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Chapter 8: Problem 9

Referring to the standard normal table (Table A, Appendix C), find the probability that a randomly selected z score will be (a) above 1.96 (b) either above 1.96 or below -1.96 (c) between -1.96 and 1.96 (d) either above 2.58 or below -2.58

Short Answer

Expert verified
(a) 0.025, (b) 0.05, (c) 0.95, (d) 0.01

Step by step solution

01

Understanding the standard normal table

The standard normal table or z-table provides the area (i.e., the probability) that falls under the standard normal curve to the left of a given z-score. Therefore, if a z-score is positive, the area to the left of it is greater than 0.5 (because more than 50% of the data falls to the left of the mean), and if a z-score is negative, the area to the left of it is less than 0.5.
02

Finding the z-score probabilities

(a) Look up a z-score of 1.96 in the z-table. This gives an area of 0.9750 to the left of the z-score. To find the probability that a randomly selected z-score is above 1.96, subtract the area from 1 (because total probability under the curve is 1), i.e., 1 - 0.9750 = 0.025.(b) This is asking for the probability that the z-score is in the tails of the distribution, either below -1.96 or above 1.96. This is 2*P(Z>1.96) = 2*0.025 = 0.05.(c) This is the probability that the z-score is between -1.96 and 1.96. This is simply 2*P(Z<1.96) - 1 = 2*0.975 - 1 = 0.95.(d) Similar to (b), this is 2*P(Z>2.58). Look up 2.58 in the z-table, this gives 0.995 (rounded off). Therefore, 2*(1-0.995) = 0.01.
03

Interpretation

The probabilities provide an understanding of the likelihood of different scenarios based on the z-score. For instance, there is a 2.5% chance of getting a z-score above 1.96.

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Key Concepts

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

Understanding the Z-Score
The z-score is a statistical measurement that indicates how many standard deviations an element is from the mean. In a standard normal distribution, the mean is 0, and the standard deviation is 1.
  • Z-scores allow comparison between scores from different distributions.
  • They help in finding the position of a score within a distribution.
For example, a z-score of 1.96 suggests that the score is 1.96 standard deviations above the mean. If you have a negative z-score, like -1.96, it indicates that the score is below the mean. This position helps determine probability concerning the normal curve.
The Basics of Probability
Probability measures the likelihood of a particular event occurring. It ranges from 0 to 1, where 0 means the event is impossible, and 1 means it's certain. In the context of the standard normal distribution, probability represents the area under the curve.
  • The total area under the curve is equal to 1.
  • Each z-score corresponds to a specific probability reflecting the area to its left on the curve.
For example, in the standard normal distribution, the probability of a z-score greater than 1.96 is calculated by subtracting the area (probability) to the left of 1.96 from 1, which gives the proportion of data lying in the upper tail of the distribution.
Navigating the Z-Table
A z-table, or standard normal table, helps find probabilities associated with specific z-scores. It provides the area to the left of a z-score on the standard normal distribution curve.
  • Use the z-table to look up areas corresponding to z-scores.
  • For a z-score of 1.96, the z-table shows an area of 0.9750, meaning 97.50% of the data is left of this point.
If you need the probability of a z-score being above a particular point, subtract the table value from 1. For a score below a negative value, just use the direct table value for negative z-scores.
The Normal Distribution Curve
The normal distribution curve is symmetrical and bell-shaped; it reflects how data points are distributed around a mean. The standard normal curve has a mean of 0 and a standard deviation of 1.
  • The peak represents the most frequent occurrence (mean).
  • The tails represent extreme values outliers but hold less probability.
The characteristics of the distribution allow for straightforward calculation of probabilities for different z-scores, simplifying the analysis of data within the context of standard deviations from the mean. Understanding this curve helps determine the likelihood of the occurrence of scores within and beyond certain z-score thresholds.

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Most popular questions from this chapter

The probability of a boy being born equals \(.50,\) or \(1 / 2\), as does the probability of a girl being born. For a randomly selected family with two children, what's the probability of (a) two boys, that is, a boy and a boy? (Reminder: Before using either the addition or multiplication rule, satisfy yourself that the various events are either mutually exclusive or independent, respectively.) (b) two girls? (c) either two boys or two girls?

As subjects arrive to participate in an experiment, tables of random numbers are used to make random assignments to either group A or group B. (To ensure equal numbers of subjects in the two groups, alternate subjects are automatically assigned to the other, smaller group.) Indicate with a Yes or No whether each of the following rules would work: (a) Assign the subject to group \(\mathrm{A}\) if the random number is even and to group \(\mathrm{B}\) if the random number is odd. (b) Assign the subject to group A if the first digit of the random number is between \(\mathbf{O}\) and 4 and to group \(\mathrm{B}\) if the first digit is between 5 and \(9 .\) (c) Assign the subject to group A if the first two digits of the random number are between 00 and 40 and to group \(\mathrm{B}\) if the first two digits are between 41 and \(99 .\) (d) Assign the subject to group A if the first three digits of the random number are between 000 and 499 and to group \(\mathrm{B}\) if the first three digits are between 500 and \(999 .\)

Continue to assume that people are equally likely to be born during any of the months. However, just for the sake of this exercise, assume that there is a tendency for married couples to have been born during the same month. Furthermore, we wish to calculate the probability of a husband and wife both being born during December. (a) It would be appropriate to use the multiplication rule for independent outcomes. True or False? (b) The probability of a married couple both being born during December is smaller than, equal to, or larger than \((1 / 12)(1 / 12)=1 / 144 .\) (c) With only the previous information, it would be possible to calculate the actual probability of a married couple both being born during December. True or False?
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