Chapter 6.2, Problem 5GP

Solution

(a)

To describe: The shape of the binomial distribution of exactly $k$ success with success probability 0.4 and given number of trails.

Shape of the distribution is skewed.

Given information:

Number of trail is 5.

Formula used:

Probability of a certain event can be found by the following formula:

  $P\left(\text{Event}\right)=\frac{\text{Number}\text{of}\text{favorable}\text{outcome}}{\text{Number}\text{of}\text{total}\text{outcome}}$

Formula of Binomial distribution of $n$ trail of an event with probability of success $p$ for exactly $k$ success is

  $P\left(X=k\right)=C{}_k{\left(p\right)}^k{\left(1-p\right)}^{n-k}$

Shape of the distribution will be symmetric if left half of the distribution is the mirror image of right half.

Shape of the distribution will be skewed if the distribution is not symmetric about any vertical line.

Calculation:

Here, number of randomly chosen family is 5.

  $n=5$

Success probability is 0.4.

  $p=0.4$

The probability distribution is shown below:

  $P\left(k=r\right)=C{}_r{\left(0.4\right)}^r{\left(0.39\right)}^{5-r}$ for $r=0,1,2,3,4,5$

The binomial distribution table is shown below:

$k$	Binomial distribution
0	0.07776
1	0.2592
2	0.3456
3	0.2304
4	0.0768
5	0.01024

The histogram for the probability distribution is shown below:

  

Shape of the distribution will be skewed since the distribution is not symmetric about any vertical line.

(b)

To describe: The shape of the binomial distribution of exactly $k$ success with success probability 0.5 and given number of trails.

Shape of the distribution is symmetric.

Given information:

Number of trail is 5.

Formula used:

Probability of a certain event can be found by the following formula:

  $P\left(\text{Event}\right)=\frac{\text{Number}\text{of}\text{favorable}\text{outcome}}{\text{Number}\text{of}\text{total}\text{outcome}}$

Formula of Binomial distribution of $n$ trail of an event with probability of success $p$ for exactly $k$ success is

  $P\left(X=k\right)=C{}_k{\left(p\right)}^k{\left(1-p\right)}^{n-k}$

Shape of the distribution will be symmetric if left half of the distribution is the mirror image of right half.

Shape of the distribution will be skewed if the distribution is not symmetric about any vertical line.

Calculation:

Here, number of randomly chosen family is 5.

  $n=5$

Success probability is 0.5.

  $p=0.5$

The probability distribution is shown below:

  $P\left(k=r\right)=C{}_r{\left(0.4\right)}^r{\left(0.39\right)}^{5-r}$ for $r=0,1,2,3,4,5$

The binomial distribution table is shown below:

$k$	Binomial distribution
0	0.03125
1	0.15625
2	0.3125
3	0.3125
4	0.15625
5	0.03125

The histogram for the probability distribution is shown below:

  

Shape of the distribution will be symmetric since left half of the distribution is the mirror image of right half..