Question

If np >= 5 and nq >= 5 , estimate P(at least 10) with n = 13 and p = 0.5 by using the normal distribution as an approximation to the binomial distribution ; if np < 5 or nq < 5 then state that the normal approximation is not suitable.

Answer 1

The question provides that:

`\begin{gathered} n=13 \\ p=0.5 \end{gathered}`

Therefore, we have that:

`q=1-p=1-0.5=0.5`

To check if we can use the normal distribution as an approximation, we will check the values of np and nq:

`\begin{gathered} np=13*0.5=6.5 \\ nq=13*0.5=6.5 \end{gathered}`

Since,

`\begin{gathered} np\ge5 \\ \text{and} \\ nq\ge5 \end{gathered}`

then we can use the normal distribution as an approximation.

To evaluate P (at least 10), we are evaluating:

`P(X\ge10)`

The standard deviation of the distribution is gotten to be:

`\sigma=\sqrt[]{np}=\sqrt[]{6.5}=2.550`

The mean is 6.5.

Therefore, the Z-score is gotten to be:

`Z=\frac{x-\bar{x}}{\sigma}`

Hence, it is calculated to be:

`Z=(10-6.5)/(2.550)=1.37`

The probability is therefore given to be:

`P(Z\ge1.37)=Pr(0\le Z)-Pr(0\le Z\le1.37)`

Using the Probability Distribution Table, we have:

`P(Z\ge1.37)=0.5-0.4147=0.0853\approx0.085`

Therefore, the answer is:

`P(at\text{ }least\text{ }10)=0.085`