Question

The mean of a B.D, is 15 and standard deviation is 5, then which one of the following is correct.
(a) $p=\dfrac{2}{3}$
(b) $q=\dfrac{5}{3}$
(c) Data’s are absolutely correct
(d) Data’s are absolutely wrong

Answer 1

Firstly, we have to check whether the given data satisfies the binomial distribution. For this, we have to check the mean and variance of the given data. We will get the variance by squaring the standard deviation. If the variance is less than the mean, then the given data is correct. If the variance is greater than mean, then the data is incorrect.

Complete step by step solution:
We are given that the mean of a binomial distribution is 15 and standard deviation is 5. We know that mean of a BD is given by
$\mu =np$
Where n is the number of experiments and p is the probability of success.
$\Rightarrow \mu =np=15$
We know that standard deviation of a BD is given by
$\Rightarrow \sigma =\sqrt{npq}=5$
Where q is the probability of failure.
We also know that variance is the square of standard deviation.
$\Rightarrow {{\sigma }^{2}}=npq=25$
We have studied that in binomial distribution, the variance is always less than the mean.
$\Rightarrow {{\sigma }^{2}}<\mu$
Here, we can see that variance 25 is not less than the mean 15.
Therefore, the given data is wrong.
Hence the correct option is d.

Note: For a binomial distribution, the variance is always less than the mean. For Poisson distribution, mean and variance are equal. For Negative Binomial distribution the variance is greater than the mean. Thus students read the question carefully to understand what type of distribution is mentioned.