/*! 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 44 Fat contents (in percentage) for... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 44

Fat contents (in percentage) for 10 randomly selected hot dogs were given in the article "Sensory and Mechanical Assessment of the Quality of Frankfurters" (Journal of Texture Studies [1990]: \(395-409\) ). Use the following data to construct a \(90 \%\) confidence interval for the true mean fat percentage of hot dogs: \(\begin{array}{llllllllll}25.2 & 21.3 & 22.8 & 17.0 & 29.8 & 21.0 & 25.5 & 16.0 & 20.9 & 19.5\end{array}\)

Short Answer

Expert verified
Based on these steps, calculate your numerical values to conclude what the 90% confidence interval for the true mean fat percentage of hot dogs is.

Step by step solution

01

Calculate the sample mean

First, find the sample mean (also known as the arithmetic mean) by adding up all the sample data values and dividing by the number of samples. In this case, add up the 10 fat contents and divide by 10.
02

Calculate the sample standard deviation

Next, calculate the sample standard deviation, which is a measure of the amount of variation within the sample data. You can use the following formula to calculate the sample standard deviation: \[ s = \sqrt{\frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2}\] where \(n\) is the number of samples, \(x_i\) are the individual sample measurements, and \(\bar{x}\) is the sample mean that we calculated in Step 1.
03

Calculate the standard error

The standard error is the standard deviation divided by the square root of the sample size. Calculate this as follows: \[SE = \frac{s}{\sqrt{n}}\] where \(s\) is the sample standard deviation and \(n\) is the number of samples.
04

Use the z-score to construct the confidence interval

A 90% confidence interval uses a z-score of 卤1.645 (z-scores represent the number of standard deviations from the mean a data point is). So, the confidence interval can be calculated as: \[(\bar{x} - z* SE, \bar{x} + z * SE)\] where \(\bar{x}\) is the sample mean, \(z\) is the z-score and \(SE\) is the standard error.

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
In statistics, the sample mean is a simple yet powerful concept. It represents the average of a set of numbers and provides a central value around which the data points are distributed. To compute the sample mean, add all the sample values together and divide by the number of values. For example, if we have fat content percentages as our data, we sum them up and divide by 10, since we are dealing with 10 samples. This measure helps in understanding the central tendency of our data.
  • Calculation: \( \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i \)
  • Purpose: Determines the center of your data set.
  • Importance: First step in statistical analysis.
Calculating the mean offers insights into the overall level of fat content in the hotdogs, simplifying the data's complexity.
Sample Standard Deviation
The sample standard deviation is a key statistic that quantifies the amount of variation or dispersion in a set of data values. It shows us how much individual scores differ from the sample mean. To calculate it, we take the square root of the variance, which is the average of the squared differences from the mean. This helps us understand the spread of the data.
  • Formula: \( s = \sqrt{\frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2} \)
  • Relevance: Shows variations and identifies outliers.
  • Role: A necessary step in further statistical procedures.
Having a higher standard deviation means the data points are spread out over a wider range, whereas a smaller standard deviation indicates that they are closer to the mean.
Standard Error
The standard error is a statistical analysis tool used to measure the accuracy of a sample mean by showing how much it deviates from the actual population mean. It is calculated by dividing the sample standard deviation by the square root of the sample size. It's vital for constructing confidence intervals.
  • Calculation: \( SE = \frac{s}{\sqrt{n}} \)
  • Significance: Indicates the precision of the sample mean as an estimate.
  • Uses: Smaller SE suggests more precise estimates.
Essentially, the standard error offers a glimpse into the reliability of the sample mean in representing the larger population.
Z-score
The z-score is an important concept in statistics that measures how many standard deviations a data point is from the mean. It is often used in the process of constructing confidence intervals, indicating where a specific data point lies in a standard normal distribution. For a 90% confidence interval, the z-score is typically 卤1.645.
  • Definition: Measures deviation from the mean.
  • Application: Calculate how typical or atypical a data point is.
  • Confidence Intervals: Necessary for interval estimation.
Applying z-scores allows statisticians to understand the probability of a data point occurring within a certain range, thus facilitating more informed decisions when analyzing and interpreting data sets.

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

The National Geographic Society conducted a study that included 3000 respondents, age 18 to 24, in nine different countries (San Luis Obispo Tribune, November 21 . 2002). The society found that \(10 \%\) of the participants could not identify their own country on a blank world map. a. Construct a \(90 \%\) confidence interval for the proportion who can identify their own country on a blank world map. b. What assumptions are necessary for the confidence interval in Part (a) to be valid? c. To what population would it be reasonable to generalize the confidence interval estimate from Part (a)?

Conducts an annual survey 6 of all households expe the past year. This esti102 randomly selected adults. The report states, "One can say with \(95 \%\) confidence that the margin of sampling error is \(\pm 3\) percentage points." Explain how this statement can be justified.

The article "The Association Between Television Viewing and Irregular Sleep Schedules Among Children Less Than 3 Years of Age" (Pediatrics [2005]: \(851-856\) ) reported the accompanying \(95 \%\) confidence intervals for average TV viewing time (in hours per day) for three different age groups. \begin{tabular}{lcc} Age Group & \(95 \%\) Confidence Interval \\ \hline Less than 12 months & \((0.8,1.0)\) \\ 12 to 23 months & \((1.4,1.8)\) \\ 24 to 35 months & \((2.1,2.5)\) \\ & & \end{tabular} a. Suppose that the sample sizes for each of the three age group samples were equal. Based on the given confidence intervals, which of the age group samples had the greatest variability in TV viewing time? Explain your choice. b. Now suppose that the sample standard deviations for the three age group samples were equal, but that the three sample sizes might have been different. Which of the three age group samples had the largest sample size? Explain your choice. c. The interval \((.768,1.302)\) is either a \(90 \%\) confidence interval or a \(99 \%\) confidence interval for the mean TV viewing time for children less than 12 months old. Is the confidence level for this interval \(90 \%\) or \(99 \% ?\) Explain your choice.

In an AP-AOL sports poll (Associated Press, December 18,2005 ), 394 of 1000 randomly selected U.S. adults indicated that they considered themselves to be baseball fans. Of the 394 baseball fans, 272 stated that they thought the designated hitter rule should either be expanded to both baseball leagues or eliminated. a. Construct a \(95 \%\) confidence interval for the proportion of U.S. adults that consider themselves to be baseball fans. b. Construct a \(95 \%\) confidence interval for the proportion of those who consider themselves to be baseball fans that think the designated hitter rule should be expanded to both leagues or eliminated. c. Explain why the confidence intervals of Parts (a) and (b) are not the same width even though they both have a confidence level of \(95 \%\).

9.14 Discuss how each of the following factors affects the width of the confidence interval for \(\pi\) : a. The confidence level b. The sample size c. The value of \(p\)
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

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.