/*! 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 4 What is meant by the \(95 \%\) c... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 4

What is meant by the \(95 \%\) confidence interval of the mean?

Short Answer

Expert verified
The 95% confidence interval for the mean is a range where the true mean is likely to be, with 95% certainty.

Step by step solution

01

Understanding the Concept

A confidence interval is a range of values that is used to estimate the true population parameter. In this exercise, it is focusing on the mean of a dataset.
02

Defining the 95% Confidence Level

The 95% confidence level means that if you were to take 100 different samples and compute a confidence interval for each sample, you would expect about 95 of the intervals to contain the true population mean.
03

Interval Interpretation

The interval has an upper and lower bound, calculated from sample data. These bounds signify where the true population mean is likely to be found, with a specified level of confidence, which in this case is 95%.
04

Mathematical Representation

Mathematically, a 95% confidence interval is often written as: \[\bar{x} \pm Z_{0.025} \left(\frac{\sigma}{\sqrt{n}}\right)\]Where: \(\bar{x}\) is the sample mean, \(Z_{0.025}\) corresponds to the critical value from the Z-distribution for 95% confidence, \(\sigma\) is the population standard deviation, \(n\) is the sample size.
05

Clarifying with Example

For instance, if you calculate a 95% confidence interval for the mean to be from 30 to 40, you interpret that you are 95% confident that the true mean of the population lies between 30 and 40.

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.

95% Confidence Level
When we talk about a 95% confidence level, we are referring to the degree of certainty we have that a specific statistical result contains the true population parameter, in this case, the mean. Imagine you were able to repeatedly sample from the population and calculate a confidence interval each time. If 95 out of 100 such intervals actually capture the true population mean, that is what a 95% confidence level signifies.

In practical terms, this means that while individual intervals might miss the mark on occasion, the technique of using a 95% confidence interval is reliable overall, capturing the population mean most of the time.
Z-distribution
The Z-distribution, also known as the standard normal distribution, is a fundamental concept in statistics. It is used to calculate probabilities and to help determine the critical values for confidence intervals.

This distribution is symmetric and bell-shaped, centered around a mean of 0, with a standard deviation of 1. Using the Z-distribution, we can find critical values which are necessary for constructing confidence intervals.
  • Pivotal for standardizing data, allowing for comparison.
  • Key in determining the \(Z_{0.025}\) value, which corresponds to the area of 0.025 in each tail, when achieving a 95% confidence level.
  • Helps define the range within which the population mean is expected to fall.
Population Mean
The population mean, often represented by \(\mu\), is the average of all observations in a population. Since it is typically not feasible to measure the entire population, we use sample data to make estimates.

We estimate this population mean using the sample mean \(\bar{x}\), which serves as an unbiased estimator. We then use tools like the confidence interval to provide a range where the true population mean might lie.
  • Helps provide insights into the average characteristic of the population.
  • Serves as a benchmark for comparing sample data.
  • Essential for many statistical analyses and interpretations.
Sample Data
Sample data consists of a subset of observations taken from a larger population. Since analyzing an entire population is often impractical, statisticians use sample data to make inferences about the population.

From sample data, we calculate statistics such as the sample mean \(\bar{x}\) and sample standard deviation \(s\). These statistics then inform the calculation of confidence intervals.
  • Provides a practical way to estimate population parameters.
  • Helps identify trends and patterns applicable to the population.
  • Serves as a foundation for constructing confidence intervals and testing hypotheses.

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

Parking Meter Revenue A one-sided confidence inter- val can be found for a mean by using $$ \mu>\bar{X}-t_{\alpha} \frac{s}{\sqrt{n}} \quad \text { or } \quad \mu<\bar{X}+t_{\alpha} \frac{s}{\sqrt{n}} $$ $$ \begin{array}{l}{\text { where } t_{\alpha} \text { is the value found under the row labeled One }} \\ {\text { tail. Find two one-sided } 95 \% \text { confidence intervals of }} \\ {\text { the population mean for the data shown, and interpret }}\end{array} $$$$ \begin{array}{l}{\text { the answers. The data represent the daily revenues in }} \\ {\text { dollars from } 20 \text { parking meters in a small municipality. }}\end{array} $$ $$ \begin{array}{llll}{2.60} & {1.05} & {2.45} & {2.90} \\ {1.30} & {3.10} & {2.35} & {2.00} \\ {2.40} & {2.35} & {2.40} & {1.95} \\ {2.80} & {2.50} & {2.10} & {1.75} \\ {1.00} & {2.75} & {1.80} & {1.95}\end{array} $$

Holiday Gifts A survey of 100 Americans found that \(68 \%\) said they find it hard to buy holiday gifts that convey their true feelings. Find the \(90 \%\) confidence interval of the population proportion.

Calculator Battery Lifetimes A confidence interval for a standard deviation for large samples taken from a normally distributed population can be approximated by $$ s-z_{\alpha / 2} \frac{s}{\sqrt{2 n}}<\sigma

Find each. $$ \begin{array}{l}{\text { a. } z_{a / 2} \text { for the } 99 \% \text { confidence interval }} \\ {\text { b. } z_{a / 2} \text { for the } 98 \% \text { confidence interval }} \\ {\text { c. } z_{a / 2} \text { for the } 95 \% \text { confidence interval }} \\ {\text { d. } z_{a / 2} \text { for the } 90 \% \text { confidence interval }} \\ {\text { e. } z_{a / 2} \text { for the } 94 \% \text { confidence interval }}\end{array} $$

State Gasoline Taxes A random sample of state gaso- line taxes (in cents) is shown here for 12 states. Use the data to estimate the true population mean gasoline tax with \(90 \%\) confidence. Does your interval contain the national average of 44.7 cents? $$ \begin{array}{llllll}{38.4} & {40.9} & {67} & {32.5} & {51.5} & {43.4} \\\ {38} & {43.4} & {50.7} & {35.4} & {39.3} & {41.4}\end{array} $$
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Geometry

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.