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Chapter 5: Problem 6

(a) The area below \(z=0.8\) (b) The area above \(z=1.2\) (c) The area between \(z=-1.75\) and \(z=-1.25\)

Short Answer

Expert verified
The areas below and above certain z-values, plus between two z-values, can be determined using a Z-table to locate the appropriate corresponding probabilities.

Step by step solution

01

Determining Area below z=0.8

Using the standard Z-table, look for the row that corresponds to \(z=0.8\), then locate the column that corresponds to \(0.00\). The value where the row and column meet is the area below \(z=0.8\).
02

Determining Area above z=1.2

First, find the area below \(z=1.2\) as in the previous step. The standard normal distribution is symmetric about zero. So, the total area above and below zero is 1. Subtract the area below \(z=1.2\) from 1 to find the area above \(z=1.2\).
03

Determining Area between z=-1.75 and z=-1.25

First, determine the areas to the left of \(z=-1.75\) and \(z=-1.25\) respectively as described in Step 1. Once these two areas are found, subtract the smaller from the larger to find the area between these two Z-values.

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Key Concepts

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

Standard Normal Distribution
In statistics, the standard normal distribution is a crucial concept that many students encounter. It is a type of probability distribution that is symmetrical around the mean. This distribution is described using a bell-shaped curve, known as the normal curve.
  • The mean of a standard normal distribution is 0.
  • The standard deviation is 1.
  • The total area under the curve is 1, which represents the total probability.
Understanding the standard normal distribution helps in various statistical applications, such as hypothesis testing and confidence intervals. By converting different normal distributions to a standard normal distribution, we can use a common scale to compare data effectively.
Area Under the Curve
The term "area under the curve" in the context of a normal distribution refers to the probability of a particular range of values. Calculating this is essential in statistics to determine the likelihood of a score falling within a certain range. To find the area: - Utilize the Z-table, which provides the area (probability) to the left of a given z-score. - This area, or probability, represents the probability of occurrences as described by the normal distribution curve. For example, if we want the area below a z-score of 0.8, we check the Z-table. The intersection provides us with the probability for all values lower than the z-score. This concept helps answer questions like those in our exercise, where we seek probabilities below, above, or between specific z-scores.
Z-scores
Z-scores, also known as standard scores, are measures that describe a value's position relative to the mean of a data set. They indicate how many standard deviations an element is from the mean.
  • A positive z-score means the value is above the mean.
  • A negative z-score means it is below the mean.
  • A z-score of zero indicates that the data point is exactly at the mean.
Z-scores are fundamental for comparing scores from different distributions. They normalize different score scales to a uniform one 鈥� the standard normal distribution. From our exercise, calculating z-scores allows for determining specific areas under the normal curve, helping understand probabilities across different sections of the curve.

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Most popular questions from this chapter

Exercises 5.21 to 5.28 ask you to convert an area from one normal distribution to an equivalent area for a different normal distribution. Draw sketches of both normal distributions, find and label the endpoints, and shade the regions on both curves. The middle \(80 \%\) for a standard normal distribution converted to a \(N(100,15)\) distribution.

How Often Do You Use Cash? In a survey \(^{14}\) of 1000 American adults conducted in April 2012 , \(43 \%\) reported having gone through an entire week without paying for anything in cash. Test to see if this sample provides evidence that the proportion of all American adults going a week without paying cash is greater than \(40 \%\). Use the fact that a randomization distribution is approximately normally distributed with a standard error of \(S E=0.016\). Show all details of the test and use a \(5 \%\) significance level.

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 \(H_{0}: p=0.25\) vs \(H_{a}: p<0.25\) when the sample has \(n=800\) and \(\hat{p}=0.235,\) with \(S E=0.018\).

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

What Proportion Have College Degrees? According to the US Census Bureau, \(^{4}\) about \(27.5 \%\) of US adults over the age of 25 have a bachelor's level (or higher) college degree. For random samples of \(n=500\) US adults over the age of 25 , the sample proportions, \(\hat{p},\) with at least a bachelor's degree follow a normal distribution with mean 0.275 and standard deviation \(0.02 .\) Draw a sketch of this normal distribution and label at least three points on the horizontal axis.
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