Answer to Question #165624 in Statistics and Probability for Dolly

Question #165624

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

 


1
Expert's answer
2021-02-24T06:01:22-0500

(i)

xˉ=i=1nxin=i=1n(xi60)n+60\bar{x}=\dfrac{\displaystyle\sum_{i=1}^nx_i}{n}=\dfrac{\displaystyle\sum_{i=1}^n(x_i-60)}{n}+60

xˉ=24570+60=63.5\bar{x}=\dfrac{245}{70}+60=63.5

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


(ii)


xˉ=i=170(xi50)70+50=63.5\bar{x}=\dfrac{\displaystyle\sum_{i=1}^{70}(x_i-50)}{70}+50=63.5

i=170(xi50)=(63.550)(70)=945\displaystyle\sum_{i=1}^{70}(x_i-50)=(63.5-50)(70)=945

(iii)


i=1n(xi50)2=i=1n(xi63.5+13.5)2\displaystyle\sum_{i=1}^n(x_i-50)^2=\displaystyle\sum_{i=1}^n(x_i-63.5+13.5)^2

=i=1n(xi63.5)2+27i=1n(xi63.5)+182.25i=1n1=\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

=nσ2+182.25n=n\sigma^2+182.25n


i=170(xi50)2\displaystyle\sum_{i=1}^{70}(x_i-50)^2

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


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