Standard Deviation of Population and of a Sample
The weight of newborn children in the United States vary according to the normal
distribution with mean 7.7 pounds and standard deviation 1.68 pounds. The government
classifies a newborn as having low birth weight if the weight is less than 5.7 pounds.
a) What is the probability that a baby chosen at random weighs less than 5.7 pounds at
birth?
b) You choose three babies at random. What is the probability that their average birth
weight is less tan 5.7 pounds?
The weight x of all female German Shepherds is approximately normally distributed with population mean u=64 pounds and population standard deviation of o=8 pounds. a)What is the probability that a randomly selected female german shepherd weighs more than 75 pounds? p(x>75)?)
To answer these questions, we can use the properties of the normal distribution and the formulas for the standard deviation of a population and a sample.
a) To find the probability that a baby chosen at random weighs less than 5.7 pounds at birth, we need to calculate the z-score and then find the corresponding area under the normal distribution curve.
The z-score can be calculated using the formula:
z = (x - μ) / σ
where x is the value we're interested in (5.7 pounds), μ is the mean (7.7 pounds), and σ is the standard deviation (1.68 pounds).
So the z-score is:
z = (5.7 - 7.7) / 1.68
z = -1.19
Now, we can use a standard normal distribution table or a calculator to find the probability associated with the z-score of -1.19. The table or calculator will give us the probability of the area under the curve to the left of -1.19.
b) To find the probability that the average birth weight of three babies chosen at random is less than 5.7 pounds, we use the concept of the central limit theorem. According to the central limit theorem, the distribution of sample means approaches a normal distribution with a mean equal to the population mean and a standard deviation equal to the population standard deviation divided by the square root of the sample size.
In this case, since we have a sample size of three, the standard deviation of the sample (also known as the standard error) is σ / √n, where n is the sample size.
So the standard deviation of the sample is 1.68 / √3.
Now, we calculate the z-score using the same formula as before:
z = (x - μ) / (σ / √n)
z = (5.7 - 7.7) / (1.68 / √3)
z = -2.244
We can use a standard normal distribution table or a calculator to find the probability associated with the z-score of -2.244. The table or calculator will give us the probability of the area under the curve to the left of -2.244.