructions prior to Exercise 7.25. "Spending on Favorite Drinks" is the title of a graph that appeared as part of the USA Snapshot series in the newspaper USA TODAY (November 4,2016 ). This graph summarized data from a survey of adult Americans. Each survey participant reported how much they spend each month on various beverages, such as coffee, wine, and beer. These data were used to learn about the mean monthly spending for each type of beverage.
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