For a given sample the computed values of variance and fourth central moment are $3$ and $63$ respectively. Then underlying frequency distribution is classified as:

Question

From a lot of 10 items, which include 3 defective items, a sample of 5 items is drawn at random. Let the random variable $ X $ denote the number of defective items in the sample. If the variance of $ X $ is $ \sigma^2 $, then $ 96\sigma^2 $ is equal to _________.

Answer 1

Step 1: Recall what kurtosis measures.
Kurtosis tells us how peaked or flat a distribution is, judged by the coefficient $\beta_2=\dfrac{\mu_4}{\mu_2^2}$, where $\mu_2$ is the variance and $\mu_4$ the fourth central moment.
Step 2: Note the given moments.
Here $\mu_2=3$ and $\mu_4=63$.
Step 3: Square the variance.
$\mu_2^2=3^2=9$.
Step 4: Compute the coefficient.
$\beta_2=\dfrac{63}{9}=7$.
Step 5: Compare with the benchmark value $3$.
A normal (mesokurtic) distribution has $\beta_2=3$. Here $\beta_2=7$, which is greater than $3$.
Step 6: Name the distribution type.
Since $\beta_2>3$ the curve is more sharply peaked than normal, which is called leptokurtic.
\[ \boxed{\text{Leptokurtic}} \]

For a given sample the computed values of variance and fourth central moment are \(3\) and \(63\) respectively. Then underlying frequency distribution is classified as:

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The Correct Option is A

Solution and Explanation

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