/*! 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 16 The number of passengers in each... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 16

The number of passengers in each of 120 randomly observed vehicles during morning rush hour was recorded, with the following results. $$ \begin{array}{c|ccccc} x & 1 & 2 & 3 & 4 & 5 \\ \hline f & 84 & 29 & 3 & 3 & 1 \end{array} $$ Find the mean, the median, and the mode of this data set.

Short Answer

Expert verified
Mean: 1.4, Median: 1, Mode: 1.

Step by step solution

01

Calculate the Mean

To find the mean number of passengers, we multiply the number of passengers \(x\) by the frequency \(f\) for each value, sum these products, and then divide by the total number of vehicles. \[\text{Mean} = \frac{1 \times 84 + 2 \times 29 + 3 \times 3 + 4 \times 3 + 5 \times 1}{120}\] Calculating each product: \(1 \times 84 = 84\), \(2 \times 29 = 58\), \(3 \times 3 = 9\), \(4 \times 3 = 12\), \(5 \times 1 = 5\). The sum is \(84 + 58 + 9 + 12 + 5 = 168\). Thus, the mean is \(\frac{168}{120} = 1.4\).
02

Determine the Median

To find the median, we need the middle value of the dataset. Since there are 120 observations, the median will be the average of the 60th and 61st values in the ordered dataset. Listing the frequencies cumulatively: there are 84 vehicles with 1 passenger, and 113 vehicles with either 1 or 2 passengers. This means both the 60th and 61st values fall within the group of vehicles with 1 passenger. Therefore, the median number of passengers is 1.
03

Identify the Mode

The mode is the number of passengers that appears most frequently in our dataset. From the table, the highest frequency \(f\) is 84, which corresponds to \(x = 1\) passenger. Thus, the mode of this dataset is 1.

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.

Mean Calculation
The mean, sometimes known simply as the average, represents the central point of a dataset. It is calculated by dividing the sum of all observations by the total number of observations. In the context of the passengers observed, we multiplied the number of passengers in a vehicle by their corresponding frequency and added these results together. This gave us 168 as the total sum of all observed passengers.
The calculation of the mean can be summarized as follows:
  • Multiply each passenger count (1, 2, 3, 4, and 5) by the number of times it was observed (its frequency).
  • Add all these products together to get the total sum.
  • Divide this sum by the total number of vehicles, which is 120 in this case.
Thus, the mean number of passengers per vehicle is calculated as \( \frac{168}{120} = 1.4 \), indicating that on average, there are about 1.4 passengers per vehicle during the morning rush hour.
Median Determination
The median is a measure of the middle value in a dataset when it is ordered from the lowest to the highest. It provides a central location of the data, less affected by extreme values compared to the mean.
To find the median in this dataset, we need to determine the middle position. Since there are 120 vehicles, we look for the 60th and 61st values. The dataset is constructed by adding frequencies cumulatively:
  • The first group (1 passenger) accounted for 84 vehicles, covering the 1st to the 84th positions.
  • The second group (2 passengers) accounted for 29 vehicles, spanning from the 85th to the 113th positions.
Given this setup, both the 60th and 61st values are part of the first group, meaning they both have 1 passenger. Therefore, the median number of passengers is determined to be 1, a true reflection that more than half the vehicles had only one passenger.
Mode Identification
The mode is the most recurrent value in a dataset. Unlike the mean and median, which are related to the numerical center, the mode highlights the most common value observed.
In our vehicle dataset, the mode is easily identified by looking at the frequencies. The number 1 passenger frequency is 84, which is the highest among all groups:
  • 1 passenger appears in 84 vehicles.
  • 2 passengers appear in 29 vehicles.
  • Others (3, 4, and 5 passengers) have even lower frequencies.
Amongst all observations, having 1 passenger in a vehicle is significantly more common. Therefore, the mode for the number of passengers per vehicle is 1, showing that it is the most typical number of passengers during the morning rush hour.

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

Five laboratory mice with thymus leukemia are observed for a predetermined period of 500 days. After 450 days, three mice have died, and one of the remaining mice is sacrificed for analysis. By the end of the observational period, the last remaining mouse still survives. The recorded survival times for the five mice are \(\begin{array}{llllll}222 & 421 & 378 & 450^{*} & 500^{*}\end{array}\) where " indicates that the mouse survived for at least the given number of days but the exact value of the observation is unknown. a. Can you find the sample mean for the data set? If so, find it. If not, explain why not. b. Can you find the sample median for the data set? If so, find it. If not, explain why not.

During a one-day blood drive 300 people donated blood at a mobile donation center. The blood types of these 300 donors are summarized below. $$ \begin{array}{c|cccc} \text { Blood Type } & O & A & B & A B \\ \hline \text { Frequency } & 136 & 120 & 32 & 12 \end{array} $$ Identify the blood type that has the highest relative frequency for these 300 people. Can you conclude that the blood type you identified is also most common for all people in the population at large? Explain.

State Chebyshev's Theorem.

Describe the conditions under which the Empirical Rule may be applied.

Rosencrantz and Guildenstern are on a weight-reducing diet. Rosencrantz, who weighs \(178 \mathrm{lb}\), belongs to an age and body-type group for which the mean weight is \(145 \mathrm{lb}\) and the standard deviation is \(15 \mathrm{lb}\). Guildenstern, who weighs \(204 \mathrm{lb}\), belongs to an age and body-type group for which the mean weight is \(165 \mathrm{lb}\) and the standard deviation is 20 lb. Assuming z-scores are good measures for comparison in this context, who is more overweight for his age and body type?
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Statistics

Read Explanation

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

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.