91Ó°ÊÓ

The accompanying data on number of minutes used for cell phone calls in one month was generated to be consistent with summary statistics published in a report of a marketing study of San Diego residents (TeleTruth, March 2009): $$ \begin{array}{rrrrrrrrrr} 189 & 0 & 189 & 177 & 106 & 201 & 0 & 212 & 0 & 306 \\ 0 & 0 & 59 & 224 & 0 & 189 & 142 & 83 & 71 & 165 \\ 236 & 0 & 142 & 236 & 130 & & & & & \end{array} $$ a. Would you recommend the mean or the median as a measure of center for this data set? Give a brief explanation of your choice. (Hint: It may help to look at a graphical display of the data.) b. Compute a trimmed mean by deleting the three smallest observations and the three largest observations in the data set and then averaging the remaining 19 observations. What is the trimming percentage for this trimmed mean? c. What trimming percentage would you need to use in order to delete all of the 0 minute values from the data set? Would you recommend a trimmed mean with this trimming percentage? Explain why or why not.

Step by step solution

01

Determine Measure of Center

To determine which measure of center is more appropriate, i.e., the mean or median, one must examine the distribution of the data. However, from the data provided, it can be observed that there are a significant number of outliers (0 values) which would skew the mean towards zero. Therefore, the median would be a more accurate measure of center for this data set.

02

Compute a Trimmed Mean

Trimmed mean can be calculated by deleting the three smallest observations and the three largest observations in the data set and then averaging the remaining 19 observations. When sorted in increasing order, the smallest observations are {0, 0, 0} and largest are {236, 236, 306}. Deleting these, the remaining data set becomes: \[ \{59, 71, 83, 106, 130, 142, 142, 165, 177, 189, 189, 189, 201, 212, 224\} \] The trimmed mean then becomes the average of these remaining values which can be calculated as the sum of these values divided by number of these values.

03

Calculate trimming percentage

The trimming percentage represents the proportion of observations removed from each end. For the trimmed mean calculated in step 2, we have removed 3 from each end out of total 25 observations. So, the trimming percentage would be \((3+3)/25*100 = 24\%\).

04

Trimmed mean by excluding 0 values

To calculate a trimmed mean by excluding all of the 0-minute values from the data set, one would need to count the number of 0 values present and consider them for trimming. This leads to 7 zero values in the data set of 25, the trimming percentage becomes \((7/25) * 100 = 28\%\). However, the decision to recommend this trimming percentage will depend on the context and desired balance between excluding extremes (0 minute values in this case) and retaining enough data to have a meaningful result.

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Key Concepts

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

Measure of Center

In descriptive statistics, the measure of center is a value that is used to indicate the central point of a data set. It helps us understand what a typical data value looks like. The two most popular measures of center are the mean and the median.
The **mean** is calculated by adding up all the values and dividing by the number of values. It represents an average, but can be heavily influenced by extremely high or low numbers.
The **median**, on the other hand, is the middle value when the data is sorted from smallest to largest. It divides the data set into two equal halves and is less affected by outliers or skewed data.
In the presence of outliers or non-normal data distribution, the median can provide a more accurate depiction of the data's center than the mean.

Trimmed Mean

The trimmed mean is a method used to calculate the mean while minimizing the effect of outliers. This process involves removing a certain percentage of the lowest and highest values before calculating the mean. It is especially useful when dealing with skewed data or outliers that might distort the average.
To calculate the trimmed mean, follow these steps:

• Sort the data from smallest to largest.
• Decide on the number of observations to remove from each end.
• Remove these observations and calculate the mean of the remaining data.

The trimming percentage is determined by the proportion of data points removed from the dataset. For example, if you remove three observations from each end of a 25-point data set, the trimming percentage is 24%.

Outliers

Outliers are data points that significantly differ from the majority of a data set. They can be much higher or lower than other entries and often result from variability in the data, measurement errors, or unusual occurrences.
Outliers can skew your data and give misleading interpretations when calculating measures like the mean. It’s essential to identify outliers so that you can decide whether they should be included in your analysis or possibly treated separately from the rest of the data.
The median or trimmed mean are often preferred measures of center in datasets with outliers because they are less sensitive to extreme values.

Data Distribution

Data distribution is the way in which data values are spread across a range. It provides an overall picture of the pattern formed by your data.
A few common types of distributions include:

• **Normal Distribution**: Data that forms a symmetric, bell-shaped curve.
• **Skewed Distribution**: Data is asymmetrical and can be skewed left or right, often due to outliers.
• **Uniform Distribution**: Data is evenly spread across the range, with no visible peaks.

Recognizing the type of data distribution is crucial as it affects the choice of statistical methods. For instance, the mean works well with normal distributions, while the median or trimmed mean is often better for skewed distributions with outliers.