91Ó°ÊÓ

Waiting The following data represent the number of customers waiting for a table at 6: 00 P.M. for 40 consecutive Saturdays at Bobak's Restaurant: $$\begin{aligned} &\begin{array}{|c|ccc|} \hline 11 & 5 & 11 & 3 \\ \hline 4 & 5 & 13 & 9 \\ \hline 13 & 10 & 9 & 6 \\ \hline 10 & 8 & 7 & 3 \\ \hline 7 & 9 & 10 & 4 \\ \hline 6 & 8 & 6 & 7 \\ \hline 6 & 4 & 14 & 11 \\ \hline 8 & 10 & 9 & 5 \\ \hline 8 & 8 & 7 & 8 \\ \hline 8 & 6 & 11 & 8 \end{array}\\\ &\end{aligned}$$ (a) Construct a frequency distribution of the data. (b) Construct a relative frequency distribution of the data. (c) What percentage of the Saturdays had 10 or more customers waiting for a table at 6: 00 P.M.? (d) What percentage of the Saturdays had five or fewer customers waiting for a table at 6: 00 P.M.? (e) Construct a frequency histogram of the data. (f) Construct a relative frequency histogram of the data. (g) Describe the shape of the distribution.

Step by step solution

01

- Tally the Frequency Distribution

Count how many times each number of customers occurs in the dataset. Organize this into a frequency distribution table. For example: \(\begin{array}{|c|c|} \hline \text{Number of Customers} & \text{Frequency} \hline 3 & 3 \hline 4 & 3 \hline \text{...} & \text{...} \hline 14 & 1 \hline \end{array}\)

02

- Construct the Relative Frequency Distribution

Calculate the relative frequency for each number of customers by dividing the frequency by the total number of observations (40). Organize this into a table as well. For instance: \(\begin{array}{|c|c|} \hline \text{Number of Customers} & \text{Relative Frequency} \hline 3 & 0.075 \hline 4 & 0.075 \hline \text{...} & \text{...} \hline 14 & 0.025 \hline \end{array}\)

03

- Calculate Percentage for 10 or More Customers

Find the sum of the relative frequencies for the number of customers 10 or more, then convert that to a percentage by multiplying by 100.

04

- Calculate Percentage for 5 or Fewer Customers

Find the sum of the relative frequencies for the number of customers 5 or fewer, then convert that to a percentage by multiplying by 100.

05

- Construct a Frequency Histogram

Create a histogram with the number of customers on the x-axis and the frequency on the y-axis. Each bar should represent how often each number of customers occurred.

06

- Construct a Relative Frequency Histogram

Create another histogram, this time with the number of customers on the x-axis and the relative frequency on the y-axis.

07

- Describe the Shape of the Distribution

Observe the histograms and determine whether the distribution is symmetric, skewed to the left, skewed to the right, or bimodal.

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 91Ó°ÊÓ!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Frequency Distribution

In a frequency distribution, we organize data into different categories based on the number of occurrences. For example, in the given exercise, we count how many times each number of customers waiting at Bobak's Restaurant appears within the dataset of 40 Saturdays. The frequency distribution helps us to clearly see which numbers are more common. For instance:
* 3 customers: 3 times
* 4 customers: 3 times
This table helps us to understand the dataset better by clearly showing the distribution pattern.

Relative Frequency Distribution

Relative frequency distribution is a way to express the data as a fraction or a percentage of the total number of observations. It provides a more proportional view of the data. To calculate relative frequency, divide the frequency of each category by the total number of observations. For example, if 3 customers were observed 3 times out of 40 Saturdays, the relative frequency is 3/40 = 0.075, or 7.5%. This percentage representation helps us to understand the portion of each category relative to the entire dataset.

Histogram

A histogram is a graphical representation of the frequency distribution. The x-axis represents the number of customers, while the y-axis shows the frequency (or relative frequency). Each bar in the histogram represents the number of times a particular number of customers was observed. For example, a bar representing 8 customers would be taller if 8 customers were frequently observed. Creating a frequency histogram gives us a visual way to identify patterns in the data, such as which range of customer numbers is most common.

Percentage Calculation

To find the percentage of Saturdays with 10 or more customers, add the frequencies for all customer numbers from 10 upwards, divide the sum by 40 (total observations), and multiply by 100. For example: percentage = (sum of frequencies for 10, 11, 13, 14) / 40 * 100.
Similarly, for calculating Saturdays with five or fewer customers, sum the frequencies for customer numbers 5 and below, divide by 40, and multiply by 100. These calculations are crucial for understanding how often specific conditions occur within the data set.

Shape of Distribution

The shape of the distribution tells us about how the data is spread or clustered. When we look at the histograms, we can determine if the data is symmetric (where data is evenly spread out), skewed to the left (more data on the right side), or skewed to the right (more data on the left side). For example, if most customers fall between the numbers 6 to 11 and only a few are outside of this range, the distribution might be symmetric. By understanding the shape, we gain insights into the data trends and patterns.