/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 26 Find the following: a. \(\quad... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 26

Find the following: a. \(\quad P(-3.052.43)\)

Short Answer

Expert verified
a. 0.4989, b. 0.9073, c. 0.0150, d. 0.0074

Step by step solution

01

Find the probability for a. \( P(-3.05

First, lookup the z-score for -3.05 in a standard Z-table, which gives 0.0011. This means that 0.0011 (or 0.11%) of the distribution falls to the left of -3.05. Because \(z<0.00\) represents the left half of the entire normal distribution, it represents 0.50 (or 50%) of the distribution. So, to find \(P(-3.05<z<0.00)\), subtract the Z-table value of -3.05 from 0.5: \(0.5 - 0.0011 = 0.4989\)
02

Find the probability for b. \(P(-2.43

First, lookup Z-table values for both -2.43 and 1.37 which are 0.0074 and 0.9147 respectively. To find the probability, subtract the smaller Z-table value from the larger one: \(0.9147 - 0.0074 = 0.9073\).
03

Find the probability for c. \(P(z

Lookup the z-score for -2.17 in a standard Z-table. The result is 0.0150 which directly provides the required probability as it falls to the left of -2.17.
04

Find the probability for d. \(P(z>2.43)\)

Lookup the z-score for 2.43 in a standard Z-table, which gives 0.9926. The required probability falls to the right of 2.43. Since the total probability is 1, subtract the Z-table value from 1 to get the probability: \(1 - 0.9926 = 0.0074\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Z-Score
The Z-score is an essential concept in statistics, especially when dealing with the standard normal distribution. It represents how many standard deviations a data point is from the mean of the distribution. To calculate a Z-score, you use the formula:\[ Z = \frac{X - \mu}{\sigma} \]where:
  • \(X\) is the value you are evaluating.
  • \(\mu\) is the mean of the distribution.
  • \(\sigma\) is the standard deviation of the distribution.
A Z-score allows us to standardize values, making it easier to compare different data points from different normal distributions. Using Z-scores, we determine the probability of a value occurring within a normal distribution or assess how extreme it is.
Probability Calculation
Probability calculation in the context of the standard normal distribution helps us understand the likelihood of a Z-score falling within a certain range. We are typically interested in:
  • The probability of a Z-score being less than a particular value.
  • The probability of a Z-score being greater than a particular value.
  • The probability of a Z-score falling between two values.
For instance, to find the probability of a Z-score lying between two values, like in step b: We first find the cumulative probabilities for each score using the Z-table and then subtract the smaller value from the larger one. This calculation helps us understand the fraction of the normal distribution that resides between those two z-scores.
Z-Table Lookup
The Z-table is a tool used to find the cumulative probability associated with a Z-score. This table gives the probability that a standard normal random variable is less than a certain value. To use the Z-table:
  • Locate the Z-score in the table's row and column (usually represented by tenths and hundredths).
  • Find the intersection that gives the cumulative probability.
For example, if you are looking for the probability \(P(z < -2.17)\), you first find the Z-score of -2.17 in the table, which provides the cumulative probability of 0.0150, indicating the area to the left of the Z-score.Understanding Z-table lookup is crucial for determining probabilities related to standard normal distributions effectively.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Given that \(z\) is the standard normal variable, find the value of \(c\) such that: a. \(P(|z|>c)=0.0384\). b. \(P(|z|< c)=0.8740\).

Find the following: a. \(\quad P(0.001.25)\) d. \(\quad P(z<1.75)\)

Assume that the distribution of data in Exercise 6.137 was exactly normally distributed, with a mean of 0.00 and a standard deviation of 0.020. a. Find the bounds of the middle \(95 \%\) of the distribution. b. What percent of the data is actually within the interval found in part a? c. Using \(z\) -scores, determine the percentage of estimated conformance to specification. That is, what percentage of the measurements would be expected to fall within the specification range of \(0.000 \pm 0.030\) units?

Find the following areas under the standard normal curve. a. To the right of \(z=3.18, P(z>3.18)\) b. To the right of \(z=1.84, P(z>1.84)\) c. To the right of \(z=0.75, P(z>0.75)\)

According to Federal Highway Administration 2007 statistics, the percent of licensed female drivers has just surpassed the percent of licensed male drivers. Of the drivers in the United States, \(50.2 \%\) are females. If a random sample of 50 drivers is to be selected for a survey, a. what is the probability that no more than half (25) of the drivers are female? b. what is the probability that at least three-fourths (38) of the drivers are female?
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.