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Chapter 6: Problem 20

Find the probability that a piece of data picked at random from a normal population will have a standard score \((z)\) that lies a. between 0 and 0.74. b. to the right of 0.74. c. to the left of 0.74. d. between -0.74 and 0.74.

Short Answer

Expert verified
a. The probability is 27.04%. \n b. The probability is 22.96%. \n c. The probability is 77.04%. \n d. The probability is 54.08%.

Step by step solution

01

Identify Areas under the Standard Normal Distribution Curve

One can refer to the Standard Normal Distribution table, also known as the z-table, to find the corresponding probability for each value of \( z \). This table lists the probability associated with each z-score.
02

Apply the Z-table for Each Case

a. For \( z \) between 0 and 0.74, find the area associated with 0.74 in the z-table, which is 0.7704. Since z-score of 0 has 50% of the data below it, calculate the difference to find the proportion between 0 and 0.74. \n b. For \( z \) to the right of 0.74, we need to understand that this is equivalent to finding the area greater than 0.74. Therefore, subtract the associated probability found in the z-table from 1 (total probability). \n c. In the case of \( z \) to the left of 0.74, the needed area under the curve is directly provided by z-table for 0.74. \n d. For \( z \) between -0.74 and 0.74, the area from the mean to z (0.74) is the same as the area from the mean to -z (-0.74), considering the symmetric of normal distribution. Hence, find the proportion of 0.74 from z-table and subtract 0.5 to find the half, then double it to get the area between -0.74 and 0.74.
03

Perform Calculations

a. Between 0 and 0.74: \(0.7704 - 0.5000 = 0.2704\). \n b. To the right of 0.74: \(1 - 0.7704 = 0.2296\). \n c. To the left of 0.74: Directly find in z-table, which is \(0.7704\). \n d. Between -0.74 and 0.74: \((0.7704 - 0.5000) * 2 = 0.5408\).
04

Express the Results in Terms of Percentages

Transform these proportions into percentages by multiplying each by 100. \n a. Between 0 and 0.74: \(0.2704 * 100 = 27.04\%\). \n b. To the right of 0.74: \(0.2296 * 100 = 22.96\%\). \n c. To the left of 0.74: \(0.7704 * 100 = 77.04\%\). \n d. Between -0.74 and 0.74: \(0.5408 * 100 = 54.08\%\).

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Key Concepts

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

Standard Score
In statistics, the standard score, commonly known as the "z-score," is a measurement that describes a value's relation to the mean of a group of values. It's expressed in terms of standard deviations from the mean. It's a simple yet powerful way to determine how unusual or common a data point is within a normal distribution. When you calculate a standard score, you鈥檙e essentially standardizing your data, allowing different datasets to be compared meaningfully.
  • The formula for calculating a z-score is: \( z = \frac{(X - \mu)}{\sigma} \), where \( X \) is the value, \( \mu \) is the mean, and \( \sigma \) is the standard deviation.
  • A z-score of 0 indicates the value is exactly at the mean.
  • A positive z-score signifies a value above the mean, while a negative z-score indicates a value below the mean.
Understanding the concept of a standard score is essential because it directly informs how we use tools like the z-table to find probabilities associated with specific values.
Z-table
The z-table is a mathematical table that contains the cumulative probability from the mean up to a given z-score. This tool is fundamental in statistics because it helps quickly determine probabilities associated with specific z-scores, which correspond to different points on a standard normal distribution curve. To use the z-table:
  • Locate the desired z-score in the table, which will give you the cumulative probability from the mean to that z-score.
  • The table usually provides the area (or probability) under the curve to the left of the z-score, covering that portion of the distribution.
  • For probabilities to the right of the z-score, simply subtract the table value from 1.
The z-table makes it easy to find the probability of a standard score occurring by translating the z-score into a probability percentage. This is why it was used in the exercises to find the probabilities between specific z-score ranges, such as between 0 and 0.74, and more.
Probability Calculation
Probability calculation using the normal distribution involves determining the likelihood that a random variable falls within a particular range. With the normal distribution being symmetric, we can use the z-table to perform these calculations effectively. Here's how probabilities were calculated in the given examples:
  • Between 0 and 0.74: The z-table gives the cumulative probability up to 0.74, which is 0.7704. Subtracting the probability up to 0 (0.5000) from this gives the area between these two points.
  • To the right of 0.74: Calculate the probability by subtracting the cumulative probability up to 0.74 from 1. This gives the area under the curve to the right of 0.74.
  • To the left of 0.74: Use the cumulative probability directly from the z-table.
  • Between -0.74 and 0.74: Since the distribution is symmetric, find the area from the mean to 0.74, multiply by 2 to cover both sides.
These calculations not only show how z-scores play into understanding probability but also how effective the z-table is in providing swift answers for normally distributed data questions.

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Most popular questions from this chapter

Understanding the \(z\) notation, \(z(\alpha),\) requires us to know whether we have a \(z\) -score or an area. Each of the following expressions uses the \(z\) notation in a variety of ways, some typical and some not so typical. Find the value asked for in each of the following, and then with the aid of a diagram explain what your answer represents. a. \(\quad z(0.08)\) b. the area between \(z(0.98)\) and \(z(0.02)\) c. \(\quad z(1.00-0.01)\) d. \(\quad z(0.025)-z(0.975)\)

6.52 According to Collegeboard.com [http://www.collegeboard.com/ \(]\) the national average salary for a plumber as of 2007 is \(\$ 47,350 .\) If we assume that the annual salaries for plumbers are normally distributed with a standard deviation of \(\$ 5250,\) find the following: a. What percentage earn below \(\$ 30,000 ?\) b. What percentage earn above \(\$ 63,000 ?\)

Find the following areas under the normal curve. a. To the right of \(z=0.00\) b. To the right of \(z=1.05\) c. To the right of \(z=-2.30\) d. To the left of \(z=1.60\) e. To the left of \(z=-1.60\)

a. Use a computer or calculator to list both the probability distribution and the cumulative probability distribution for the binomial probability experiment with \(n=40\) and \(p=0.4.\) b. Explain the relationship between the two distributions found in part a. c. If you could use only one of these lists when solving problems, which one would you use and why?

Find the normal approximation for the binomial probability \(P(x \geq 9),\) where \(n=13\) and \(p=0.7\) Compare this to the value of \(P(x \geq 9)\) obtained from Table 2.
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