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Chapter 6: Problem 42

Let \(z\) be a random variable with a standard normal distribution. Find the indicated probability, and shade the corresponding area under the standard normal curve. $$ P(-2.20 \leq z \leq 1.40) $$

Short Answer

Expert verified
The probability is approximately 0.9053.

Step by step solution

01

Understanding the Problem

We need to find the probability that the standard normal random variable, denoted as \(z\), falls between -2.20 and 1.40. This is represented as \(P(-2.20 \leq z \leq 1.40)\). The standard normal distribution has a mean of 0 and a standard deviation of 1, represented by the normal curve.
02

Using the Standard Normal Table

To find the probability, we use the standard normal distribution table (also known as the Z-table) which provides the cumulative probability from \(-\infty\) to a particular \(z\) value. We look up \(-2.20\) and \(1.40\) separately.
03

Finding Cumulative Probabilities

From the Z-table, locate \(P(z \leq -2.20)\). The table gives a value of approximately 0.0139. Next, locate \(P(z \leq 1.40)\), which is approximately 0.9192.
04

Calculating the Desired Probability

The probability for \(P(-2.20 \leq z \leq 1.40)\) is found by subtracting the cumulative probability for \(z\leq -2.20\) from the cumulative probability for \(z\leq 1.40\):\[ P(-2.20 \leq z \leq 1.40) = P(z \leq 1.40) - P(z \leq -2.20) = 0.9192 - 0.0139 = 0.9053. \]
05

Shading the Area

This solution corresponds to the area under the standard normal curve between \(z = -2.20\) and \(z = 1.40\). To visualize it, shade the area on the curve 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.

Probability Calculation
Probability calculation is key to understanding the behavior of random variables within a certain range. In this exercise, we are interested in finding the probability that a standard normal random variable, denoted as \(z\), falls between two specific values, \(-2.20\) and \(1.40\). This is symbolized as \(P(-2.20 \leq z \leq 1.40)\).

To calculate this probability, we focus on the area under the standard normal distribution curve between these two points. The standard normal distribution is symmetric and centered around 0 with a standard deviation of 1. Using a Z-table, which is a reference for cumulative probabilities, helps us find
  • the cumulative probability from \(-\infty\) up to \(-2.20\) and
  • the cumulative probability from \(-\infty\) up to \(1.40\).
This method allows us to determine the probability or the 'area' between these two values. By subtracting the lower cumulative probability from the upper one, we arrive at the probability of \(z\) being in that range. This calculated probability represents the shaded region on the standard normal curve.
Z-Table
The Z-table, also known as the standard normal distribution table, is a crucial tool in probability theory. It provides the cumulative probability of a standard normal random variable being less than or equal to a specific \(z\) value. In other words, it shows us the area under the curve from \(-\infty\) to that \(z\) value.

When given a problem, like finding \(P(-2.20 \leq z \leq 1.40)\), we use the Z-table to lookup up each individual \(z\) value:
  • For \(z = -2.20\), the table tells us that approximately 0.0139 of the data falls to the left of this point.
  • For \(z = 1.40\), about 0.9192 of the data lies to the left.
These values are essential for calculating probabilities because they show the likelihood of a \(z\) value being less than or equal to certain negative and positive standardized scores. Understanding how to use the Z-table simplifies finding probabilities related to standard normal distributions.
Normal Curve
The normal curve, often referred to simply as the bell curve, represents the standard normal distribution graphically. This bell-shaped curve is symmetric about the mean, which is 0 for a standard normal distribution. The curve's spread is determined by the standard deviation, which is 1 in this case.

The area under this curve corresponds to probability. It's why shading a specific portion, like between \(z = -2.20\) and \(z = 1.40\), visualizes the probability of \(z\) values falling within that range. The entire area under the curve sums up to 1, representing the total probability for all possible outcomes.

Because the curve is uniform in its ascent and descent, moving from the center to either end, probabilities around the mean are larger and taper off as you move toward the tails. This aspect of the normal curve is fundamental in statistical analysis and probability calculations involving standard normal variables.

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Most popular questions from this chapter

Let \(x\) be a random variable that represents checkout time (time spent in the actual checkout process) in minutes in the express lane of a large grocery. Based on a consumer survey, the mean of the \(x\) distribution is about \(\mu=2.7\) minutes, with standard deviation \(\sigma=0.6\) minute. Assume that the express lane always has customers waiting to be checked out and that the distribution of \(x\) values is more or less symmetrical and mound-shaped. What is the probability that the total checkout time for the next 30 customers is less than 90 minutes? Let us solve this problem in steps. (a) Let \(x_{i}\) (for \(\left.i=1,2,3, \ldots, 30\right)\) represent the checkout time for each customer. For example, \(x_{1}\) is the checkout time for the first customer, \(x_{2}\) is the checkout time for the second customer, and so forth. Each \(x_{i}\) has mean \(\mu=2.7\) minutes and standard deviation \(\sigma=0.6\) minute. Let \(w=x_{1}+x_{2}+\cdots+x_{30}\) Explain why the problem is asking us to compute the probability that \(w\) is less than 90 . (b) Use a little algebra and explain why \(w<90\) is mathematically equivalent to \(w / 30<3 .\) Since \(w\) is the total of the \(30 x\) values, then \(w / 30=\bar{x}\). Therefore, the statement \(\bar{x}<3\) is equivalent to the statement \(w<90\). From this we conclude that the probabilities \(P(\bar{x}<3)\) and \(P(w<90)\) are equal. (c) What does the central limit theorem say about the probability distribution of \(\bar{x} ?\) Is it approximately normal? What are the mean and standard deviation of the \(\bar{x}\) distribution? (d) Use the result of part (c) to compute \(P(\bar{x}<3)\). What does this result tell you about \(P(w<90)\) ?

Let \(z\) be a random variable with a standard normal distribution. Find the indicated probability, and shade the corresponding area under the standard normal curve. $$ P(-2.18 \leq z \leq-0.42) $$

What is a population parameter? Give three examples.

Let \(z\) be a random variable with a standard normal distribution. Find the indicated probability, and shade the corresponding area under the standard normal curve. $$ P(z \leq 3.20) $$

Porphyrin is a pigment in blood protoplasm and other body fluids that is significant in body energy and storage. Let \(x\) be a random variable that represents the number of milligrams of porphyrin per deciliter of blood. In healthy adults, \(x\) is approximately normally distributed with mean \(\mu=38\) and standard deviation \(\sigma=12\) (see reference in Problem 25). What is the probability that (a) \(x\) is less than 60 ? (b) \(x\) is greater than \(16 ?\) (c) \(x\) is between 16 and 60 ? (d) \(x\) is more than 60 ? (This may indicate an infection, anemia, or another type of illness.)
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