1.E: Introduction to Statistics (Exercises)
- Page ID
- 1093
These are homework exercises to accompany the Textmap created for "Introductory Statistics" by Shafer and Zhang.
1.1: Basic Definitions and Concepts
Exercises
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Explain what is meant by the term population.
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Explain what is meant by the term sample.
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Explain how a sample differs from a population.
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Explain what is meant by the term sample data.
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Explain what a parameter is.
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Explain what a statistic is.
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Give an example of a population and two different characteristics that may be of interest.
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Describe the difference between descriptive statistics and inferential statistics. Illustrate with an example.
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Identify each of the following data sets as either a population or a sample:
- The grade point averages (GPAs) of all students at a college.
- The GPAs of a randomly selected group of students on a college campus.
- The ages of the nine Supreme Court Justices of the United States on \(\text{January}\; 1,\; 1842\).
- The gender of every second customer who enters a movie theater.
- The lengths of Atlantic croakers caught on a fishing trip to the beach.
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Identify the following measures as either quantitative or qualitative:
- The \(30\) high-temperature readings of the last \(30\) days.
- The scores of \(40\) students on an English test.
- The blood types of \(120\) teachers in a middle school.
- The last four digits of social security numbers of all students in a class.
- The numbers on the jerseys of \(53\) football players on a team.
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Identify the following measures as either quantitative or qualitative:
- The genders of the first \(40\) newborns in a hospital one year.
- The natural hair color of \(20\) randomly selected fashion models.
- The ages of \(20\) randomly selected fashion models.
- The fuel economy in miles per gallon of \(20\) new cars purchased last month.
- The political affiliation of \(500\) randomly selected voters.
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A researcher wishes to estimate the average amount spent per person by visitors to a theme park. He takes a random sample of forty visitors and obtains an average of \(\$28\) per person.
- What is the population of interest?
- What is the parameter of interest?
- Based on this sample, do we know the average amount spent per person by visitors to the park? Explain fully.
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A researcher wishes to estimate the average weight of newborns in South America in the last five years. He takes a random sample of \(235\) newborns and obtains an average of \(3.27\) kilograms.
- What is the population of interest?
- What is the parameter of interest?
- Based on this sample, do we know the average weight of newborns in South America? Explain fully.
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A researcher wishes to estimate the proportion of all adults who own a cell phone. He takes a random sample of \(1,572\) adults; \(1,298\) of them own a cell phone, hence \(1298/1572 \approx 0.83\) or about \(83\%\) own a cell phone.
- What is the population of interest?
- What is the parameter of interest?
- What is the statistic involved?
- Based on this sample, do we know the proportion of all adults who own a cell phone? Explain fully.
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A sociologist wishes to estimate the proportion of all adults in a certain region who have never married. In a random sample of \(1,320\) adults, \(145\) have never married, hence \(145/1320 \approx 0.11\) or about \(11\%\) have never married.
- What is the population of interest?
- What is the parameter of interest?
- What is the statistic involved?
- Based on this sample, do we know the proportion of all adults who have never married? Explain fully.
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- What must be true of a sample if it is to give a reliable estimate of the value of a particular population parameter?
- What must be true of a sample if it is to give certain knowledge of the value of a particular population parameter?
Answers
- A population is the total collection of objects that are of interest in a statistical study.
- A sample, being a subset, is typically smaller than the population. In a statistical study, all elements of a sample are available for observation, which is not typically the case for a population.
- A parameter is a value describing a characteristic of a population. In a statistical study the value of a parameter is typically unknown.
- All currently registered students at a particular college form a population. Two population characteristics of interest could be the average GPA and the proportion of students over \(23\) years.
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- Population.
- Sample.
- Population.
- Sample.
- Sample.
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- Qualitative.
- Qualitative.
- Quantitative.
- Quantitative.
- Qualitative.
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- All newborn babies in South America in the last five years.
- The average birth weight of all newborn babies in South America in the last five years.
- No, not exactly, but we know the approximate value of the average.
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- All adults in the region.
- The proportion of the adults in the region who have never married.
- The proportion computed from the sample, \(0.1\).
- No, not exactly, but we know the approximate value of the proportion.
1.2: Overview
1.3: Presentation of Data
Exercises
1. List all the measurements for the data set represented by the following data frequency table.
\[\begin{array}{c|ccccc}x & 31 & 32 & 33 & 34 & 35 \\ \hline f & 1 & 5 & 6 & 4 & 2\end{array}\]
2. List all the measurements for the data set represented by the following data frequency table
\[\begin{array}{c|ccccccc}x & 97 & 98 & 99 & 100 & 101 & 102 & 103 & 105 \\ \hline f & 7 & 5 & 3 & 4 & 2 & 2 & 1 & 1\end{array}\]
3. Construct the data frequency table for the following data set.
\[\begin{array}22 & 25 & 22 & 27 & 24 & 23 \\ 26 & 24 & 22 & 24 & 26 &\end{array}\]
4. Construct the data frequency table for the following data set.
\[ \{1,\, 5,\, 2,\, 3,\, 5,\, 1,\, 4,\, 4,\, 4,\, 3,\, 2,\, 5,\, 1,\, 3,\, 2,\, 1,\, 1,\, 1,\, 2\} \]
Answers
- \(\{31,\, 32,\, 32,\, 32,\, 32,\, 32,\, 33,\, 33,\, 33,\, 33,\, 33,\, 33,\, 34,\, 34,\, 34,\, 34,\, 35,\, 35\}\)
- \(\begin{array}{c|ccccc}x & 22 & 23 & 24 & 25 & 26 & 27 \\ \hline f & 3 & 1 & 3 & 1 & 2 & 1\end{array}\)
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