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Chapter 6: Problem 44

Assume a standard Normal distribution. Draw a separate, well-labeled Normal curve for each part. a. Find the \(z\) -score that gives a left area of \(0.9774\). b. Find the \(z\) -score that gives a left area of \(0.8225\).

Short Answer

Expert verified
The z-score corresponding to a left area value of 0.9774 is approximately 2.00, while the z-score corresponding to a left area value of 0.8225 is approximately 0.90.

Step by step solution

01

Determine the z-score for a left area of 0.9774

In order to find the z-score, use a standard normal distribution table or a calculator that can find this value. Looking up in a standard normal distribution table and finding the closest value to 0.9774 gives a z-score of approximately 2.00.
02

Determine the z-score for a left area of 0.8225

Similarly, you'll want to use a standard normal distribution table or calculator to find the z-score corresponding to a left area value of 0.8225. By doing so, you'll find that the closest value gives a z-score of approximately 0.90.

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Key Concepts

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

Z-score
Understanding a Z-score is key when working with the standard normal distribution. A Z-score represents the number of standard deviations a data point is from the mean.
In a standard normal distribution:
  • The mean is 0.
  • The standard deviation is 1.
A positive Z-score indicates a data point that is above the mean, while a negative Z-score means it's below the mean.
To find a Z-score for a specific left area, you can use a probability table or a calculator. In probability terms, the left area represents the cumulative probability or the probability of a data point being less than a certain value. For example, a Z-score of 2.00 means the point is 2 standard deviations above the mean, encompassing about 97.74% of the data to the left.
Normal Curve
The normal curve, or bell curve, is a graphical representation of the normal distribution.
It is called a bell curve due to its shape, which is highest in the middle and tails off at both ends.
Here are some key features:
  • The peak represents the mean of the dataset.
  • It is symmetric about the mean.
  • The total area under the curve is 1, representing all possible probabilities.
  • Most data falls within 1, 2, or 3 standard deviations from the mean.
In a standard normal distribution, 68% of data falls within 1 standard deviation, 95% within 2, and 99.7% within 3 standard deviations. This symmetry and proportionate spread make it a powerful statistical tool.
Probability Table
A probability table, also known as a Z-table, is an essential tool for finding Z-scores that correspond to specific probabilities in a standard normal distribution.
These tables provide the probability or area under the normal curve to the left of a given Z-score.
To use a probability table:
  • Locate the nearest value to your specific left area or probability.
  • Identify the corresponding Z-score from the table's row and column headers.
Probability tables are invaluable for quick lookups and are widely used in statistics to find values that aren't immediately calculable by hand. Understanding how to read them can help solve complex statistical problems with more ease.

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Most popular questions from this chapter

The Normal model \(N(500,100)\) describes the distribution of critical reading SAT scores in the United States. Which of the following questions asks for a probability and which asks for a measurement (and is thus an inverse Normal question)? a. What reading SAT score is at the 65 th percentile? b. What is the probability that a randomly selected person will score 550 or more?

A study of human body temperatures using healthy women showed a mean of \(98.4^{\circ} \mathrm{F}\) and a standard deviation of about \(0.70^{\circ} \mathrm{F}\). Assume the temperatures are approximately Normally distributed. a. Find the percentage of healthy women with temperatures below \(98.6^{\circ} \mathrm{F}\) (this temperature was considered typical for many decades). b. What temperature does a healthy woman have if her temperature is at the 76 th percentile?

Extreme Negative z-Scores For each question, find the area to the right of the given z-score in a standard Normal distribution. In this question, round your answers to the nearest \(0.000 .\) Include an appropriately labeled sketch of the \(N(0,1)\) curve. a. \(z=-4.00\) b. \(z=-8.00\) c. \(z=-30.00\) d. If you had the exact probability for these right proportions, which would be the largest and which would be the smallest? e. Which is equal to the area in part b: the area below (to the left of) \(z=8.00\) or the area above (to the right of) \(z=8.00 ?\)

The average birth weight of elephants is 230 pounds. Assume that the distribution of birth weights is Normal with a standard deviation of 50 pounds. Find the birth weight of elephants at the 95 th percentile.

Use a table or technology to answer each question. Include an appropriately labeled sketch of the Normal curve for each part. Shade the appropriate region. a. Find the probability that a z-score will be \(1.76\) or less. b. Find the probability that a z-score will be \(1.76\) or more. c. Find the probability that a \(z\) -score will be between \(-1.3\) and \(-1.03\).
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