Questions: Suppose that the probability distribution for birth weights is normal with a mean of 120 ounces and a standard deviation of 20 ounces. The probability that a randomly selected infant has a birth weight between 100 ounces and 140 ounces is [ Select ] 68 %. The probability that a randomly selected infant has a birth weight between 110 and 130 is [Select] 68 %.

Solution Steps

To find the probability that a randomly selected infant has a birth weight between 100 ounces and 140 ounces, we calculate the Z-scores for the bounds:

$$Z_{start} = \frac{100 - 120}{20} = -1.0$$ $$Z_{end} = \frac{140 - 120}{20} = 1.0$$

Using the cumulative distribution function $\Phi$, we find:

$$P = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(1.0) - \Phi(-1.0) = 0.6827$$

Thus, the probability is:

$$P \approx 68.27\%$$

Next, we calculate the probability that a randomly selected infant has a birth weight between 110 ounces and 130 ounces. The Z-scores for these bounds are:

$$Z_{start} = \frac{110 - 120}{20} = -0.5$$ $$Z_{end} = \frac{130 - 120}{20} = 0.5$$

Again, using the cumulative distribution function $\Phi$:

$$P = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(0.5) - \Phi(-0.5) = 0.3829$$

Thus, the probability is:

$$P \approx 38.29\%$$

Final Answer