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Chapter 9: Problem 8

A human resources manager for a large company takes a random sample of 50 employees from the company database. She calculates the mean time that they have been employed. She records this value and then repeats the process: She takes another random sample of 50 names and calculates the mean employment time. After she has done this 1000 times, she makes a histogram of the mean employment times. Is this histogram a display of the population distribution, the distribution of a sample, or the sampling distribution of means?

Short Answer

Expert verified
The histogram represents a Sampling Distribution of Means.

Step by step solution

01

Understand the Definitions

Population Distribution: This represents the total set or the complete data. In this context, it would refer to the employment times of all employees in the company. \n\n Sample Distribution: This represents a subset of the population, which can offer insights about the overall population. In this context, if the manager selected a single sample of 50 employees and created a histogram of their employment times, that would be a sample distribution. \n\n Sampling Distribution: This is a probability distribution of a statistic (in this case, the mean), obtained from a large number of samples drawn from a specific population. The histogram that the manager creates is the plotted mean employment times derived from 1000 samples (each of 50 employees). So, it is a representative of the sampling distribution of means.
02

Identify the Correct Type of Distribution

In light of the above definitions, it is clear that the histogram the Human Resource manager created is neither illustrating the population distribution nor a single sample distribution. She has taken a large number of samples (1000), calculated their means, and made a histogram of those means. Hence, this histogram represents a sampling distribution of means.

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Key Concepts

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

Population Distribution
When we talk about the population distribution, we are referring to the entire set of data that one is interested in. In this case, it would involve the employment times of every single employee at the company. Each employee's employment time is a data point, and together these data points create a complete picture or distribution.

Population distributions are foundational in statistics because they represent the true variability and characteristics of a full dataset. They always remain the same until there is an actual change in the measured group.
  • Total dataset
  • Contains all members of the set
  • Displays true measures of central tendency (mean, median, mode)
Understanding population distribution helps us make predictions about the data and is crucial in determining how to appropriately draw samples for studying smaller, manageable parts of the population.
Sample Distribution
A sample distribution is essentially a snapshot or a smaller, manageable segment of the population distribution. It consists of a subset of observations taken from the larger group. In our human resources scenario, the sample distribution would be evident if the manager chose just one group of 50 employees and recorded their employment times in a histogram.

Taking a sample helps in making inferences about the population without needing to record data from every single member, saving time and resources.
  • Smaller subset of a population
  • Used to derive insights about the overall population
  • Has its own measures like a smaller mean or median
Each sample distribution can show unique characteristics, often influenced by the randomness and variability found within different sample selections.
Probability Distribution
Probability distribution is key when dealing with sampling and statistics. It provides a mathematical function, showing all possible values of a random variable within a population and how likely they are to occur. In simple terms, it assigns probabilities to different outcomes.

This concept comes into play significantly when analyzing sampling distributions, as it helps predict how data points obtained from samples (like the means in the given scenario) are spread out over a range.
  • Shows likelihood of different outcomes in an experiment
  • Useful for calculating probabilities within a sample
  • Helps in understanding data spread and variability
Probability distributions are pivotal for understanding the underlying structure and behavior of data, enabling statisticians to estimate parameters and make informed predictions.

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Most popular questions from this chapter

A study of all the students at a small college showed a mean age of \(20.7\) and a standard deviation of \(2.5\) years. a. Are these numbers statistics or parameters? Explain. b. Label both numbers with their appropriate symbol (such as \(\bar{x}, \mu, s\), or \(\sigma)\).

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The distribution of the scores on a certain exam is \(N(100,10)\) which means that the exam scores are Normally distributed with a mean of 100 and a standard deviation of \(10 .\) a. Sketch or use technology to create the curve and label on the \(x\) -axis the position of the mean, the mean plus or minus one standard deviation, the mean plus or minus two standard deviations, and the mean plus or minus three standard deviations. b. Find the probability that a randomly selected score is between 90 and \(110 .\) Shade the region under the Normal curve whose area corresponds to this probability.

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Assume women's heights are approximately Normally distributed with a mean of 65 inches and a standard deviation of \(2.5\) inches. Which of the following questions can be answered using the Central Limit Theorem for sample means as needed? If the question can be answered, do so. If the question cannot be answered, explain why the Central Limit Theorem cannot be applied. a. Find the probability that a randomly selected woman is less than 63 inches tall. b. If five women are randomly selected, find the probability that the mean height of the sample is less than 63 inches. c. If 30 women are randomly selected, find the probability that the mean height of the sample is less than 63 inches.
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