/*! 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 10 A sample data set with a bell-sh... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 10

A sample data set with a bell-shaped distribution and size \(n=128\) has mean \(x^{\wedge}=2\) and standard deviation \(s=1.1\). Find the approximate number of observations in the data set that lie: a. below-0.2; b. below 3.1; c. between -1.3 and 0.9 .

Short Answer

Expert verified
3 below -0.2; 108 below 3.1; 20 between -1.3 and 0.9.

Step by step solution

01

Understand the Empirical Rule

The Empirical Rule, or the 68-95-99.7 rule, tells us that for a bell-shaped distribution, about 68% of the data lies within one standard deviation of the mean, 95% within two, and 99.7% within three. We'll use this rule to estimate the number of observations.
02

Calculate Z-Scores for Each Boundary

The Z-score formula is \( Z = \frac{X - \bar{x}}{s} \).- For \(-0.2\), \( Z = \frac{-0.2 - 2}{1.1} \approx -2.0 \).- For \(3.1\), \( Z = \frac{3.1 - 2}{1.1} \approx 1.0 \).- For \(-1.3\), \( Z = \frac{-1.3 - 2}{1.1} \approx -3.0 \).- For \(0.9\), \( Z = \frac{0.9 - 2}{1.1} \approx -1.0 \).
03

Use the Standard Normal Table

Refer to a standard normal distribution table or calculator for the probabilities associated with the calculated Z-scores.- For \(Z = -2.0\), probability \( \approx 0.0228 \).- For \(Z = 1.0\), probability \( \approx 0.8413 \).- For \(Z = -3.0\), probability \( \approx 0.0013 \).- For \(Z = -1.0\), probability \( \approx 0.1587 \).
04

Calculate Observations for Each Interval

Multiply the probabilities by the total number of observations \(n = 128\) to find the number of observations.- Below \(-0.2\): \(0.0228 \times 128 = 2.9 \approx 3 \) observations.- Below \(3.1\): \(0.8413 \times 128 = 107.7 \approx 108 \) observations.- Between \(-1.3\) and \(0.9\): \((0.1587 - 0.0013) \times 128 = 20.1 \approx 20 \) observations.

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.

Bell-Shaped Distribution
A bell-shaped distribution, often referred to as a normal distribution, is a type of probability distribution that is symmetrical and has a shape resembling a bell curve. This shape signifies that the data points are distributed in such a way that most of the data is concentrated around the mean.
  • The highest point of the bell curve is the mean, which is the average value of the data set.
  • The data falls away symmetrically on either side of the mean, with equal numbers of data points to the left and right.
The bell shape is crucial in statistics because it indicates that the data can be analyzed using standard statistical methods, such as the Empirical Rule. Knowing that a data set is bell-shaped allows us to estimate the spread of data across different intervals.
Z-Score Calculation
Z-score calculation is an essential statistical tool used to determine how far away a particular data point is from the mean, measured in terms of standard deviations. The formula to calculate a Z-score is:\[ Z = \frac{X - \bar{x}}{s} \]where:
  • \(X\) is the data point.
  • \(\bar{x}\) is the mean of the data set.
  • \(s\) is the standard deviation of the data set.
The Z-score is beneficial since it helps in standardizing different data sets, making them comparable. For instance, in the original problem, calculating Z-scores for values such as -0.2, 3.1, -1.3, and 0.9 allowed for a clear understanding of their positions relative to the mean.
Standard Normal Distribution
The standard normal distribution is a special case of the normal distribution. It has a mean of 0 and a standard deviation of 1. This unique distribution is also known as the Z-distribution.
  • Every normal distribution can be converted to a standard normal distribution using Z-scores.
  • This transformation allows us to use a standard normal table, also known as the Z-table, to find probabilities.
In the problem, after calculating the Z-scores, the next step involved referring to the standard normal distribution table to extract probabilities related to each Z-score. This standardizes data analysis, making it easier to estimate how data is distributed.
Probability Estimation
Probability estimation involves calculating the likelihood that a data point falls within a particular range in a distribution. The Empirical Rule helps in estimating these probabilities for normal distributions. Once Z-scores are calculated, we use these scores to find probabilities in the standard normal distribution table. This table provides the area under the curve to the left of a specific Z-score, representing the probability.
  • In our example, we find probabilities for Z-scores such as -2.0 and 1.0 to estimate numbers below specific values.
  • We also estimate ranges between different Z-scores, such as between -1.3 and 0.9.
These probabilities are then multiplied by the total number of observations to estimate the actual count of data points in those intervals. This approach aids in understanding the data distribution and making informed predictions.

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

A sample data set with a bell-shaped distribution has mean \(x^{\sim}=6\) and standard deviation \(s=2\). Find the approximate proportion of observations in the data set that lie: a. between 4 and 8 ; b. between 2 and 10 ; c. between 0 and 12 .

Find the mean, the median, and the mode for the number of vehicles owned in a survey of 52 households. \begin{tabular}{c|cccccccc} \(x\) & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline\(f\) & 2 & 12 & 15 & 11 & 6 & 3 & 1 & 2 \end{tabular}

Construct a stem and leaf diagram, a frequency histogram, and a relative frequency histogram for the following data set. For the histograms use classes \(51-60,61-70,\) and so on. \(\begin{array}{lllll}63 & 92 & 68 & 77 & 80\end{array}\) \(\begin{array}{lllll}70 & 85 & 88 & 85 & 36\end{array}\) \(\begin{array}{lllll}33 & 75 & 76 & 82 & 100 \\ 53 & 70 & 70 & 82 & 85\end{array}\)

Describe one difference between a frequency histogram and a relative frequency histogram.

Hockey pucks used in professional hockey games must weigh between 5.5 and 6 ounces. If the weight of pucks manufactured by a particular process is bell- shaped, has mean 5.75 ounces and standard deviation 0.125 ounce, what proportion of the pucks will be usable in professional games?
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

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.