Question

Continuity Correction In testing the assumption that the probability of a baby boy is a geneticist obtains a random sample of 1000 births and finds that 502 of them are boys. Using the continuity correction, describe the area under the graph of a normal distribution corresponding to the following. (For example, the area corresponding to "the probability of at least 502 boys" is this: the area to the right of

a. The probability of 502 or fewer boys

b. The probability of exactly 502 boys

c. The probability of more than 502 boys

Short answer

a. The probability of 502 or fewer boys is represented by the area under the graph to the left of 502.5.

b.The probability of 502 boys is represented by the area under the graph between the values 501.5 and 502.5.

c. Therefore, the probability of more than 502 boys is represented by the area under the graph to the right of 502.5.

Step-by-step solution

Given information

A sample of 1000 births is selected, out of which 502 are boys. The probability of a baby boy is equal to 0.512.

Continuity correction

Let X denote the event of a baby boy being born.

Thus, success is the event of the birth of a baby born.

The continuity correction for approximation using normal distribution is an interval with the following expression:

$\left(x-0.5,x+0.5\right)$where

x is the required number of successes.

Here, the number of successes is given to be equal to 502.

Thus, x=502.

By applying the continuity correction, the following interval is obtained:

$$\begin{gathered} \left(x-0.5,x+0.5\right)=\left(502-0.5,502+0.5\right) \\ =\left(501.5,502.5\right) \end{gathered}$$

Area under the graph

a.

The probability value required is as follows:

$P\left(x⩽502\right)$

The probability of 502 or fewer boys can be represented as the probability of at the most 502 boys.

In terms of the bounds of the continuity correction interval, the above probability has the following expression:

$$\begin{gathered} P\left(\text{at}\text{most}x\right)=\text{Area}\text{to}\text{the}\text{left}\text{of}\text{the}\text{upper}\text{bound} \\ P\left(\text{at}\text{most}502\right)=\text{Area}\text{to}\text{the}\text{left}\text{of}502.5 \end{gathered}$$

Therefore, the probability of 502 or fewer boys is represented by the area under the graph to the left of 502.5.

b.

The probability value required is as follows:

$P\left(x=502\right)$

In terms of the bounds of the continuity correction interval, the above probability

$$\begin{gathered} P\left(\text{exactly}x\right)=\text{Area}\text{between}\text{lower}\text{bound}\text{and}\text{upper}\text{bound} \\ P\left(\text{exactly}502\right)=\text{Area}\text{between}501.5\text{and}502.5 \end{gathered}$$

Therefore, the probability of 502 boys is represented by the area under the graph between the values 501.5 and 50.2.5.

c.

The probability value required is as follows:

$P\left(x>502\right)$

In terms of the bounds of the continuity correction interval, the above probability

$$\begin{gathered} P\left(\text{more}\text{than}x\right)=\text{Area}\text{to}\text{the}\text{right}\text{of}\text{the}\text{upper}\text{bound} \\ P\left(\text{more}\text{than}502\right)=\text{Area}\text{to}\text{the}\text{right}\text{of}502.5 \end{gathered}$$

Therefore, the probability of more than 502 boys is represented by the area under the graph to the right of 502.5.