The lengths of pregnancies are normally distributed with a mean of 268 days and a standard deviation of 15 days. Out of 50 pregnancies, how many would you expect to last less than 250 days?
Assume that the heights of women are normally distributed with a mean of 63.9 inches and a standard deviation of 2.5 inches. The U.S. Army requires that the heights of women be between 58 and 80 inches. If 200 women want to enlist in the U.S. Army, how many would you expect to meet the height requirements?
Use z-scores. For your first problem, determine z using the formula:
z = (x - mean)/sd
...where x = 250, mean = 268, and sd = 15.
Once you have z, check the z-table for the probability (remember the first problem is asking less than 250, so keep that in mind when checking the table). Once you have the probability, multiply 50 by that value you find to get the number you need.
The second problem is similar to the first, with the exception of finding two z-scores instead of one.
Find z using 58 for x and also find z using 80 for x. Then use the z-table to find the probability between the two z-scores. Once you have the probability, multiply 200 by that value you find to get the number you need.
I hope this will help get you started.
To find out how many pregnancies would be expected to last less than 250 days, we can use the z-score formula and standard normal distribution.
First, we need to calculate the z-score, which measures how many standard deviations a value is from the mean. The formula for calculating the z-score is:
z = (x - μ) / σ
where:
x = the value we are interested in (250 days)
μ = the mean (268 days)
σ = the standard deviation (15 days)
Plugging in the values, we get:
z = (250 - 268) / 15
z = -18 / 15
z = -1.2
Next, we need to look up the cumulative probability associated with the z-score of -1.2 in a standard normal distribution table. This gives us the probability of a pregnancy lasting less than 250 days.
Looking up the z-score of -1.2 in the table, we find that the cumulative probability is approximately 0.1151.
Since we have 50 pregnancies, we can multiply the probability by the number of pregnancies to find the expected number of pregnancies lasting less than 250 days:
Expected number = 0.1151 * 50
Expected number = 5.755
Rounding to the nearest whole number, we would expect around 6 pregnancies to last less than 250 days out of 50 pregnancies.
To find out how many pregnancies would be expected to last less than 250 days, we need to use the concept of z-scores and the standard normal distribution.
First, we calculate the z-score for a pregnancy with a length of 250 days using the formula:
z = (X - μ) / σ
where X is the observed value, μ is the mean, and σ is the standard deviation.
Plugging in the values, we get:
z = (250 - 268) / 15
z = -1.2
Next, we need to find the probability associated with this z-score. Since the distribution is assumed to be normal, we can use the standard normal distribution table or a calculator to find this probability.
Looking up the z-score of -1.2 in a standard normal distribution table, we find that the corresponding probability is approximately 0.1151.
This means that there is a 0.1151 probability that a randomly chosen pregnancy lasts less than 250 days.
To find the number of pregnancies expected to last less than 250 days out of 50 pregnancies, we multiply the probability by the number of pregnancies:
Expected number = Probability * Number of pregnancies
Expected number = 0.1151 * 50
Expected number ≈ 5.755
Therefore, we would expect approximately 6 pregnancies to last less than 250 days out of 50 pregnancies.