Question

Low Birth Weight The University of Maryland Medical Center considers “low birth weights” to be those that are less than 5.5 lb or 2495 g. Birth weights are normally distributed with a mean of 3152.0 g and a standard deviation of 693.4 g (based on Data Set 4 “Births” in Appendix B).

a. If a birth weight is randomly selected, what is the probability that it is a “low birth weight”?

b. Find the weights considered to be significantly low, using the criterion of a birth weight having a probability of 0.05 or less.

c. Compare the results from parts (a) and (b).

Short answer

a.The probability of Low birth weight is 0.1711.

b. The birth weight having a probability of 0.05 or less is 2011.4 g weight is considered significantly low.

c.The birth weights between 2011.4g and 2496g are considered “low birth weights” but are not significantly low.

Step-by-step solution

Given information

Low birth weights are recognized as weights lesser than 5.5lb or 2495g.

The birth weights follow Normal distribution with a mean of 3152.0g and a standard deviation of 693.4g.

Describe the distribution of birth weights

a.

Let X be the random variable for birth weight which follows a normal distribution

$$\begin{gathered} X~N\left(\mu ,{\sigma }^2\right) \\ ~ N\left(3152,{693.4}^2\right) \end{gathered}$$

Calculation for z score

The probability of birth weight lesser than 2495g can be expressed as follows:

$P\left(X<2495\right)$

The z score associated with the birth weights 2495 computed as:

$$\begin{gathered} z= \frac{x-\mu }{\sigma } \\ = \frac{2495-3152.0}{693.4} \\ = -0.95 \end{gathered}$$

Calculate the probability of getting a low birth weight

The probability that weight is less than 2496gis expressed as:

$P\left(X<2495\right)= P\left(Z<-0.95\right) ...1)$

From the standard normal table, the cumulative probability of z = -0.95 is obtained from the cell intersection of row value -0.9 and column value 0.05, which is 0.1711.

Thus,

$P\left(X<2496\right)=0.1711$

Thus, the probability of weight lesser than 2495 g is 0.1711.

Finding the weights considered to be significantly low

b.

Significantly low weights are all those birth weights that have the probability of lesser than 0.05 to appear.

That is,

$P\left(X<x\right)⩽0.05$, where x is the maximum weight below which the weights are significantly low.

The probability of birth weight having a probability of 0.05 or less can be expressed as:

$$\begin{gathered} P\left(X<x\right)=P\left(Z<z\right) \\ = 0.05 \end{gathered}$$

From the standard normal table, the area of 0.05 is observed corresponding to the row value -1.6 and column value between 0.04 and 0.05, which implies the z score is -1.645.

Substitute the value of z as follows:

$$\begin{gathered} z= \frac{x-\mu }{\sigma } \\ x= \mu +\sigma \times z \\ = 3152-1.645\times 693.4 \\ =2011.357 \end{gathered}$$

Therefore, all birth weights lesser than 2011.4 g are considered to be significantly low.

Comparing the results 

c.

The birth weights are significantly low if they are 2011.35 g or less, which have only a 5% chance to appear in the selection.

Also, birth weights lower than 2496 g are considered low birth rates with 17.11% chances to appear.

Thus, any weight that lies between 2011.5 g and 2496 g is recognized as low birth weight but actually is not significantly low (rare).