Question

1. here is a data set 1 2 3 3 4 4 4 4 5 5 6 7 a. what happens to the mean and standard deviation of the data set when the 7 is changed to a 70? b. for the data set with the value of 70, why would the median be a better choice for the measure of center than the mean? 8. which of these best estimates the standard deviation of points in a card game? a. 5 points b. 20 points c. 50 points d. 500 points

Explanation:

Step1: Calculate original mean

The original data - set is \(1,2,3,3,4,4,4,4,5,5,6,7\). The mean \(\bar{x}_1=\frac{1 + 2+3+3+4+4+4+4+5+5+6+7}{12}=\frac{48}{12} = 4\).

Step2: Calculate new mean

When \(7\) is changed to \(70\), the new sum is \(48-7 + 70=111\). The new mean \(\bar{x}_2=\frac{111}{12}=9.25\). So the mean increases.

Step3: Analyze standard - deviation

The standard deviation measures the spread of the data. When \(7\) is changed to \(70\), the data becomes more spread out. The formula for the sample standard deviation is \(s=\sqrt{\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}}\). Since the value of \(7\) is replaced with a much larger value (\(70\)), the squared - differences \((x_i-\bar{x})^2\) will be larger on average, so the standard deviation increases.

Step4: Answer part b

The median is the middle value of a sorted data - set. For a data - set with \(n = 12\) values, the median is the average of the 6th and 7th ordered values. The original and new data - sets have the same number of values and the same middle two values (when ordered) because the extreme value change (\(7\) to \(70\)) does not affect the middle values. The mean is affected by extreme values (outliers). In the data - set with \(70\), \(70\) is an outlier. The median is not affected by outliers, so it gives a better representation of the "center" of the data.

Step5: Estimate standard deviation for card - game data

The data points are centered around \(500\) with values ranging from \(490\) to \(510\). The standard deviation is a measure of the spread of the data. A value of \(5\) points seems reasonable as an estimate of the standard deviation since most of the data is within a few points of the center (\(500\)).

Answer:

a. The mean increases and the standard deviation increases.
b. The median is not affected by outliers, while the mean is affected by the outlier value of \(70\), so the median gives a better representation of the center of the data.

1. A. 5 points