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Chapter 5: Problem 8

(a) The area above \(z=1.35\). (b) The area below \(z=-0.8\). (c) The area between \(z=-1.23\) and \(z=0.75\).

Short Answer

Expert verified
The area above \(z=1.35\) equals \(1 - P1\), where \(P1\) is the cumulative probability for \(z=1.35\). The area below \(z=-0.8\) equals the cumulative probability for \(z=-0.8\), and the area between \(z=-1.23\) and \(z=0.75\) equals \(P3 - P2\), where \(P2\) and \(P3\) are the cumulative probabilities for \(z=-1.23\) and \(z=0.75\) respectively.

Step by step solution

01

Calculating The Area Above \(z=1.35\)

First, find the cumulative probability for \(z=1.35\). This can be found in a standard normal distribution table or can be calculated using statistical software. Assume this probability is \(P1\). The area above \(z=1.35\) can be found by subtracting \(P1\) from 1: \(Area = 1 - P1\).
02

Calculating The Area Below \(z=-0.8\)

Find the cumulative probability for \(z=-0.8\). This can also be found in a standard normal distribution table or can be calculated using statistical software. Such cumulative probability is the area below \(z=-0.8\).
03

Calculating The Area Between \(z=-1.23\) And \(z=0.75\)

Find and both cumulative probabilities for \(z=-1.23\) and \(z=0.75\), respectively. Assume these probabilities are \(P2\) and \(P3\). The area between \(z=-1.23\) and \(z=0.75\) can then be found by subtracting \(P2\) from \(P3\): \(Area = P3 - P2\).

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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 powerful statistical tool that measures the distance between an individual data point and the mean, expressed in terms of standard deviations. This is a crucial concept in statistics, especially when dealing with normal distributions.
The z-score formula is: \[ z = \frac{(X - \mu)}{\sigma} \]where:
  • \(X\) is the data point,
  • \(\mu\) is the mean of the distribution,
  • \(\sigma\) is the standard deviation of the distribution.
A z-score tells us how many standard deviations a data point is from the mean.
For example, a z-score of 1.35 means the value is 1.35 standard deviations above the mean. A negative z-score indicates a data point below the mean.
Decoding Cumulative Probability
Cumulative probability helps us understand the probability that a random variable will take a value less than or equal to a specific value.
In the context of a normal distribution, it tells us the probability of a value occurring within a certain range.
  • For areas to the right of a z-score (i.e., above), subtract the cumulative probability from 1.
  • For areas to the left (i.e., below), the cumulative probability itself is the area under the curve to that z-score.
  • For areas between two z-scores, subtract the cumulative probabilities at each z-score.
This concept is crucial because it allows us to find probabilities for normal distributions without long calculations.
It's often used in conjunction with the standard normal distribution table.
Navigating the Standard Normal Distribution Table
The standard normal distribution table is an essential tool for finding cumulative probabilities associated with z-scores.
It helps us quickly identify the probability of a z-score occurring by providing cumulative probabilities for each z-score value.
  • To use the table, find the row corresponding to the z-score's whole number and first decimal.
  • Then, find the column matching the second decimal.
  • The intersection of the row and column provides the cumulative probability.
However, remember that the table generally provides values for positive z-scores.
For negative z-scores, understand symmetry: a z-score of -0.8 has the same probability as 0.8 but for values to the left of the mean.
Thus, cumulative probability understanding and table use go hand in hand for accurate probability evaluations.

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Most popular questions from this chapter

Correlation between Time and Distance in Commuting In Exercise 3.85 on page \(204,\) we find an interval estimate for the correlation between Distance (in miles) and Time (in minutes) for Atlanta commuters, based on the sample of size \(n=500\) in Commute Atlanta. The correlation in the original sample is \(r=0.807\). The file BootAtlantaCorr contains 1000 values of bootstrap correlations produced from the original sample. (a) Use technology and the bootstrap samples to estimate the standard error of sample correlations between Distance and Time for samples of 500 Atlanta commutes. (b) Assuming that the bootstrap correlations can be modeled with a normal distribution, use the result of part (a) to find and interpret a \(90 \%\) confidence interval for the correlation between distance and time of Atlanta commutes.

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(a) The area below \(z=1.04\) (b) The area above \(z=-1.5\) (c) The area between \(z=1\) and \(z=2\)

Exercises 5.50 to 5.55 include a set of hypotheses, some information from one or more samples, and a standard error from a randomization distribution. Find the value of the standardized \(z\) -test statistic in each situation. Test \(H_{0}: \mu_{1}=\mu_{2}\) vs \(H_{a}: \mu_{1}>\mu_{2}\) when the samples have \(n_{1}=n_{2}=50, \bar{x}_{1}=35.4, \bar{x}_{2}=33.1,\) \(s_{1}=1.28,\) and \(s_{2}=1.17 .\) The standard error of \(\bar{x}_{1}-\bar{x}_{2}\) from the randomization distribution is \(0.25 .\)

Boys Heights Heights of 10-year-old boys (5th graders) follow an approximate normal distribution with mean \(\mu=55.5\) inches and standard deviation \(\sigma=2.7\) inches \({ }^{6}\) (a) Draw a sketch of this normal distribution and label at least three points on the horizontal axis. (b) According to this normal distribution, what proportion of 10 -year-old boys are between \(4 \mathrm{ft} 4\) in and \(5 \mathrm{ft}\) tall (between 52 inches and 60 inches)? (c) A parent says his 10 -year-old son is in the 99 th percentile in height. How tall is this boy?
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