Suppose that the birth weight of newborn babies in the U.S is normally distributed with a mean of 7 pounds and a standard deviation of 1.6 pounds. Let the random variable X be the birth weight of a randomly selected record of live birth.
a) What is the distribution of X?
b) What is the probability that a randomly selected record shows a birth weight less than 5 pounds?
c) What is the probability that a randomly selected record shows a birth weight greater than eleven pounds?
d) What is the probability that a randomly selected record shows a birth weight between 6 and 9 pounds
a) The distribution of X is a normal distribution with a mean of 7 pounds and a standard deviation of 1.6 pounds. This means that the birth weights of newborn babies in the U.S. follow a bell-shaped curve where the majority of the birth weights are close to the mean (7 pounds) and become less likely as they deviate further from the mean.
b) To find the probability that a randomly selected record shows a birth weight less than 5 pounds, we need to calculate the z-score first. The z-score measures how many standard deviations a particular value is from the mean. The formula for calculating the z-score is:
z = (x - mu) / sigma
where x is the value we want to find the probability for, mu is the mean, and sigma is the standard deviation.
In this case, x = 5 pounds, mu = 7 pounds, and sigma = 1.6 pounds. Plugging these values into the formula, we get:
z = (5 - 7) / 1.6 = -1.25
From a standard normal distribution table or using a calculator, we can find that the probability of a z-score of -1.25 or less is approximately 0.1056. Therefore, the probability that a randomly selected record shows a birth weight less than 5 pounds is approximately 0.1056 or 10.56%.
c) Similarly, to find the probability that a randomly selected record shows a birth weight greater than eleven pounds, we need to calculate the z-score. In this case, x = 11 pounds. Using the same formula as before, we get:
z = (11 - 7) / 1.6 = 2.5
The probability of a z-score of 2.5 or greater can be found using the standard normal distribution table or a calculator. This probability is approximately 0.0062 or 0.62%. Therefore, the probability that a randomly selected record shows a birth weight greater than eleven pounds is approximately 0.0062 or 0.62%.
d) To find the probability that a randomly selected record shows a birth weight between 6 and 9 pounds, we need to calculate the z-scores for both values and find the area under the curve between those z-scores.
For 6 pounds:
z1 = (6 - 7) / 1.6 = -0.625
For 9 pounds:
z2 = (9 - 7) / 1.6 = 1.25
Using the standard normal distribution table or a calculator, we can find the probabilities associated with these z-scores. The probability of a z-score of -0.625 or less is approximately 0.2660, and the probability of a z-score of 1.25 or less is approximately 0.8944.
To find the probability between these two z-scores, we subtract the smaller probability from the larger probability:
0.8944 - 0.2660 = 0.6284
Therefore, the probability that a randomly selected record shows a birth weight between 6 and 9 pounds is approximately 0.6284 or 62.84%.