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Chapter 6: Problem 44

In a standard Normal distribution, if the area to the left of a \(z\) -score is about \(0.1000\), what is the approximate \(z\) -score?

Short Answer

Expert verified
The approximate z-score for an area to the left of 0.1000 in a standard normal distribution is -1.27.

Step by step solution

01

Understand the problem

We need to find the value of the z-score that approximates an area to the left of 0.1000 in a standard normal distribution.
02

Referencing the Z-table

Since the normal distribution table typically gives the area to the left of a given z-score, we need to look up the given area in the body of the table and find the corresponding z-score. Searching for the 0.1000 area in the standard Normal Distribution Table doesn't lead to an exact match, but we find two values 0.0995 and 0.1003 that bracket our desired area. These two values correspond to z-scores of -1.28 and -1.27, respectively.
03

Interpreting the results

Given that our desired area (0.1000) falls between the two found areas (0.0995 and 0.1003), we conclude that the z-score falls between -1.28 and -1.27. However, we've been asked to approximate the z-score, so we need to round to the closest value. The z-score of 0.0995 is slightly smaller than 0.1000 while the z-score of 0.1003 is slightly larger. Since 0.1000 is closer to 0.1003, we'll choose the corresponding z-score, -1.27, as the approximate z-score.

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Key Concepts

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

Understanding Z-Score
The concept of a z-score is crucial when dealing with the normal distribution. A z-score is a numerical measurement that describes a value's relationship to the mean of a group of values. Specifically, it represents how many standard deviations an element is from the mean.
  • A positive z-score indicates the value is above the mean.
  • A negative z-score shows the value is below the mean.
  • A z-score of 0 signifies the value is exactly at the mean.
Calculating a z-score involves using the formula:\[z = \frac{(X - \mu)}{\sigma}\]where \(X\) is the value in question, \(\mu\) is the mean, and \(\sigma\) is the standard deviation. Understanding z-scores helps us determine the position of any data point within a distribution.
Area Under the Curve
The area under the normal distribution curve is fundamental to understanding probabilities and percentages in statistics. In a standard normal distribution, these probabilities are represented by the area under the curve, which always totals 1 (or 100%).
The standard normal distribution is symmetric, so each half of the curve accounts for 50% of the total area. This area can be used to signify the probability of a data point falling below a certain value.
  • For example, an area of 0.1000 to the left of a z-score means there is a 10% probability that a randomly selected data point falls below this z-score.
The area also helps in visualizing how scores are distributed around the mean, making it a key concept in interpreting statistical data.
Introduction to Standard Normal Distribution
The standard normal distribution is a specific normal distribution with a mean of 0 and a standard deviation of 1. It's the foundation for statistical calculations because it allows mathematicians to standardize different data sets for easy comparison.
  • When we look at normal distributions graphically, the standard normal distribution appears as a classic bell-shaped curve centered at zero.
  • This distribution enables the use of z-scores, providing a universal scale for measuring scores.
By converting any normal distribution into a standard normal distribution, one can make precise statistical interpretations and predictions. This conversion facilitates the use of statistical tables, like the z-table, to find probabilities associated with z-scores.
Navigating the Z-Table
The z-table, also known as the standard normal distribution table, is an invaluable tool in statistics for finding the probability linked to any z-score.
To use the z-table effectively:
  • Identify the z-score for which you want to find the probability.
  • Locate the z-score along the table's axis.
  • The corresponding value in the body of the table reveals the area to the left of that z-score.
In the given exercise, we needed a z-score whose area to the left was 0.1000. By consulting the z-table, we identified two bracketing values: 0.0995 and 0.1003, associated with z-scores of -1.28 and -1.27. Since 0.1000 is closer to 0.1003, we selected -1.27 as the approximate z-score. Knowing how to read and interpret the z-table is essential for determining probabilities and making well-informed decisions in statistical analysis.

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Most popular questions from this chapter

According to the National Health Center, the heights of 5 -year-old boys are Normally distributed with a mean of 43 inches and a standard deviation of \(1.5\) inches. a. In which percentile is a 5 -year-old boy who is \(46.5\) inches tall? b. If a 5 -year-old boy who is \(46.5\) inches tall grows up to be a man at the same percentile of height, what height will he be? Assume adult men's heights (inches) are distributed as \(N(69,3)\).

According to the British Medical Journal, the distribution of weights of newborn babies is approximately Normal, with a mean of 3390 grams and a standard deviation of 550 grams. Use a technology or a table to answer these questions. For each include an appropriately labeled and shaded Normal curve. a. What is the probability at newborn baby will weigh more than 4000 grams? b. What percentage of newborn babies weigh between 3000 and 4000 grams? c. A baby is classified as "low birth weight" if the baby weighs less than 2500 grams at birth. What percentage of newborns would we expect to be "low birth weight"?

Voice-controlled video assistants are being incorporated into a wide variety of consumer products, including smartphones, tablets, and stand-alone devices such as the Amazon Echo or Google Home. A Pew Research poll found that \(46 \%\) of Americans reported using a voice-controlled digital assistant. Suppose a group of 50 Americans is randomly selected. a. Find the probability that more than half of the sample uses a voice- controlled digital assistant. b. Find the probability that at most 20 use a voice-controlled digital assistant. c. In a group of 50 Americans, how many would we expect use one of these devices? d. Find the standard deviation for this binomial distribution. Using your answers to parts \(\mathrm{c}\) and \(\mathrm{d}\), would it be surprising to find that fewer than 10 used one of these devices? Why or why not?

Babies in the United States have a mean birth length of \(20.5\) inches with a standard deviation of \(0.90\) inch. The shape of the distribution of birth lengths is approximately Normal. a. How long is a baby born at the 20 th percentile? b. How long is a baby born at the 50 th percentile? c. How does your answer to part b compare to the mean birth length? Why should you have expected this?

According to the American Veterinary Medical Association, \(36 \%\) of Americans own a dog. a. Find the probability that exactly 4 out of 10 randomly selected Americans own a dog. b. In a random sample of 10 Americans, find the probability that 4 or fewer own a dog.
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