/*! 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 51 Use the following information to... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 51

Use the following information to answer the next three exercises: State whether the data are symmetrical, skewed to the left, or skewed to the right. 87; 87; 87; 87; 87; 88; 89; 89; 90; 91

Short Answer

Expert verified
The data is skewed to the right.

Step by step solution

01

Organize the Data

First, list the data set in order from least to greatest: 87, 87, 87, 87, 87, 88, 89, 89, 90, 91.
02

Calculate the Mean

Calculate the mean (average) by adding all the data points together and dividing by the number of data points. \[ \text{Mean} = \frac{87 + 87 + 87 + 87 + 87 + 88 + 89 + 89 + 90 + 91}{10} = \frac{882}{10} = 88.2 \]
03

Identify the Median

Find the middle value of the ordered data set. Since there are 10 values (an even number), the median will be the average of the 5th and 6th values: \[ \text{Median} = \frac{87 + 88}{2} = 87.5 \]
04

Determine Skewness

Compare the mean and the median to evaluate skewness. If the mean is greater than the median, the data is skewed to the right. If the mean is less than the median, it is skewed to the left. If they are approximately equal, the data is symmetric. Here, the mean (88.2) is greater than the median (87.5), indicating the data is skewed to the right.

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.

Symmetrical Data
In statistics, when we talk about symmetrical data, we refer to a situation where the left and right halves of the data distribution are mirror images of each other. This means that if you were to fold the distribution in half at the central point, each side would align perfectly. Symmetrical data is typically characterized by specific features:
  • The mean, median, and mode all coincide at the center of the distribution.
  • The shape of the distribution on a graph, such as a histogram, is bell-shaped.
  • Outliers are minimal or balanced across the center point.
Symmetrical data is often described by the normal distribution (bell curve), which is an important concept in probability and statistics. Knowing whether your data is symmetrical helps in choosing the right analysis techniques and in predicting the behavior of the dataset more accurately.
Mean and Median Relationship
The relationship between the mean and median plays a vital role in determining the skewness of a dataset. This relationship helps us understand the shape and central tendency of the data distribution. Let's break down how it works:
  • If the mean and median are similar or close, the data tends to be symmetrical, implying a balanced distribution around the central point.
  • If the mean is greater than the median, the data is generally skewed to the right (positively skewed). This means the tail is longer on the right side.
  • If the mean is less than the median, the data is skewed to the left (negatively skewed), indicating a longer tail on the left side.
In our exercise, the mean of 88.2 was higher than the median of 87.5. This suggests that the dataset is skewed to the right, meaning most values cluster on the lower end, with a few higher values stretching out on the higher side.
Evaluating Data Distribution
Evaluating a data distribution involves analyzing its shape, spread, and central tendency to fully understand its characteristics. This evaluation is crucial for selecting the right statistical methods and interpreting data correctly. Here's how you can do it:
  • **Shape**: Look at the shape depicted by a histogram or a frequency plot. Is it symmetrical, skewed, bell-shaped, or unimodal/bimodal?
  • **Central Tendency**: Evaluate the mean, median, and mode to understand the center of your distribution. Comparing these can reveal the skewness.
  • **Spread**: Consider the range, variance, and standard deviation to comprehend how spread out the data points are.
  • **Outliers**: Identify any significantly deviating values that might impact your analysis.
By understanding these elements, you can determine how best to use your data for analysis, hypotheses testing, or predictive modeling. In the given dataset, the skewness to the right is identified through the relationship between the mean and median and verified by the distribution's spread and possible outlier influence.

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 students in Ms. Ramirez鈥檚 math class have birthdays in each of the four seasons. Table 2.40 shows the four seasons, the number of students who have birthdays in each season, and the percentage (%) of students in each group. Construct a bar graph showing the number of students. $$ \begin{array}{|l|l|}\hline \text { Seasons } & {\text { Number of students }} & {\text { Proportion of population }} \\ \hline \text { Spring } & {8} & {24 \%} \\ \hline \text { Summer } & {9} & {26 \%} \\ \hline \text { Autumn } & {11} & {32 \%} \\ \hline \text { Winter } & {6} & {18 \%} \\\ \hline\end{array} $$

Listed are 29 ages for Academy Award winning best actors in order from smallest to largest. 18; 21; 22; 25; 26; 27; 29; 30; 31; 33; 36; 37; 41; 42; 47; 52; 55; 57; 58; 62; 64; 67; 69; 71; 72; 73; 74; 76; 77 a. Find the \(40^{\text { th }}\) percentile. b. Find the \(78^{\text { th }}\) percentile.

For each of the following data sets, create a stem plot and identify any outliers. The miles per gallon rating for 30 cars are shown below (lowest to highest). 19, 19, 19, 20, 21, 21, 25, 25, 25, 26, 26, 28, 29, 31, 31, 32, 32, 33, 34, 35, 36, 37, 37, 38, 38, 38, 38, 41, 43, 43

Use the following information to answer the next three exercises: The following data show the lengths of boats moored in a marina. The data are ordered from smallest to largest: 16; 17; 19; 20; 20; 21; 23; 24; 25; 25; 25; 26; 26; 27; 27; 27; 28; 29; 30; 32; 33; 33; 34; 35; 37; 39; 40 Identify the median.

The mean and median for the data are the same. 3; 4; 5; 5; 6; 6; 6; 6; 7; 7; 7; 7; 7; 7; 7 Is the data perfectly symmetrical? Why or why not?
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Calculus

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.