Answer to Question #165624 in Statistics and Probability for Dolly

Question

Delip measured the speeds, x km per hour, of 70 cars on a road where the 

speed limit is 60 km per hour.

 His results are summarized by Σ(x − 60) = 245.

 (i) Calculate the mean speed of these 70 cars. 

 His friend Sachim used values of (x − 50) to calculate the mean.

 (ii) Find Σ(x − 50). 

 (iii) The standard deviation of the speeds is 10.6 km per hour. Calculate 

  Σ (x- 50)^2

 

Answer 1

(i)

$$\bar{x}=\dfrac{\displaystyle\sum_{i=1}^nx_i}{n}=\dfrac{\displaystyle\sum_{i=1}^n(x_i-60)}{n}+60$$

$$\bar{x}=\dfrac{245}{70}+60=63.5$$

The mean speed of these 70 cars is 63.5 km/hr

(ii)

$$\bar{x}=\dfrac{\displaystyle\sum_{i=1}^{70}(x_i-50)}{70}+50=63.5$$

$$\displaystyle\sum_{i=1}^{70}(x_i-50)=(63.5-50)(70)=945$$

(iii)

$$\displaystyle\sum_{i=1}^n(x_i-50)^2=\displaystyle\sum_{i=1}^n(x_i-63.5+13.5)^2$$

$$=\displaystyle\sum_{i=1}^n(x_i-63.5)^2+27\displaystyle\sum_{i=1}^n(x_i-63.5)+182.25\displaystyle\sum_{i=1}^n1$$

$$=n\sigma^2+182.25n$$

$$\displaystyle\sum_{i=1}^{70}(x_i-50)^2$$

$$=70(10.6)^2+182.25(70)=20622.7$$