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Chapter 5: Problem 27

Give the \(z\) -score for a measurement from a normal distribution for the following: a. 1 standard deviation above the mean b. 1 standard deviation below the mean c. Equal to the mean d. 2.5 standard deviations below the mean e. 3 standard deviations above the mean

Short Answer

Expert verified
a. 1, b. -1, c. 0, d. -2.5, e. 3.

Step by step solution

01

Understanding the z-score Formula

The z-score is calculated as \( z = \frac{(X - \mu)}{\sigma} \), where \( X \) is the value of the measurement, \( \mu \) is the mean, and \( \sigma \) is the standard deviation. The z-score indicates how many standard deviations an element is from the mean.
02

Calculate z-score for 1 Standard Deviation Above the Mean

For a value 1 standard deviation above the mean, \( X = \mu + \sigma \). So, the z-score is \( z = \frac{(\mu + \sigma - \mu)}{\sigma} = \frac{\sigma}{\sigma} = 1 \).
03

Calculate z-score for 1 Standard Deviation Below the Mean

For a value 1 standard deviation below the mean, \( X = \mu - \sigma \). Thus, the z-score is \( z = \frac{(\mu - \sigma - \mu)}{\sigma} = \frac{-\sigma}{\sigma} = -1 \).
04

Calculate z-score Equal to the Mean

For a value equal to the mean, \( X = \mu \). Therefore, the z-score is \( z = \frac{(\mu - \mu)}{\sigma} = \frac{0}{\sigma} = 0 \).
05

Calculate z-score for 2.5 Standard Deviations Below the Mean

For a value 2.5 standard deviations below the mean, \( X = \mu - 2.5\sigma \). The z-score then is \( z = \frac{(\mu - 2.5\sigma - \mu)}{\sigma} = \frac{-2.5\sigma}{\sigma} = -2.5 \).
06

Calculate z-score for 3 Standard Deviations Above the Mean

For a value 3 standard deviations above the mean, \( X = \mu + 3\sigma \). The z-score is \( z = \frac{(\mu + 3\sigma - \mu)}{\sigma} = \frac{3\sigma}{\sigma} = 3 \).

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Key Concepts

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

Understanding Normal Distribution
A normal distribution, often depicted as a bell-shaped curve, is a fundamental concept in statistics. It describes how data points are symmetrically distributed around the mean. The highest point on the curve represents the mean, median, and mode, which are all equal in a true normal distribution.

Key characteristics include:
  • The curve is symmetrical about the mean.
  • It follows the empirical rule:
    • Approximately 68% of data falls within one standard deviation of the mean.
    • About 95% falls within two standard deviations.
    • 99.7% lies within three standard deviations.
  • The tails of the curve extend indefinitely without touching the axis.
Real-life phenomena, such as heights of individuals or test scores, often exhibit normal distribution, making it an essential tool for statistical analysis and interpretation.
What Is Standard Deviation?
Standard deviation is a statistical measurement that reveals the amount of variability or dispersion in a set of values. It explains how spread out the values are from the mean. In a normal distribution, most of the data points fall close to the mean, leading to a smaller standard deviation. Conversely, if data points are spread widely, the standard deviation is large.

Understanding standard deviation:
  • Mathematically, it's the square root of the variance, calculated as follows:
    • First, find the mean ( \( \mu \) ).
    • Subtract the mean from each data point and square the result.
    • Average these squared differences to get the variance.
    • Finally, take the square root of the variance to find the standard deviation ( \( \sigma \) ).
  • It provides insights into how much the data deviates from the average value.
  • An essential measure for understanding data consistency, often used in finance, research, and quality control.
By understanding standard deviation, you can better interpret the consistency and reliability of data.
The Significance of Mean
The mean, commonly referred to as the average, is a critical statistical measurement that summarizes data by providing a central value. Calculating the mean involves adding up all the data points and dividing by the number of points.

Mean's importance:
  • It serves as a point of comparison for other statistical measures, like standard deviation.
  • Provides a straightforward method for summarizing data sets of various sizes.
  • Used globally for a wide variety of applications, from GPA calculations to financial analyses.
Although highly useful, the mean can be skewed by outliers (extremely high or low values). Therefore, it's important, when using the mean, to consider the possible presence of such exceptional values, which might not be representative of the overall data trend.
Understanding Statistical Measurement
Statistical measurement is the process of collecting, analyzing, interpreting, and presenting data in a methodical way. It uses tools like normal distribution, mean, and standard deviation to summarize and make sense of data.

Why it matters:
  • Enables informed decision-making by providing insights into data trends and relationships.
  • In fields such as science, economics, and social sciences, statistics are essential for validating hypotheses and understanding phenomena.
  • Through statistical measurements, complex datasets can be simplified, aiding in communication and interpretation.
With robust statistical measurement, individuals and organizations can derive meaningful conclusions, paving the way for data-driven strategies and decisions.

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Most popular questions from this chapter

Waiting for an elevator. The manager of a large department store with three floors reports that the time a customer on the second floor must wait for an elevator has a uniform distribution ranging from 0 to 4 minutes. If it takes the elevator 15 seconds to go from floor to floor, find the probability that a hurried customer can reach the first floor in less than 1.5 minutes after pushing the second-floor elevator button.

Find a z-score, say, \(z_{0}\), such that a. \(P\left(z \leq z_{0}\right)=.8708\) b. \(P\left(z \geq z_{0}\right)=.0526\) c. \(P\left(z \leq z_{0}\right)=.5\) d. \(P\left(-z_{0} \leq z \leq z_{0}\right)=.8164\) e. \(P\left(z \geq z_{0}\right)=.8023\) f. \(P\left(z \geq z_{0}\right)=.0041\)

Machine repair times. An article in IEEE Transactions (Mar. 1990 ) gave an example of a flexible manufacturing system with four machines operating independently. The repair rates for the machines (i.e., the time, in hours, it takes to repair a failed machine) are exponentially distributed with means \(\mu_{1}=1, \mu_{2}=2, \mu_{3}=.5,\) and \(\mu_{4}=.5,\) respectively. a. Find the probability that the repair time for machine 1 exceeds 1 hour. b. Repeat part a for machine 2 . c. Repeat part a for machines 3 and 4 . d. If all four machines fail simultaneously, find the probability that the repair time for the entire system exceeds 1 hour.

What is a normal probability plot and how is it used?

Tropical island temperatures. Records indicate that the daily high January temperatures on a tropical island tend to have a uniform distribution over the interval from \(75^{\circ} \mathrm{F}\) to \(90^{\circ} \mathrm{F}\). A tourist arrives on the island on a randomly selected day in January. a. What is the probability that the temperature will be above \(80^{\circ} \mathrm{F} ?\) b. What is the probability that the temperature will be between \(80^{\circ} \mathrm{F}\) and \(85^{\circ} \mathrm{F} ?\) c. What is the expected temperature?
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