/*! 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 2 A sample of 81 observations is t... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 2

A sample of 81 observations is taken from a normal population with a standard deviation of \(5 .\) The sample mean is \(40 .\) Determine the 95 percent confidence interval for the population mean.

Short Answer

Expert verified
The 95% confidence interval for the population mean is (38.91, 41.09).

Step by step solution

01

Identify the Known Values

From the problem, we know the sample size \( n = 81 \), the sample standard deviation \( \sigma = 5 \), and the sample mean \( \bar{x} = 40 \).
02

Determine the Z-score for 95% Confidence

A 95% confidence interval corresponds to the Z-score that captures 95% of the data from a standard normal distribution. For a two-tailed test, this Z-score is \( z = 1.96 \).
03

Calculate the Standard Error

The standard error (SE) of the sample mean is calculated using the formula \( SE = \frac{\sigma}{\sqrt{n}} \). Substituting the known values gives \( SE = \frac{5}{\sqrt{81}} = \frac{5}{9} \approx 0.556 \).
04

Compute the Margin of Error

The margin of error (ME) is found by multiplying the Z-score by the standard error. So, \( ME = 1.96 \times 0.556 \approx 1.0896 \).
05

Determine the Confidence Interval

The 95% confidence interval is calculated using the formula \( \bar{x} \pm ME \). Here, it becomes \( 40 \pm 1.0896 \). Simplifying this, the interval is \( (38.9104, 41.0896) \).

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.

Sample Mean
The sample mean is a crucial concept in statistics and represents the average value of a set of observations. It is denoted as \( \bar{x} \), and it is calculated by summing all the observations in a sample and then dividing by the total number of observations in that sample.
The formula for calculating the sample mean is:\[\bar{x} = \frac{\sum{x_i}}{n}\]Where:
  • \( \sum{x_i} \) is the sum of all observed values.
  • \( n \) is the number of observations.
In the context of the exercise, the sample mean \( \bar{x} = 40 \) is calculated from 81 observations. The sample mean is often used in inferential statistics to estimate the population mean, especially when direct measurement of the population is impractical.
Standard Error
The concept of standard error (SE) is fundamental in understanding how the sample mean represents the population mean. It provides an estimate of the variability of the sample mean from the population mean.

The formula for calculating the standard error is:\[SE = \frac{\sigma}{\sqrt{n}}\]Where:
  • \( \sigma \) is the standard deviation of the population.
  • \( n \) is the number of observations in the sample.
In the exercise example, the standard deviation \( \sigma \) is given as \( 5 \), and the standard error is calculated using the sample size of 81, resulting in an SE of approximately \( 0.556 \). This small standard error indicates that there is little variability between the sample mean and the true population mean, suggesting a reliable sample.
Z-score
The Z-score ties the observed data to the standard normal distribution, allowing statisticians to understand probabilities and make inferences about the population. In this context, the Z-score is crucial for calculating the confidence interval.
A 95% confidence interval means that we expect 95% of sample means to lie within this interval when repeated samples are taken. For a 95% confidence level in a standard normal distribution, the Z-score is commonly \( 1.96 \).
This Z-score reflects the number of standard deviations an observation is from the mean. By using the Z-score, we can scale the standard error to capture the desired level of confidence around the sample mean.
Margin of Error
The margin of error is an essential component of confidence intervals, quantifying the range within which we expect the true population mean to lie, relative to the sample mean.
The margin of error is calculated by multiplying the Z-score by the standard error:\[ME = Z \times SE\]In our exercise, the margin of error is computed as \( 1.96 \times 0.556 \approx 1.0896 \).
This value is then used to establish the range of the confidence interval around the sample mean. For example, with a sample mean of 40, the confidence interval would fall between \( 38.9104 \) and \( 41.0896 \), indicating that we are 95% confident that the true population mean is within this range.

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

16\. Ms. Maria Wilson is considering running for mayor of the town of Bear Gulch, Montana. Before completing the petitions, she decides to conduct a survey of voters in Bear Gulch. A sample of 400 voters reveals that 300 would support her in the November election. a. Estimate the value of the population proportion. b. Develop a 99 percent confidence interval for the population proportion. c. Interpret your findings.

A sample of 352 subscribers to Wired magazine shows the mean time spent using the Internet is 13.4 hours per week, with a sample standard deviation of 6.8 hours. Find the 95 percent confidence interval for the mean time Wired subscribers spend on the Internet.

Police Chief Edward Wilkin of River City reports 500 traffic citations were issued last month. A sample of 35 of these citations showed the mean amount of the fine was \(\$ 54,\) with a standard deviation of \(\$ 4.50 .\) Construct a 95 percent confidence interval for the mean amount of a citation in River City.

A student conducted a study and reported that the 95 percent confidence interval for the mean ranged from 46 to \(54 .\) He was sure that the mean of the sample was 50 that the standard deviation of the sample was \(16,\) and that the sample was at least \(30,\) but could not remember the exact number. Can you help him out?

It is estimated that 60 percent of U.S. households subscribe to cable TV. You would like to verify this statement for your class in mass communications. If you want your estimate to be within 5 percentage points. with a 95 percent level of confidence, how large of a sample is required?
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

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.