The correct answer is: Option (A) The new standard deviation is greater than $27.
Explanation:
The given sample set is the following:
75, 82, 100, 120, 140.
To find the new standard deviation, add the sixth skateboard to the above sample set, as follows:
75, 82, 100, 120, 140, 450.
Now we have 6 elements in the sample set.
Step-1: Find the mean of the new sample set.
[tex]Mean = \overline{x}=\frac{\underset{i}\sum x_i}{n}[/tex]
Where, n is the total number of elements in the sample set. In this case, n=6.
[tex]\overline{x}=\frac{75+82+100+120+140+450}{6}\\ \overline{x} \approx 161.167[/tex]
Step-2: Find the variance ([tex]s^2[/tex]).
[tex]s^2 = \frac{\underset{i} \sum (x_i - \overline x)^2}{n-1}[/tex]
[tex]s^2 = \frac{(75-161.167)^2+(82-161.167)^2+(100-161.167)^2+(120-161.167)^2+(140-161.167)^2+(450-161.167)^2}{6-1} \\s^2 = 20600.167[/tex]
Step-3: Find the new standard deviation.
Standard deviation is the square-root of variance.
[tex]\sqrt{s^2} = \sqrt{20600.167} \\s \approx 143.53[/tex]
New standard deviation ($143.53) is greater than the standard deviation ($27) without the sixth skateboard sample in sample set.
Conclusion: The Option (A) The new standard deviation is greater than $27 is the right answer.
