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Chapter 2: Problem 121

Use the following information to answer the next two exercises. X = the number of days per week that 100 clients use a particular exercise facility. $$\begin{array}{|l|l|}\hline x & {\text { Frequency }} \\ \hline 0 & {3} \\\ \hline 1 & {12} \\ \hline 2 & {33} \\ \hline 3 & {28} \\ \hline 4 & {11} \\\ \hline 5 & {9} \\ \hline 6 & {4} \\ \hline\end{array}$$ The number that is 1.5 standard deviations BELOW the mean is approximately _____ a. 0.7 b. 4.8 c. \(-2.8\) d. Cannot be determined

Short Answer

Expert verified
The correct answer is (a) 0.7.

Step by step solution

01

Understand the Problem

We need to find the value 1.5 standard deviations below the mean of the number of days clients use the facility.
02

Compute the Mean

To find the mean, calculate the sum of the product of each value of \(x\) and its frequency, and then divide by the total number of clients. \[ \text{Mean} = \frac{0 \times 3 + 1 \times 12 + 2 \times 33 + 3 \times 28 + 4 \times 11 + 5 \times 9 + 6 \times 4}{100} \] Calculating this gives:\[ \text{Mean} = \frac{0 + 12 + 66 + 84 + 44 + 45 + 24}{100} = \frac{275}{100} = 2.75 \]
03

Compute the Variance

To compute the variance, calculate the sum of the squared difference between each value of \(x\) and the mean, weighted by the frequency, and divide by the total number of clients.\[ \text{Variance} = \frac{(0-2.75)^2 \times 3 + (1-2.75)^2 \times 12 + (2-2.75)^2 \times 33 + (3-2.75)^2 \times 28 + (4-2.75)^2 \times 11 + (5-2.75)^2 \times 9 + (6-2.75)^2 \times 4}{100} \] Calculate the squared deviations and multiply by frequencies, summing these gives:\[ \text{Variance} = \frac{22.6875 + 36.75 + 18.5625 + 1.75 + 16.4375 + 51.1875 + 26.8125}{100} = \frac{174.1875}{100} = 1.741875 \]
04

Compute the Standard Deviation

The standard deviation is the square root of the variance:\[ \text{Standard Deviation} = \sqrt{1.741875} \approx 1.32 \]
05

Determine the Value 1.5 Standard Deviations Below Mean

Calculate the value 1.5 standard deviations below the mean by subtracting \(1.5 \times \text{Standard Deviation}\) from the mean:\[ 2.75 - 1.5 \times 1.32 = 2.75 - 1.98 = 0.77 \]
06

Choose the Closest Multiple Choice Answer

The closest answer option to 0.77 is (a) 0.7.

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Key Concepts

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

Mean
In simple terms, the mean is the average of a set of numbers. Calculating the mean helps us understand the central value within a dataset. To find the mean, sum up all the numbers and then divide this total by the count of numbers. In the scenario of our exercise facility, we're interested in how often clients visit the gym on average per week.
  • First, multiply each number of days (x) by the corresponding frequency (how many clients visited that many days per week).
  • Add up all these products.
  • Finally, divide this total by the total number of clients, which in our case is 100.
From the calculated values, the mean is found to be 2.75 days per week. This figure represents the average number of days clients are likely to visit the exercise facility.
Variance
Variance measures how much the numbers in a dataset differ from the mean. It's a central concept in statistics because it tells us about the spread or "spread-outedness" of the dataset.
  • To compute variance, subtract the mean from each number in the dataset to find the deviation.
  • Square each of these deviations.
  • Then, multiply each squared deviation by the frequency of its corresponding x value.
  • Add up these values to get the sum of all squared differences.
  • Divide this total by the number of data points (or clients, in this scenario, which is 100).
The result in our exercise is a variance of approximately 1.741875. This value indicates the degree of variation from the mean over the dataset of client visits.
Standard Deviation
Standard deviation is closely linked to the concept of variance, as it is simply the square root of the variance. This measure gives us a more intuitive understanding, as it returns us to the same units as our original data.
  • Calculate standard deviation by taking the square root of the variance.
For our dataset, the standard deviation came out to be approximately 1.32. Standard deviation can provide insights into the overall consistency of the dataset. The smaller the standard deviation, the more closely the data clusters around the mean. When we say that something is "1.5 standard deviations below the mean," we are measuring a precise distance from the average.
Frequency Distribution
Frequency distribution is an overview of how often different values occur within a dataset. It offers a snapshot of the dataset, showing the count of data points within specific range categories.
  • A frequency distribution table lists each individual value of x with the frequency, meaning how many times each occurs.
  • This gives a clearer understanding of the entire dataset at a glance.
  • It helps in visualizing patterns, trends, and potential anomalies.
In our exercise, the frequency distribution tells us, for example, how many clients visit the exercise facility 0 days, 1 day, 2 days, and so on. It's crucial in forming the basis for further statistical analysis, such as calculating the mean, variance, and standard deviation.

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