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Chapter 7: Problem 6

Two symbols are used for the standard deviation: \(\sigma\) and s. a. Which represents a parameter, and which represents a statistic? b. To estimate the commute time for all students at a college, 100 students are asked to report their commute times in minutes. The standard deviation for these 100 commute times was \(13.9\) minutes. Is this standard deviation \(\sigma\) or s?

Short Answer

Expert verified
a. \(\sigma\) represents a parameter, and s represents a statistic. b. This standard deviation is denoted as s because the data is sourced from a sample, not an entire population.

Step by step solution

01

Identify Parameter and Statistic

In statistics: \n- A parameter is a value that refers to a characteristic of a population. The symbol \(\sigma\) is used to denote the standard deviation of a population; therefore, \(\sigma\) represents a parameter. \n- A statistic is a value that refers to a characteristic of a sample. The symbol s is used to denote the standard deviation of a sample; therefore, s represents a statistic.
02

Identify Standard Deviation

100 students at a college were asked to report their commute times. This group of students serves as a sample representing the entire student population at the college. The standard deviation of these students' commute times was 13.9 minutes. In this case, the standard deviation represents a statistic of a sample, therefore it is denoted as s.

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Key Concepts

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

Parameter vs Statistic
In the world of statistics, understanding the difference between a parameter and a statistic is key to analyzing data correctly.

A **parameter** is a value that describes a characteristic of an entire population. This could be the average age of everyone in a country or the standard deviation, which measures how spread out numbers are, of an entire population. In mathematical notation, parameters often use Greek letters like \( \sigma \) (sigma) for standard deviation.

On the other hand, a **statistic** describes a characteristic of a sample, which is a subset of the population. Statistics use English letters; for example, \( s \) for standard deviation. This could be the mean commute time of a group of 100 students, which is used to estimate the average for all students in a college.

  • A **parameter** is fixed, but it may be unknown since it refers to the whole population.
  • A **statistic** can vary from sample to sample, providing an estimate of a population parameter.
This distinction helps in deciding whether to extend findings from a sample to a larger population.
Population vs Sample
In statistics, it's important to distinguish between a population and a sample. This distinction affects how we interpret data and calculate measures like the standard deviation.

A **population** refers to the entire group that you are interested in studying. For example, if you are researching all students at a college, the population includes every single student there.

In practice, examining every member of a population isn't always possible or practical. This is where a **sample** comes in. A sample is a smaller, manageable group chosen from the population. For instance, surveying 100 students about their commute times can serve as a representation of the whole student body.

  • **Population:** the complete group you're examining; often large and comprehensive.
  • **Sample:** a subset drawn from the population; more practical for analysis.
Understanding whether you are dealing with a population or a sample is crucial for selecting the right statistical formulas and correctly interpreting results.
Statistical Notation
Statistical notation is a language of symbols used to represent data, making it easier to perform and communicate statistical analyses effectively. Familiarizing yourself with these symbols enhances clarity in understanding statistical concepts.

  • **\( \sigma \) (sigma):** Represents the standard deviation of an entire population, and is considered a parameter.
  • **\( s \) :** Indicates the standard deviation of a sample, and is viewed as a statistic.
Additionally, other notations exist to represent fundamental concepts:
  • **\( \mu \) (mu):** Denotes the mean of a population.
  • **\( \bar{x} \) :** Represents the mean of a sample.
These notations are standard in statistics and help simplify communication across statistical work. Consistency in use ensures efficiency in understanding and conveying statistical outcomes.

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Most popular questions from this chapter

The Centers for Disease Control and Prevention (CDC) conducts an annual Youth Risk Behavior Survey, surveying over 15,000 high school students. The 2015 survey reported that, while cigarette use among high school youth had declined to its lowest levels, \(24 \%\) of those surveyed reported using e-cigarettes. Identify the sample and population. Is the value \(24 \%\) a parameter or a statistic? What symbol would we use for this value?

The website scholarshipstats.com collected data on all 5341 NCAA basketball players for the 2017 season and found a mean height of 77 inches. Is the number 77 a parameter or a statistic? Also identify the population and explain your choice.

According to data released in 2016 , \(69 \%\) of students in the United States enroll in college directly after high school graduation. Suppose a sample of 200 recent high school graduates is randomly selected. After verifying the conditions for the Central Limit Theorem are met, find the probability that at most \(65 \%\) enrolled in college directly after high school graduation. (Source: nces.ed.gov)

The Perry Preschool Project was created in the early \(1960 \mathrm{~s}\) by David Weikart in Ypsilanti, Michigan. In this project, 123 African American children were randomly assigned to one of two groups: One group enrolled in the Perry Preschool, and the other group did not. Follow-up studies were done for decades. One research question was whether attendance at preschool had an effect on high school graduation. The table shows whether the students graduated from regular high school or not and includes both boys and girls (Schweinhart et al. 2005 ). Find a \(95 \%\) confidence interval for the difference in proportions, and interpret it. $$ \begin{array}{|lcc|} \hline & \text { Preschool } & \text { No Preschool } \\ \hline \text { Grad HS } & 37 & 29 \\ \hline \text { No Grad HS } & 20 & 35 \\ \hline \end{array} $$

Statistics student Hector Porath wanted to find out whether gender and the use of turn signals when driving were independent. He made notes when driving in his truck for several weeks. He noted the gender of each person that he observed and whether he or she used the turn signal when turning or changing lanes. (In his state, the law says that you must use your turn signal when changing lanes, as well as when turning.) The data he collected are shown in the table. $$ \begin{array}{|lcc|} \hline & \text { Men } & \text { Women } \\ \hline \text { Turn signal } & 585 & 452 \\ \hline \text { No signal } & 351 & 155 \\ \hline & 936 & 607 \\ \hline \end{array} $$ a. What percentage of men used turn signals, and what percentage of women used them? b. Assuming the conditions are met (although admittedly this was not a random selection), find a \(95 \%\) confidence interval for the difference in percentages. State whether the interval captures 0, and explain whether this provides evidence that the proportions of men and women who use turn signals differ in the population. c. Another student collected similar data with a smaller sample size: $$ \begin{array}{|l|l|l|} \hline & \text { Men } & \text { Women } \\ \hline \text { Turn Signal } & 59 & 45 \\ \hline \text { No Signal } & 35 & 16 \\ \hline & 94 & 61 \\ \hline \end{array} $$ First find the percentage of men and the percentage of women who used turn signals, and then, assuming the conditions are met, find a \(95 \%\) confidence interval for the difference in percentages. State whether the interval captures 0 , and explain whether this provides evidence that the percentage of men who use turn signals differs from the percentage of women who do so. d. Are the conclusions in parts \(\mathrm{b}\) and \(\mathrm{c}\) different? Explain.
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