Question

The lengths of pregnancies are normally distributed with a mean of 266 days and a standard deviation of 15 days. a. Find the probability of a pregnancy lasting 307 days or longer. b. If the length of pregnancy is in the lowest 3​%, then the baby is premature. Find the length that separates premature babies from those who are not premature.

Answer 1

Answer:

The probability of a pregnancy longer than 307 days is P=0.00317=0.3%.

The pregnancy length that separates premature babies from those who are not premature is 238 days.

Step-by-step explanation:

a) To find the probability of a pregnancy lasting more than 307 days, we calculate the z-value and then calculate the probability in a standard normal distribution table.

$z=(X-\mu)/\sigma=(307-266)/15=2.73\\\\P(X>307)=P(z>2.73)=0.00317$

The probability of a pregnancy longer than 307 days is P=0.00317=0.3%.

b) First, we have to know the z-value that P(z<z₁)=0.03. This value of z is z=-1.88.

$P(z$

Then we can calculate the cut off value for the pregnancy days as:

$X=\mu+z*\sigma=266+(-1.88)*15=266-28=238$

The pregnancy length that separates premature babies from those who are not premature is 238 days.

Answer 2

Answer:

a. 0.313% (0.003134842261), b. 237.79 days (237.788095878)

Step-by-step explanation:

In this case, the length of pregnancies is a normally distributed variable, with a mean of 266 days, and a standard deviation of 15 days.  

A graph showing the distribution, with regions of interest for the answer, is presented below.

First Part: Find the probability of a pregnancy lasting 307 days or longer.

To answer the question regarding the probability of a pregnancy lasting 307 days or longer, it is necessary to calculate what the cumulative probability distribution value is at 307 days. By the way, according to the graph below, 307 days are quite far from the population mean (266 days).

Using the function normaldist(266,15).cdf(307), from free Desmos software on Internet, we find that, at this length (307 days), the sum of all probabilities for all cases at this value is 99.69%  (0.996865157739).

Considering that the total area of the curve is 1, then the probability of pregnancy lasting 307 days or longer is 1 - 0.996865157739 or 0.003134842261 (or 0.00313), approximately 0.313%, a very low probabilty.

This probability is showed as the "light blue" region at the right extreme of the graph.

Second Part: Find the length that separates premature babies from those who are not premature.

To find the length that separates premature babies from those who are not premature, it is a question about find the days related with the probability of 3% (or 0.03) to find such premature babies. So, it is a question of finding a percentile (or 100-quantiles): given the cumulative normal distribution curve, what is the value (length of pregnancies) that represents this 3%.

Using the function quantile(normaldist(266,15), 0.03), from free Desmos software on Internet, we obtained a value of 237.79 days (237.788095878) for the length of pregnacies of premature babies. In other words, those babies whose mothers have a length of pregnancy lower than 237.79 days are considered premature, or this is "the length that separates premature babies from those who are not premature".

The area below 237.79 days is the blue shaded region in the graph below, at the left extreme of it.