Question

question 1 (1 point)
here is a data set:
1 2 3 3 4 4 4 4
5 5 6 7

1. what happens to the mean and standard deviation of the data set when the 7 is changed to a 70?

the mean increases from ____ to __ and the standard deviation __ (decreases, increases) from about 1.58 to ____

1. 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?

the median would be a better choice because the data set with the 70 is a ______ (bell - shaped, uniform, skewed) distribution.
blank 1:
blank 2:
blank 3:
blank 4:
blank 5:

Explanation:

Step1: Calculate original mean

The original data set is \(1,2,3,3,4,4,4,4,5,5,6,7\). The sum of the data is \(1 + 2+3+3 + 4+4+4+4+5+5+6+7=48\), and there are \(n = 12\) data - points. The original mean \(\bar{x}_1=\frac{48}{12}=4\).

Step2: Calculate new mean

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

Step3: Analyze standard - deviation change

The standard deviation measures the spread of the data. When a value is changed from \(7\) to \(70\), the data becomes more spread out. So the standard deviation increases. To calculate the new standard deviation, we use the formula \(s=\sqrt{\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}}\). For the new data set, first calculate \(\sum_{i = 1}^{12}(x_i - 9.25)^2\). After calculation, the new standard deviation is approximately \(18.97\).

Step4: Analyze measure of center

The original data set is relatively symmetric. But when \(7\) is changed to \(70\), the data set is skewed because of the out - lier \(70\). In a skewed distribution, the median is a better measure of the center than the mean as the mean is affected by extreme values.

Answer:

Blank 1: \(4\)
Blank 2: \(9.25\)
Blank 3: increases
Blank 4: \(18.97\)
Blank 5: skewed