A Numerical Value Used As A Summary Measure For A Sample ✓ Solved

· Updated on December 8, 2025

Question 11 A Numerical Value Used As A Summary Measure For A Sample

QUESTION 1 1. A numerical value used as a summary measure for a sample, such as sample mean, is known as a population parameter sample parameter sample statistic population mean QUESTION 2 1. The 50th percentile is the mode median mean third quartile QUESTION 3 1. The difference between the largest and the smallest data values is the variance interquartile range range coefficient of variation QUESTION 4 1. When data are positively skewed, the mean will usually be greater than the median smaller than the median equal to the median positive QUESTION 5 1. The numerical value of the standard deviation can never be larger than the variance zero negative smaller than the variance QUESTION 6 1. Which of the following symbols represents the standard deviation of the population? σ2 σ μ x ̅ QUESTION 7 1. Which of the following symbols represents the mean of the population? σ2 σ μ x ̅ QUESTION 8 1. Which of the following symbols represents the mean of the sample? σ2 σ μ x ̅ QUESTION 9 1. Two events are mutually exclusive if their intersection is 1 if they have no sample points in common if their intersection is 0.5 None of these alternatives is correct. QUESTION 10 1. The range of probability is any value larger than zero any value between minus infinity to plus infinity zero to one any value between -1 to 1 QUESTION 11 1. The sum of the probabilities of two complementary events is .. QUESTION 13 1. Events A and B are mutually exclusive. Which of the following statements is also true? A and B are also independent. P(A ∪ B) = P(A)P(B) P(A ∪ B) = P(A) + P(B) P(A ∩ B) = P(A) + P(B) QUESTION 14 1. The set of all possible sample points (experimental outcomes) is called a sample an event the sample space a population QUESTION 15 1. If A and B are independent events, then P(A) must be equal to P(B) P(A) must be greater than P(B) P(A) must be less than P(B) P(A) must be equal to P(A│B)

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The concept of a numerical value used as a summary measure for a sample is fundamental in statistics, serving as a concise representation of data characteristics. The most common example is the sample mean, which estimates the population mean and acts as a crucial descriptive statistic in summarizing data sets. The sample mean, denoted as x̄, provides a central tendency measure and is calculated by summing all sample observations and dividing by the number of observations (Freedman, Pisani, & Purves, 2007). Alongside the mean, other summary measures such as median, mode, and quartiles offer additional perspectives on data distribution.

The median, representing the 50th percentile, is the middle value that separates the lower half from the upper half of the data. Unlike the mean, the median is resistant to outliers and skewed data, making it useful for skewed distributions (Lohr, 2009). The mode, the most frequently occurring data point, often complements measures of central tendency, especially when data are nominal or categorical. Quartiles divide data into four equal parts, with the third quartile indicating the 75th percentile.

In descriptive statistics, measures such as the range and interquartile range (IQR) quantify the spread of data. The range, defined as the difference between the maximum and minimum values, offers a simple measure of variability but is sensitive to outliers (Floyd & Widaman, 1995). The variance and standard deviation are more robust measures of dispersion, with the standard deviation providing an average amount of deviation from the mean. Notably, the standard deviation of a population is represented by the Greek letter sigma (σ), and the variance by σ² (Ott & Longnecker, 2015).

Understanding skewness is vital when interpreting the effects of data distribution on statistical measures. When data are positively skewed, the mean tends to be greater than the median because the tail on the right pulls the mean upward. Conversely, in negatively skewed data, the mean is less than the median. This relationship helps analysts choose appropriate measures for central tendency depending on the data distribution (Helsel & Hirsch, 2002).

Probability values, ranging from 0 to 1, encapsulate the likelihood of events occurring. The sum of probabilities for two mutually exclusive events is equal to the probability that either event occurs, which mathematically is expressed as P(A ∪ B) = P(A) + P(B). When events are mutually exclusive, they cannot occur simultaneously, meaning their intersection is zero (Ross, 2014). The concept of independence indicates that the occurrence of one event does not influence the probability of another, formalized as P(A | B) = P(A). If A and B are independent, then P(A) = P(A | B), emphasizing the constancy of A's probability regardless of B's occurrence (Casella & Berger, 2002).

In probability theory, the sample space comprises all possible outcomes of an experiment. Understanding the sample space is essential because it forms the basis for calculating probabilities and analyzing events (Johnson & Luden, 2014). Recognizing whether events are mutually exclusive, independent, or complementary allows statisticians to compute combined probabilities accurately and interpret relationships between different events.

References

  • Casella, G., & Berger, R. L. (2002). Statistical inference (2nd ed.). Duxbury.
  • Floyd, F. J., & Widaman, K. F. (1995). Factor analysis in the development and refinement of clinical assessment instruments. Psychological Assessment, 7(3), 286-299.
  • Freedman, D., Pisani, R., & Purves, R. (2007). Statistics (4th ed.). W. W. Norton & Company.
  • Helsel, D. R., & Hirsch, R. M. (2002). Statistical methods in water resources. U.S. Geological Survey.
  • Lohr, S. L. (2009). Sampling: Design and analysis. Cengage Learning.
  • Ott, R. L., & Longnecker, M. (2015). An introduction to statistical methods and data analysis (7th ed.). Cengage Learning.
  • Ross, S. M. (2014). Introduction to probability models (11th ed.). Academic Press.
  • Johnson, R. A., & Luden, R. H. (2014). Probability and statistics for engineering and the sciences. Cengage Learning.