Question

consider the data set 25,26,27,28,29,30,31,32,33. complete parts (a) through (c) below. (type an integer or a decimal. do not round.) the median is 29 (type an integer or a decimal. do not round.) b. replace the 33 in the data set by 123 and again compute the mean and median. decide which measure of center works better here, and explain your answer. the mean is 39. (type an integer or a decimal. do not round.) the median is 29. (type an integer or a decimal. do not round.) which center of measure works better here? a. both centers of measure work equally well here. they are both typical of most of the data. b. the mean works better here since it is more typical of most of the data. c. neither measure of center works for this data set. neither measure of center is typical of most of the data. d. the median works better here since it is more typical of most of the data. c. for the data set in part (b), the mean is neither central nor typical for the data. the lack of what property of the mean accounts for this result? a. the mean is not constant. since the mean changes depending on the observations in the data set and/or the number of observations, the mean cannot be representative of the underlying population. b. the mean is not resistant to outliers. since the 33 in the data set was replaced by 123, the mean is pulled in that direction due to that single observation, and is therefore neither central nor typical for the data. c. the mean has a lower probability of being observed. whereas observations in the data set have been observed and are likely to be obtained in repeated sampling, the mean is not necessarily observable, and is therefore neither central nor typical for the data.

Explanation:

Part (b)

To find the mean after replacing 33 with 123:

1. First, find the sum of the original data set: \(25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33\). Let's calculate that: \(25+26 = 51\), \(51+27 = 78\), \(78+28 = 106\), \(106+29 = 135\), \(135+30 = 165\), \(165+31 = 196\), \(196+32 = 228\), \(228+33 = 261\).
2. Now, replace 33 with 123. So the new sum is \(261 - 33 + 123 = 261 + 90 = 351\).
3. There are 9 data points. So the mean is \(\frac{351}{9} = 39\).

To find the median:

1. The data set after replacement is \(25, 26, 27, 28, 29, 30, 31, 32, 123\).
2. Since there are 9 (odd number) data points, the median is the middle value, which is the 5th value. The 5th value is 29.

Now, for the measure of center: The mean was affected by the outlier (123) and became 39, which is not typical of most data points (most are between 25 - 32). The median remained 29, which is in the middle of the original - like data. So the median works better as it's more typical.

Part (c)

• Option A: The mean being "not constant" is not the issue here. The mean changes because of an outlier, not just the number of observations or general data change.
• Option B: The mean is not resistant to outliers. When we replaced 33 (a relatively small number) with 123 (a large outlier), the mean was pulled up (from original mean, let's check original mean: \(\frac{261}{9}=29\) to 39) because of that single outlier. This makes the mean not central or typical.
• Option C: The "probability of being observed" is not a property related to the mean's lack of centrality here. The issue is the outlier affecting the mean, not observability.

Answer:

s:

Part (b)

• The mean is \(\boldsymbol{39}\).
• The median is \(\boldsymbol{29}\).
• The correct option for which center works better: \(\boldsymbol{D}\). The median works better here since it is more typical of most of the data.

Part (c)

• The correct option: \(\boldsymbol{B}\). The mean is not resistant to outliers. Since the 33 in the data set was replaced by 123, the mean is pulled in that direction due to that single observation, and is therefore neither central nor typical for the data.