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Practice each question, then open the answer or worked solution.
The mean and variance of eight observations are 9 and 9.25, respectively. If six of the observations are 6, 7, 10, 12, 12 and 13, find the remaining two observations.
Solution:-
The mean and variance of 7 observations are 8 and 16, respectively. If five of the observations are 2, 4, 10, 12, and 14. Find the remaining two observations.
Solution:-
The mean and standard deviation of six observations are 8 and 4, respectively. If each observation is multiplied by 3, find the new mean and new standard deviation of the resulting observations.
Solution:-
We know that,
Given that x̅ is the mean and σ2 is the variance of n observations x1, x2, …,xn . Prove that the mean and variance of the observations ax1, ax2, ax3, …., axn are ax̅ and a2σ2, respectively, (a ≠ 0).
Solution:-
From the question, it is given that n observations are x1, x2,…..xn
The mean of the n observation = x̅
The variance of the n observation = σ2
As we know,
The mean and standard deviation of 20 observations are found to be 10 and 2, respectively. On rechecking, it was found that observation 8 was incorrect. Calculate the correct mean and standard deviation in each of the following cases: (i) If the wrong item is omitted. (ii) If it is replaced by 12.
Solution:-
(i) If the wrong item is omitted.
From the question, it is given that
The number of observations, i.e., n = 20
The incorrect mean = 20
The incorrect standard deviation = 2
(ii) If it is replaced by 12.
From the question, it is given that
The number of incorrect sum observations, i.e., n = 200
The correct sum of observations n = 200 – 8 + 12
n = 204
Then, correct mean = correct sum/20
= 204/20
= 10.2
The mean and standard deviation of marks obtained by 50 students of a class in three subjects, Mathematics, Physics and Chemistry, are given below.
| Subject | Mathematics | Physics | Chemistry |
| Mean | 42 | 32 | 40.9 |
| Standard deviation | 12 | 15 | 20 |
Which of the three subjects shows the highest variability in marks, and which shows the lowest?
Solution:-
From the question, it is given that
The mean of Mathematics = 42
The standard deviation of Mathematics = 12
The mean of Physics = 32
The standard deviation of physics = 15
The mean of Chemistry = 40.9
The standard deviation of chemistry = 20
As we know,
The mean and standard deviation of a group of 100 observations were found to be 20 and 3, respectively. Later on, it was found that three observations were incorrect, which were recorded as 21, 21 and 18. Find the mean and standard deviation if the incorrect observations are omitted.
Solution:-
From the question, it is given that
The total number of observations (n) = 100
The incorrect mean, (x̅) = 20
And the incorrect standard deviation (σ) = 3
