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Chapter 11: Problem 3

In Problems 3鈥�8, determine whether the sampling is dependent or independent. Indicate whether the response variable is qualitative or quantitative. A sociologist wishes to compare the annual salaries of married couples in which both spouses work and determines each spouse鈥檚 annual salary.

Short Answer

Expert verified
The sampling is dependent, and the response variable is quantitative.

Step by step solution

01

- Understand the Sampling

Identify whether the sampling method involves one group or two separate groups. In this case, the sociologist is collecting data from married couples where both spouses work. Since data is collected from pairs (each married couple), this implies that the sampling is dependent.
02

- Determine the Response Variable

Identify what is being measured or compared. The sociologist is comparing the annual salaries of spouses. Annual salary is a measurable quantity that can be expressed numerically.
03

- Classify the Response Variable

Determine if the response variable is qualitative or quantitative. Quantitative data can be measured and expressed numerically (like height, weight, or salary). Qualitative data describes categories or qualities (like gender or eye color). In this case, annual salary is quantitative.

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Key Concepts

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

Dependent Sampling
In statistics, **dependent sampling** occurs when the data collected from one member of a pair or group is related to the data collected from another member of the same pair or group. This often happens when the focus is on matched pairs or naturally paired subjects. For example, in the exercise, the sociologist is studying married couples where both spouses work. Here, the data points (each spouse's annual salary) are naturally paired because they belong to the same household.

Dependent sampling is useful in studies where it's crucial to account for the relationship between data points. Examples include:
  • Before-and-after studies on the same subjects
  • Paired comparison studies (like taste tests)
  • Studies on twins or matched siblings
In these cases, the inherent relationship between the pairs helps understand the subject better by reducing variability caused by individual differences.
Quantitative Data
Quantitative data refers to numerical information that can be measured and expressed mathematically. It is distinct from qualitative data, which describes characteristics or attributes that cannot be measured with numbers. In the given exercise, the sociologist is collecting data on **annual salaries**, which is a classic example of quantitative data.

Quantitative data can be analyzed using various statistical methods to find patterns, trends, and relationships. Here's why quantitative data is important:
  • Helps in comparing and contrasting different data points numerically
  • Allows application of mathematical models to predict future trends
  • Facilitates precise measurement and hypothesis testing
Some common examples of quantitative data include height, weight, age, test scores, and income.
Response Variable Identification
Identifying the **response variable** in a study is essential as it is the primary outcome that is measured or observed. The response variable, also known as the dependent variable, is what researchers are interested in explaining or predicting. In the exercise provided, the response variable is the **annual salary** of both spouses in a household where both work.

To correctly identify the response variable, follow these steps:
  • Determine what is being measured or compared
  • Understand the context of the study and what the research aims to focus on
  • Ensure clarity by distinguishing the response variable from other types of variables, such as explanatory variables (independent variables).
The response variable is central to the analysis because it will demonstrate the effect or the result of the study, allowing the researcher to draw meaningful conclusions.

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Most popular questions from this chapter

A Secchi disk is an 8 -inch-diameter weighted disk that is painted black and white and attached to a rope. The disk is lowered into water and the depth (in inches) at which it is no longer visible is recorded. The measurement is an indication of water clarity. An environmental biologist interested in determining whether the water clarity of the lake at Joliet Junior College is improving takes measurements at the same location on eight dates during the course of a year and repeats the measurements on the same dates five years later. She obtains the following results: $$ \begin{array}{lcccccccc} \text { Observation } & \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} \\ \text { Date } & \mathbf{5 / 1 1} & \mathbf{6 / 7} & \mathbf{6 / 2 4} & \mathbf{7 / 8} & \mathbf{7 / 2 7} & \mathbf{8 / 3 1} & 9 / 30 & \mathbf{1 0 / 1 2} \\ \hline \begin{array}{l} \text { Initial } \\ \text { depth, } X_{i} \end{array} & 38 & 58 & 65 & 74 & 56 & 36 & 56 & 52 \\ \hline \begin{array}{l} \text { Depth five } \\ \text { years later, } Y_{i} \end{array} & 52 & 60 & 72 & 72 & 54 & 48 & 58 & 60 \\ \hline \end{array} $$ (a) Why is it important to take the measurements on the same date? (b) Does the evidence suggest that the clarity of the lake is improving at the \(\alpha=0.05\) level of significance? Note: A normal probability plot and boxplot of the data indicate that the differences are approximately normally distributed with no outliers. (c) Draw a boxplot of the differenced data. Does this visual evidence support the results obtained in part (b)?

For each study, explain which statistical procedure (estimating a single proportion; estimating a single mean; hypothesis test for a single proportion; hypothesis test for a single mean; hypothesis test or estimation of two proportions, hypothesis test or estimation of two means, dependent or independent) would most likely be used for the research objective given. Assume all model requirements for conducting the appropriate procedure have been satisfied. Does Marriott Courtyard charge more than Holiday Inn Express for a one-night stay?

Researcher Seth B. Young measured the walking speed of travelers in San Francisco International Airport and Cleveland Hopkins International Airport. The standard deviation speed of the 35 travelers who were departing was 53 feet per minute. The standard deviation speed of the 35 travelers who were arriving was 34 feet per minute. Assuming walking speed is normally distributed, does the evidence suggest the standard deviation walking speed is different between the two groups? Use the \(\alpha=0.05\) level of significance.

Aluminum Bottles The aluminum bottle, first introduced in 1991 by CCL Container for mainly personal and household items such as lotions, has become popular with beverage manufacturers. Besides being lightweight and requiring less packaging, the aluminum bottle is reported to cool faster and stay cold longer than typical glass bottles. A small brewery tests this claim and obtains the following information regarding the time (in minutes) required to chill a bottle of beer from room temperature \(\left(75^{\circ} \mathrm{F}\right)\) to serving temperature \(\left(45^{\circ} \mathrm{F}\right) .\) Construct and interpret a \(90 \%\) confidence interval for the mean difference in cooling time for clear glass versus aluminum. $$ \begin{array}{lcc} & \text { Clear Glass } & \text { Aluminum } \\ \hline \text { Sample size } & 42 & 35 \\ \hline \begin{array}{l} \text { Mean time to } \\ \text { chill (minutes) } \end{array} & 133.8 & 92.4 \\ \hline \begin{array}{l} \text { Sample standard } \\ \text { deviation (minutes) } \end{array} & 9.9 & 7.3 \\ \hline \end{array} $$

MythBusters In a MythBusters episode, the question was asked,"Which is better? A four-way stop or a roundabout?" "Better" was determined based on determining the number of vehicles that travel through the four-way stop over a 5 -minute interval of time. Suppose the folks at MythBusters conducted this experiment 15 times for each intersection design. (a) What is the variable of interest in this study? Is it qualitative or quantitative? (b) Explain why the data might be analyzed by comparing two independent means. Include in this explanation what the null and alternative hypotheses would be and what each mean represents (c) A potential improvement on the experimental design might be to identify 15 groups of different drivers and ask each group to drive through each intersection design for the 5 - minute time interval. Explain why this is a better design and explain the role randomization would play. What would be the null and alternative hypotheses here and how would the mean be computed?
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