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Chapter 14: Problem 64

Informal surveys. In everyday life we are constantly involved in activities that can be described as informal surveys, often without even realizing it. Here are some examples. (i) Al gets up in the morning and wants to know what kind of day it is going to be, so he peeks out the window. He doesn't see any dark clouds, so he figures it's not going to rain. (ii) Betty takes a sip from a cup of coffee and burns her lips. She concludes that the coffee is too hot and decides to add a tad of cold water to it. (iii) Carla got her first Math 101 exam back with a C grade on it. The students sitting on each side of her also received C grades. She concludes that the entire Math 101 class received a \(C\) on the first exam. For each of the preceding examples, (a) describe the population. (b) discuss whether the sample is random or not. (c) discuss the validity of the conclusions drawn. (There is no right or wrong answer to this question, but you should be able to make a reasonable case for your position.)

Short Answer

Expert verified
Al's population is the day's weather, his sample is not random, and his conclusion validity can be debated. Betty's population is the cup's coffee temperature, her sample is not random, and her conclusion's validity can also be argued. Carla's population is Math 101's grades, her sample is not random, and her conclusion is probably not valid, as it's based on a small, non-random sample.

Step by step solution

01

Analyzing Al's scenario

(a) The population is the weather condition of the day. (b) The sample is not random as it's taken through a single observation of looking out the window. (c) The validity of the conclusion could be challenged since weather conditions may change during the day and also, regional weather patterns influence the validity of his conclusion.
02

Analyzing Betty's scenario

(a) The population in this case is the overall temperature of the coffee in the cup. (b) The sample here is not random since it's based on her first sip, it may not be representative of the entire coffee's temperature. (c) The conclusion she made depends on her tolerance to hot substances which varies from person to person, hence, generalizing it might not be entirely valid.
03

Analyzing Carla's scenario

(a) The population for Carla's situation is the grades of the entire Math 101 class. (b) The sample was not random as it included only her score and the scores of the two students next to her. (c) Her conclusion might not be valid since three students' grades are not an adequate representation of the entire class.

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Key Concepts

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

Population Samples
In the context of surveys, a population refers to the entire set of subjects or instances that you want to study or gather information about. In informal surveys, identifying the correct population is crucial because it helps to ensure that the survey results are relevant and applicable to the group you're interested in. Let's delve into the examples shared earlier:

- **Al's Scenario**: The population is the weather conditions of the entire day. Al's interest is not just in the current weather, but potentially how it may affect his day. - **Betty's Scenario**: The population involves the total temperature of the contents of the coffee cup. A quick sip only samples a part of this population. - **Carla's Scenario**: The grades of all students in the Math 101 class form the population. Carla's conclusion about the class relying on just a few grades is an oversimplification.

Understanding the population helps in recognizing the scope of your survey and aligning your questions to address that full set, not just a limited perspective. By clearly defining the population, you ensure that your survey isn't inadvertently skewed or misrepresentative of broader truths.
Random Sampling
Random sampling is a technique where every member of a population has an equal chance of being chosen. This method is vital for gathering unbiased, representative samples that can accurately reflect the population at large. It minimizes bias and improves the reliability of survey conclusions.

In the given examples, none of the samples utilized is truly random:

- **Al's Weather Observation**: Peeking out the window at a single moment isn鈥檛 random; it lacks variance. To improve, Al could check weather forecasts or take observations at various times. - **Betty's Coffee Temperature Check**: Her sampling is just a quick sip, which isn鈥檛 random. Stirring the coffee could provide a more consistent temperature sample. - **Carla's Grade Conclusion**: The sample is restricted to her immediate neighbors' grades. Ideally, she would need to check a random selection of grades from the entire class.

Random sampling helps in achieving a representative subset that reflects the larger population, providing a sound basis for drawing conclusions and making decisions.
Validity of Conclusions
The validity of conclusions refers to how well the drawn conclusions reflect the true state of the population being studied. It takes into account the sampling method and the adequacy of the sample size. Conclusions have validity if they appropriately infer from the sample to the population.

In each of our examples, validity issues arise due to the nature of the sampling:

- **Al鈥檚 Weather Prediction**: Since his method wasn't random and didn鈥檛 consider changes, the validity is questionable. More accurate means like forecasts or multiple observations would enhance validity.
- **Betty鈥檚 Temperature Judgement**: Her conclusion on the coffee鈥檚 temperature might not hold if she sampled only a particular spot. A homogenous mix would have provided a valid assessment.
- **Carla's Grade Analysis**: With only three grades considered, the sample is insufficient to make a valid conclusion about all the students. Surveying a more representative portion of the class would offer validity.

Assessing and ensuring the validity of survey conclusions involves carefully considering sample randomness, size, and method to ensure that the conclusions are a fair reflection of the population.

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Refer to the following story: The 1250 students at Eureka High School are having an election for Homecoming King. The candidates are Tomlinson (captain of the football team), Garcia (class president), and Marsalis (member of the marching band). At the football game a week before the election, a pre- election poll was taken of students as they entered the stadium gates. Of the students who attended the game, 203 planned to vote for Tomlinson, 42 planned to vote for Garcia, and 105 planned to vote for Marsalis. Name the sampling method used for this survey.

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Refer to the following story: The manufacturer of a new vitamin (vitamin \(X\) ) decides to sponsor a study to determine the vitamin's effectiveness in curing the common cold. Five hundred college students having a cold were recruited from colleges in the San Diego area and were paid to participate as subjects in this study. The subjects were each given two tablets of vitamin \(X\) a day. Based on information provided by the subjects themselves, 457 of the 500 subjects were cured of their colds within 3 days. (The average number of days a cold lasts is 4.87 days.) As a result of this study, the manufacturer launched an advertising campaign based on the claim that "vitamin \(X\) is more than \(90 \%\) effective in curing the common cold." (a) Describe as specifically as you can the target population for the study. (b) Describe the sampling frame for the study. (c) Describe the sample used for the study.
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