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**Miscellaneous Exercise **

**1. 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.**

**Ans. **Let two required observations be and Then,

According to question,

……….(i)

Also

……….(ii)

When then on solving and we get and

When then on solving and we get and

**2. The mean and variance of 7 observations are 8 and 16 respectively. If five of the observations are 2, 4, 10, 12, 14, find the remaining two observations.**

**Ans. **Let two required observations be and Then,

According to question,

……….(i)

Also

……….(ii)

When then on solving and we get and

When then on solving and we get and

**3. 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.**

**Ans. **Let six observations be then

According to questions,

Now, if each observation is multiplied by 3, then

New mean = = 24

Also

Now, if each observation is multiplied by 3, then

New Variance =

= 720 – 576 = 144

Therefore, New S.D. = = 12

**4. Given that **** is the mean and **** is the variance of **** observations **** Prove that the mean and variance of the observations **** are **** and **** respectively **

**Ans. **Given:

Also

New Mean =

Also

New Variance =

=

=

**5. The mean and standard deviation of 20 observations are found to be 10 and 2 respectively. On rechecking, it was found that an observation 8 was incorrect. Calculate the correct mean and standard deviation in each of the following cases:**

**(i) if wrong item is omitted **

**(ii) if it is replaced by 12.**

**Ans. **Given: and

Now,

Incorrect

Now

**(i)** If wrong item is omitted: then we have 19 observations.

Correct = Incorrect

Correct = 200 – 8 = 192

Correct Mean = = 10.1

Also Correct Incorrect

Correct Variance = (Correct ) – (Correct mean)^{2}

= = =

Correct S.D. = = 1.997

**(ii)** If it is replaced by 12: then

Correct = Incorrect

Correct = 200 – 8 + 12 = 204

Correct Mean = = 10.2

Also Correct Incorrect

Correct Variance = (Correct )

= = =

Correct S.D. = = 11.989

**6. The mean and standard deviation of marks obtained by 50 students of a class in three subjects, Mathematics, Physics and Chemistry are given below:**

**Which of the three subjects shows the highest variability in marks and which shows the lowest?**

**Ans. **For Mathematics, = 42 and = 12

C.V. of Mathematics = = 28.57%

For Physics, = 32 and = 15

C.V. of Physics = = 46.88%

For Chemistry, = 40.9 and = 20

C.V. of Chemistry = = 48.9%

Therefore, Chemistry with highest C.V. shows highest variability and Mathematics with lowest C.V. shows lowest variability.

**7. 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 12, 21 and 18. Find the mean and standard deviation if the incorrect observations are omitted.**

**Ans. **Given: and

Now,

Incorrect

Now

When wrong items 21, 21 and 18 are omitted from the data, then there will be 97 observations.

Correct = Incorrect

Correct = 2000 – 21 – 21 – 18 = 1940

Correct Mean = = 20

Also Correct Incorrect

Correct Variance = (Correct ) – (Correct mean)^{2}

= = =

Correct S.D. = = 3.036