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Chapter 2: Problem 76

Fifteen randomly selected college students were asked to state the number of hours they slept the previous night. The resulting data values are 5,6,6,8,7,7,9,5 \(4,8,11,6,7,8,7 .\) Find the following: a. \(\operatorname{mean}_{1} \bar{x}\) b. \(\operatorname{median}, \widetilde{x}\) c. mode d. midrange

Short Answer

Expert verified
The mean \(\bar{x}\) is 6.73 hours, median \(\widetilde{x}\) is 7 hours, mode is 7 hours and the midrange is 7.5 hours.

Step by step solution

01

Calculating the mean

The mean is calculated by adding up all the values and dividing by the number of values. The sum of the values is \(5+6+6+8+7+7+9+5+4+8+11+6+7+8+7 = 101\). And there are 15 values, so the mean is calculated as \(\bar{x} = \frac{101}{15} = 6.73\)
02

Calculating the median

First, you need to arrange the values in ascending order: 4,5,5,6,6,6,7,7,7,7,8,8,8,9,11. The median is the middle value. Since there are an odd number (15) of values, the median is the value in the middle, which is the eighth value: \(\widetilde{x} = 7\).
03

Identifying the mode

The mode is the most frequently occurring value in the data set. By inspecting the list, it can be seen that '7' appears more frequently than any other value in the data set, making it the mode.
04

Calculating the midrange

The midrange of a dataset is calculated by adding the highest and lowest values and dividing by 2. In this case, it will be: midrange = \(\frac{4+11}{2} = 7.5\)

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Key Concepts

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

Understanding the Mean
The mean is often referred to as the average and is one of the most common statistical measures. It represents the central value of a dataset. To find the mean, follow these simple steps:
  • Sum up all the data values.
  • Count the total number of values.
  • Divide the sum by the number of values.
These steps help calculate the mean accurately, showing a balanced point where the sums of deviations on either side are equal. In this exercise, calculating the mean of the students' sleep hours involves adding all the hours together, which gives us 101, and dividing by the total count of 15, thus giving a mean of approximately 6.73.
Delving into the Median
The median is another measure of central tendency but differs from the mean. It signifies the middle value in a dataset when all numbers are arranged in order. Here's how you find the median:
  • Arrange all numbers in ascending or descending order.
  • Identify the middle number if there is an odd amount of data points.
  • If even, calculate the average of the two middle numbers.
Often, the median gives a better sense of a typical value when your data has outliers or skewed values. In our case, arranging the sleep hours from smallest to largest makes the eighth number 7, which is the median.
Identifying the Mode
The mode is a simple yet unique statistic that identifies the most frequently occurring value in a dataset. Unlike the mean and median, the mode can have more than one value or even none if no number repeats. To find the mode:
  • Count how many times each number appears.
  • Identify the number or numbers with the highest frequency.
In some datasets with varied values, identifying a mode may not be straightforward, especially when every number appears with equal frequency. However, here, the number 7 appears most often, making it the mode of student sleep hours.
Exploring the Midrange
The midrange is less commonly used compared to the mean, median, and mode, but it provides a simplistic calculation of the spread and center of a dataset. It is calculated by the formula:
\[ \text{Midrange} = \frac{\text{Smallest Value} + \text{Largest Value}}{2}\]
The midrange gives a rough estimate of the middle by considering only the extreme values in the dataset. While simple, it can be heavily influenced by outliers. In the given sleep hours, the smallest value is 4 and the largest is 11, resulting in a midrange of 7.5.

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Most popular questions from this chapter

Using the empirical rule, determine the approximate percentage of a normal distribution that is expected to fall within the interval described. a. Less than the mean b. Greater than 1 standard deviation above the mean c. Less than 1 standard deviation above the mean d. Between 1 standard deviation below the mean and 2 standard deviations above the mean

Chebyshev's theorem can be stated in an equivalent form to that given on page \(98 .\) For example, to say "at least \(75 \%\) of the data fall within 2 standard deviations of the mean" is equivalent to stating "at most, \(25 \%\) will be more than 2 standard deviations away from the mean." a. At most, what percentage of a distribution will be 3 or more standard deviations from the mean? b. At most, what percentage of a distribution will be 4 or more standard deviations from the mean?

Demonstrates the effect that the number of classes or bins has on the shape of a histogram. a. What shape distribution does using one class or bin produce? b. What shape distribution does using two classes or bins produce? c. What shape distribution does using 10 or 20 bins produce?

The Office of Coal, Nuclear, Electric and Alternate Fuels reported the following data as the costs (in cents) of the average revenue per kilowatt- hour for sectors in Arkansas: $$\begin{array}{lllllllll} \hline 6.61 & 7.61 & 6.99 & 7.48 & 5.10 & 7.56 & 6.65 & 5.93 & 7.92 \\ 5.52 & 7.47 & 6.79 & 8.27 & 7.50 & 7.44 & 6.36 & 5.20 & 5.48 \\ 7.69 & 8.74 & 5.75 & 6.94 & 7.70 & 6.67 & 4.59 & 5.96 & 7.26 \\ 5.38 & 8.88 & 7.49 & 6.89 & 7.25 & 6.89 & 6.41 & 5.86 & 8.04 \\ \hline \end{array}$$ a. Prepare a grouped frequency distribution for the average revenue per kilowatt-hour using class boundaries 4,5,6,7,8,9 b. Find the class width. c. List the class midpoints. d. Construct a relative frequency histogram of these data.

Consider these two sets of data: $$\begin{array}{llllll} \hline \text { Set 1 } & 46 & 55 & 50 & 47 & 52 \\ \text { Set 2 } & 30 & 55 & 65 & 47 & 53 \\ \hline \end{array}$$ Both sets have the same mean, \(50 .\) Compare these measures for both sets: \(\Sigma(x-\bar{x}), \operatorname{SS}(x),\) and range. Comment on the meaning of these comparisons.
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