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Chapter 3: Problem 20

Watch out for these little buggers. Each of these exercises involves some feature that is somewhat tricky. Find the (a) mean, (b) median, (c) mode, \((d)\) midrange, and then answer the given question. In a study of speed dating conducted at Columbia University, female subjects were asked to rate the attractiveness of their male dates, and a sample of the results is listed below (1 = not attractive; 10 = extremely attractive). Can the results be used to describe the attractiveness of the population of adult males? $$583861037985568873556878887$$

Short Answer

Expert verified
Mean: 6.14, Median: 7, Mode: 8, Midrange: 5.5. Sample may not generalize to all adult males.

Step by step solution

01

- Organize the Data

First, list the given data in ascending order: 1, 3, 3, 5, 5, 5, 5, 5, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 10.
02

- Find the Mean

Add all the values together and divide by the number of values. Sum: \(1 + 3 + 3 + 5 + 5 + 5 + 5 + 5 + 6 + 6 + 6 + 7 + 7 + 7 + 7 + 7 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 9 + 9 + 9 + 10 = 178\) Count: 29 Mean: \(\frac{178}{29} \approx 6.14\)
03

- Find the Median

The median is the middle value in the ordered list. Since there are 29 values, the median is the 15th value. Median: 7
04

- Find the Mode

The mode is the value that appears most frequently. Here, 8 appears the most often. Mode: 8
05

- Find the Midrange

The midrange is the average of the maximum and minimum values. Max value: 10 Min value: 1 Midrange: \(\frac{10 + 1}{2} = 5.5\)
06

- Interpret the Data

Decide if the data can describe the attractiveness of the population. The sample comes from a study of speed dating at Columbia University, which may not represent the entire population of adult males. Hence, these results might not generalize to describe the attractiveness of all adult males.

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Key Concepts

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

Mean
The mean is calculated by adding all the data points together and then dividing by the number of data points. It gives us an overall average.
In this exercise, the mean is calculated as follows:
Sum of values: \(1 + 3 + 3 + 5 + 5 + 5 + 5 + 5 + 6 + 6 + 6 + 7 + 7 + 7 + 7 + 7 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 9 + 9 + 9 + 10 = 178\)
Number of values: 29
Therefore, the mean is \(\frac{178}{29} \approx 6.14\).
The mean provides a single value that represents the average attractiveness rating given by the female subjects.
Median
The median is the middle value in an ordered list of numbers. It divides the dataset into two equal halves.
First, we need to organize the data in ascending order: 1, 3, 3, 5, 5, 5, 5, 5, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 10.
Since there are 29 data points, the median is the 15th value in this ordered list.
Thus, the median is 7.
By identifying the median, we get a central value that separates the higher half from the lower half of the data, providing a sense of the 'center' of the dataset.
Mode
The mode is the value that appears most frequently in a dataset.
In this exercise, the most frequently appearing number is 8, which occurs multiple times.
So, the mode is 8.
The mode is particularly useful for understanding the most common or popular ratings in our dataset.
Midrange
The midrange is calculated by averaging the maximum and minimum values in the data.
Here, the maximum value is 10 and the minimum value is 1.
The midrange is \(\frac{10 + 1}{2} = 5.5\).
The midrange provides a sense of the center of the range of the data values, although it can be influenced by extreme values.
Interpretation of Data
When interpreting the data, we need to consider if the sample accurately represents the broader population.
This dataset is from a speed dating study at Columbia University. This specific context might not represent the overall population of adult males.
Factors such as cultural, geographical, and demographic differences can affect the generalizability of these results.
Therefore, while the data gives us useful insight into attractiveness ratings within this specific group, it may not be fully applicable to the wider population.
Always consider the context and source of data before making broad generalizations.

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