91Ó°ÊÓ

Conditions For each situation described below, identify the population and the sample, explain what \(p\) and \(\hat{p}\) represent, and tell whether the methods of this chapter can be used to create a confidence interval. a. Police set up an auto checkpoint at which drivers are stopped and their cars inspected for safety problems. They find that 14 of the 134 cars stopped have at least one safety violation. They want to estimate the percentage of all cars that may be unsafe. b. A TV talk show asks viewers to register their opinions on prayer in schools by logging on to a website. Of the 602 people who voted, 488 favored prayer in schools. We want to estimate the level of support among the general public. c. A school is considering requiring students to wear uniforms. The PTA surveys parent opinion by sending a questionnaire home with all 1245 students; 380 surveys are returned, with 228 families in favor of the change. d. A college admits 1632 freshmen one year, and four years later, 1388 of them graduate on time. The college wants to estimate the percentage of all their freshman enrollees who graduate on time.

Step by step solution

01

Understanding Situation 'a'

Population: All cars. Sample: The 134 cars stopped at the checkpoint. \(p\): The true percentage of all cars that may be unsafe. \(\hat{p}\): The percentage of cars stopped at the auto-checkpoint that are unsafe i.e., \(\frac{14}{134}\). The methods of this chapter can be used to create a confidence interval because the sample can be assumed to be random.

02

Understanding Situation 'b'

Population: All the viewers of the TV talk show. Sample: The 602 people who voted. \(p\): The true support level for prayer in schools among the general public. \(\hat{p}\): The percentage of people who voted in favor of prayer in schools i.e., \(\frac{488}{602}\). The methods of this chapter cannot be used to create a confidence interval because the sample is biased (only includes viewers who were motivated to vote).

03

Understanding Situation 'c'

Population: All parents of students at the school. Sample: The 380 parents who returned the survey. \(p\): The true percentage of all parents who support the uniform change. \(\hat{p}\): The percentage of parents who returned the survey and support the change i.e., \(\frac{228}{380}\). The methods of this chapter can be used to create a confidence interval because the sample can be assumed to be random.

04

Understanding Situation 'd'

Population: All freshman enrollees at the college. Sample: The 1632 freshmen admitted that year. \(p\): The true percentage of all freshmen enrollees who graduate on time. \(\hat{p}\): The percentage of freshmen admitted that year who graduated on time i.e., \(\frac{1388}{1632}\). The methods of this chapter can be used to create a confidence interval because the sample can be assumed to be random.

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Key Concepts

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

Population and Sample

In statistics, understanding the difference between a population and a sample is crucial. The **population** refers to the entire group you're interested in studying. For example, in situation 'a', the population is all cars. The **sample** is a smaller group selected from the population. For instance, the 134 cars stopped at a checkpoint serve as the sample. There are a few key reasons why we use samples:

• Observing the entire population can be impractical or impossible.
• Samples save time and resources.
• When chosen correctly, a sample can provide insights about the population.

Identifying the population and sample helps you apply your findings to a larger group, making statistical inferences possible.

Proportion Estimation

Proportion estimation allows statisticians to predict percentages for an entire population based on sample data. Here, we focus on two key symbols: **\(p\)** and **\(\hat{p}\)**.

• \(p\): Represents the true proportion in the population, which is often unknown and what we're estimating.
• \(\hat{p}\): The sample proportion, calculated from sample data, used to estimate \(p\).

In the original exercise:

• For situation 'a', \(\hat{p}\) is calculated as \(\frac{14}{134}\).
• This provides an estimate of the percentage of cars that may be unsafe.

By analyzing \(\hat{p}\), you draw conclusions about \(p\), which helps in decision-making processes.

Random Sampling

Random sampling is a technique that ensures each member of the population has an equal chance of being chosen. This method is key for reducing bias in your results. In the exercise:

• Situation 'a' assumes random sampling as all cars have the same chance of being stopped.
• This assumption allows the use of confidence interval methods to estimate a population parameter.

Random sampling is vital because:

• It provides a representative sample leading to valid inferences.
• It helps maintain objectivity in the selection process.

If random sampling isn't possible, any conclusions drawn can be unreliable, as seen in situation 'b' where the sample isn't random.

Survey Methodology

Survey methodology involves selecting, collecting, and analyzing data to understand the larger population's traits. Consider situation 'c' and 'd', where specific survey methods are applied:

• In 'c', surveys are sent home, and a portion is returned.
• In 'd', data from an entire admitted class is used for analysis.

Good survey methodology should ensure:

• The sample size is large enough to be statistically significant.
• The method of data collection doesn't introduce bias.
• The questions are clear and unbiased.

Sound survey methods lead to reliable data, making your findings applicable to the broader population.