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Chapter 1: Problem 24

A sample of midterm grades for five students showed the following results: 72,65,82,90 76\. Which of the following statements are correct, and which should be challenged as being too generalized? a. The average midterm grade for the sample of five students is 77 b. The average midterm grade for all students who took the exam is 77 . c. An estimate of the average midterm grade for all students who took the exam is 77 . d. More than half of the students who take this exam will score between 70 and 85 e. If five other students are included in the sample, their grades will be between 65 and 90 .

Short Answer

Expert verified
Statement a and c are correct, b, d, and e should be challenged.

Step by step solution

01

Calculate the Average Midterm Grade

To find the average midterm grade, sum the grades and divide by the number of grades. The grades are: 72, 65, 82, 90, and 76. Formula: Average = \( \frac{72 + 65 + 82 + 90 + 76}{5} \) First, calculate the sum: 72 + 65 + 82 + 90 + 76 = 385 Now, divide by 5: 385 / 5 = 77. The average midterm grade for the sample of five students is 77.
02

Evaluate Statement a

Statement a claims that the average midterm grade for the sample of five students is 77, which is correct. We calculated in Step 1 that the average is indeed 77.
03

Evaluate Statement b

Statement b claims that the average midterm grade for all students who took the exam is 77. This statement cannot be made because it generalizes from a sample of five students to all students without further data.
04

Evaluate Statement c

Statement c suggests an estimate of the average midterm grade for all students who took the exam is 77. This statement is valid, as a sample mean can serve as an estimate for the population mean.
05

Evaluate Statement d

Statement d claims that more than half the students will score between 70 and 85. This is a generalized statistical claim based on a small sample and isn't supported by the given data, as the distribution for all students is unknown.
06

Evaluate Statement e

Statement e suggests if five other students are included, their grades would be between 65 and 90. This is not statistically sound as new samples may differ widely from the initial sample range.

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Key Concepts

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

Sample Mean
The sample mean is a measure of the central tendency of a sample, which is calculated by summing all the observations in your sample and then dividing by the number of observations. In this exercise, the sample mean represents the average midterm grade among the five students. To compute it, simply add up all the given grades (72, 65, 82, 90, and 76) and then divide by 5, resulting in a sample mean of 77.
  • The sample mean provides a snapshot of what is happening in your particular sample set.
  • It can give insights into the typical performance of the group studied.
However, it is crucial not to mistake this value for the average of the entire student population unless further information supports such a claim.
Generalization Error
Generalization error is a critical concept to understand when analyzing how well your sample data represents the larger population. It refers to the error that occurs when a sample statisic is used to estimate a population parameter. In this exercise, when statement b attempts to generalize the average for all students based on just five data points, it risks introducing a significant generalization error. This is because:
  • The sample size is too small to confidently assert it represents the wider population.
  • The range and distribution of the grades within the whole student population may differ from those in the sample.
Generalization error reminds us to be cautious and seek additional data or analysis before extending our conclusions beyond the sample data.
Statistical Inference
Statistical inference allows us to make predictions or generalizations about a population based on a sample. This concept is applied when statement c suggests that the sample mean (77) could estimate the average for all students. While statistical inference enables this estimation, it requires:
  • Correct sampling techniques to ensure that the sample accurately reflects the entire population.
  • Knowledge of the potential variability or distribution within the overall population.
Inference is powerful, but we must remember that our estimates may not be precise or perfect. It's always important to consider the confidence interval and possible errors when making such predictions.
Average Calculation
The basic computation of an average, or mean, is a fundamental tool in statistics and involves simply summing all values and dividing by the count of those values. In our exercise, we performed the average calculation to determine the mean grade of the five students. Although this process is straightforward, some critical factors deserve attention:
  • The average provides a single number that summarizes the data set, offering quick insights into the group's overall performance.
  • There are different types of averages, like median and mode, which might be more suitable in different contexts.
Despite its simplicity, when applying average calculation to small or biased samples, misunderstandings can arise if it's used to generalize without adequate support as seen with other statements in the exercise.

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Most popular questions from this chapter

A Business Week North American subscriber study collected data from a sample of 2861 subscribers. Fifty-nine percent of the respondents indicated an annual income of \(\$ 75,000\) or more, and \(50 \%\) reported having an American Express credit card. a. What is the population of interest in this study? b. Is annual income a qualitative or quantitative variable? c. Is ownership of an American Express card a qualitative or quantitative variable? d. Does this study involve cross-sectional or time series data? e. Describe any statistical inferences Business Week might make on the basis of the survey.

The Hawaii Visitors Bureau collects data on visitors to Hawaii. The following questions were among 16 asked in a questionnaire handed out to passengers during incoming airline flights in June 2003 This trip to Hawaii is my: 1 st, 2 nd \(, 3\) rd, 4 th, etc. The primary reason for this trip is: (10 categories including vacation, convention, honeymoon) Where I plan to stay: (11 categories including hotel, apartment, relatives, camping) Total days in Hawaii a. What is the population being studied? b. Is the use of a questionnaire a good way to reach the population of passengers on incoming airline flights? c. Comment on each of the four questions in terms of whether it will provide qualitative or quantitative data.

A survey of 430 business travelers found 155 business travelers used a travel agent to make the travel arrangements (USA Today, November 20,2003 ). a. Develop a descriptive statistic that can be used to estimate the percentage of all business travelers who use a travel agent to make travel arrangements. b. The survey reported that the most frequent way business travelers make travel arrangements is by using an online travel site. If \(44 \%\) of business travelers surveyed made their arrangements this way, how many of the 430 business travelers used an online travel site? c. Are the data on how travel arrangements are made qualitative or quantitative?

The marketing group at your company developed a new diet soft drink that it claims will capture a large share of the young adult market. a. What data would you want to see before deciding to invest substantial funds in introducing the new product into the marketplace? b. How would you expect the data mentioned in part (a) to be obtained?

A survey of 131 investment managers in Barron's Big Money poll revealed the following \((\text {Barron's,} \text { October } 28,2002)\) \(43 \%\) of managers classified themselves as bullish or very bullish on the stock market. The average expected return over the next 12 months for equities was \(11.2 \%\) \(21 \%\) selected health care as the sector most likely to lead the market in the next 12 months. When asked to estimate how long it would take for technology and telecom stocks to resume sustainable growth, the managers' average response was 2.5 years. a. Cite two descriptive statistics. b. Make an inference about the population of all investment managers concerning the average return expected on equities over the next 12 months. c. Make an inference about the length of time it will take for technology and telecom stocks to resume sustainable growth.
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