Question

3] The Mean and S.D. for group of
100 observations are 65 and 7.03
respectively If 60 of these
observations have mean and S.D.
as 70 and 3 respectively, what is
the S.D. for the group comprising
40 observations?​

Answer 1

Answer:

S.D. for the group comprising  40 observations is 4.

Step-by-step explanation:

We are given that Mean and S.D. for group of  100 observations are 65 and 7.03  respectively. Also, 60 of these  observations have mean and S.D.  as 70 and 3 respectively.

Mean formula is given by, $Xbar$ = $\frac{\sum X}{n}$

Mean of 100 observations = $\frac{\sum X_1_0_0}{100}$

                                65     = $\frac{\sum X_1_0_0}{100}$

Therefore sum of values of 100 observations, $\sum X_1_0_0$ = 65 * 100 = 6500

Similarly, sum of values of 60 observations, $\sum X_6_0$ = 70 * 60 = 4200

So, sum of values of remaining 40 observations, $\sum X_4_0$ = 6500-4200 = 2300  

Therefore, Mean of remaining 40 observations, $Xbar_4_0$ = $\frac{\sum X_4_0}{40}$ = $\frac{2300}{40}$ = 57.5 .

Now, Standard deviation formula is given by, S.D. = $\sqrt{\frac{\sum X^{2} - n*Xbar^{2} }{n-1} }$

Standard deviation of 100 observations, $S.D._1_0_0$ = $\sqrt{\frac{\sum X_1_0_0^{2} - 100*65^{2} }{100-1} }$

                 7.03 = $\sqrt{\frac{\sum X^{2}_1_0_0 - 100*65^{2} }{100-1} }$

                 $7.03^{2}$ = $\frac{\sum X^{2}_1_0_0 - 100*65^{2} }{100-1}$

Therefore, $\sum X^{2}__1_0_0$ = $(7.03^{2} * 99) + (100*65^{2})$ = 427392.67

Similarly, $S.D._6_0$ = $\frac{\sum X^{2}_6_0 - 60*70^{2} }{60-1}$

            $\sum X^{2}__6_0$ = $(3^{2} * 59) + (60*70^{2})$ = 294531

So,sum of squares of remaining 40 observations,$\sum X^{2}__4_0$ = 427392.67-294531

                                                                                             = 132861.67

Therefore, S.D. of remaining 40 observations = $\sqrt{\frac{\sum X^{2}_4_0 - 40*(Xbar_4_0)^{2} }{40-1} }$

                                                                            = $\sqrt{\frac{132861.67 - 40*57.5^{2} }{39} }$ = 3.96 ≈ 4.

Hence, S.D. for the group comprising  40 observations is 4.

Answer 2

Answer:

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