Questions: (a) Which measures of central tendency do not exist for this data set? Choose all that apply. Mean Median Mode All of these measures exist (b) Suppose that the measurement 182 (the largest measurement in the data set) were replaced by 245 . Which measures of central tendency would be affected by the change? Choose all that apply. Mean Median Mode None of these measures (c) Suppose that, starting with the original data set, the largest measurement were removed. Which measures of central tendency would be changed from those of the original data set? Choose all that apply. Mean Median Mode None of these measures (d) The relative values of the mean and median for the original data set are typical of data that have a significant skew to the right. What are the relative values of the mean and median for the original data set? Choose only one. Mean is greater Median is greater Cannot be determined

(a) Which measures of central tendency do not exist for this data set? Choose all that apply. Mean Median Mode All of these measures exist (b) Suppose that the measurement 182 (the largest measurement in the data set) were replaced by 245 . Which measures of central tendency would be affected by the change? Choose all that apply. Mean Median Mode None of these measures (c) Suppose that, starting with the original data set, the largest measurement were removed. Which measures of central tendency would be changed from those of the original data set? Choose all that apply. Mean Median Mode None of these measures (d) The relative values of the mean and median for the original data set are typical of data that have a significant skew to the right. What are the relative values of the mean and median for the original data set? Choose only one. Mean is greater Median is greater Cannot be determined
Transcript text: (a) Which measures of central tendency do not exist for this data set? Choose all that apply. & Mean Median Mode All of these measures exist \\ (b) Suppose that the measurement 182 (the largest measurement in the data set) were replaced by 245 . Which measures of central tendency would be affected by the change? Choose all that apply. & Mean Median Mode None of these measures \\ (c) Suppose that, starting with the original data set, the largest measurement were removed. Which measures of central tendency would be changed from those of the original data set? Choose all that apply. & Mean Median Mode None of these measures \\ (d) The relative values of the mean and median for the original data set are typical of data that have a significant skew to the right. What are the relative values of the mean and median for the original data set? Choose only one. & Mean is greater Median is greater Cannot be determined
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Solution

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Solution Steps

Step 1: Measures of Central Tendency Existence

For the original data set, we calculated the following measures of central tendency:

  • Mean: \[ \mu = \frac{\sum_{i=1}^N x_i}{N} = \frac{2007}{13} \approx 154.38 \]
  • Median: \( 155 \)
  • Mode: None (as all values are unique).

Thus, all measures of central tendency exist for the original data set.

Step 2: Effect of Replacing the Largest Measurement

When the largest measurement \( 182 \) is replaced by \( 245 \), the new calculations yield:

  • New Mean: \[ \mu = \frac{\sum_{i=1}^N x_i}{N} = \frac{2070}{13} \approx 159.23 \]
  • New Median: \( 155 \)
  • New Mode: None.

The mean is affected by the change, while the median and mode remain unchanged.

Step 3: Effect of Removing the Largest Measurement

When the largest measurement \( 182 \) is removed from the original data set, the calculations are as follows:

  • New Mean: \[ \mu = \frac{\sum_{i=1}^N x_i}{N} = \frac{1825}{12} \approx 152.08 \]
  • New Median: \( 152.5 \)
  • New Mode: None.

Both the mean and median are affected by the removal of the largest measurement, while the mode remains unchanged.

Final Answer

  • (a) All measures exist.
  • (b) The affected measure is Mean.
  • (c) The affected measures are Mean and Median.

Thus, the answers are:

  • (a): \(\text{All of these measures exist}\)
  • (b): \(\text{Mean}\)
  • (c): \(\text{Mean, Median}\)

\(\boxed{\text{(a): All of these measures exist, (b): Mean, (c): Mean, Median}}\)

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