the birth of healthy, full term infants in the United States is nearly normally distributed. The mean weight is 3,500 grams, and the standard deeviation is 500 grams.
(a) what percentage of the healthy newborns will weigh more than 3,250 grams?
(b) what weightswould 95% of all healthy newborns tend to fall between?
(c)what is the z score for an infant who weighs 2,750 grams?
(c) Z = -750/500 = -1.25
For the others, you need a normal distribution probability calculator or table.
There are many online. There is a good one at
http://www.stat.tamu.edu/~west/applets/normaldemo.html
It says that the answer to (a) is 69.15%
To answer these questions, we will use the concept of standard deviation and z-scores in the context of a normal distribution. Here's how you can get the answers:
(a) To find the percentage of healthy newborns weighing more than 3,250 grams, we need to calculate the area under the normal curve to the right of that weight value.
Step 1: Calculate the z-score using the formula:
z = (X - μ) / σ
where X is the weight value, μ is the mean weight, and σ is the standard deviation.
z = (3,250 - 3,500) / 500
z = -0.25
Step 2: Use a z-table or a statistical calculator to find the area under the curve to the right of the z-score (-0.25). This will give you the percentage of healthy newborns who weigh more than 3,250 grams.
(b) To find the weight range within which 95% of healthy newborns fall, we need to find the two z-scores that correspond to the area of 0.025 (on the left side) and 0.975 (on the right side) under the curve.
Step 1: Find the z-score corresponding to 0.025. This z-score represents the weight value where only 2.5% of newborns weigh less than:
z1 = invNorm(0.025)
Step 2: Find the z-score corresponding to 0.975. This z-score represents the weight value where only 2.5% of newborns weigh more than:
z2 = invNorm(0.975)
Step 3: Convert the z-scores back to weight values using the formula:
X = z * σ + μ
The weights that 95% of healthy newborns tend to fall between will be X1 and X2 obtained from the z-scores.
(c) To find the z-score for an infant who weighs 2,750 grams, we will use the same formula as before:
z = (X - μ) / σ
Substituting the given values:
z = (2,750 - 3,500) / 500
z = -1.5
The z-score for an infant who weighs 2,750 grams is -1.5.
Note: For accuracy and convenience, you may choose to use statistical software or tables to calculate these values.