Suppose the lengths of the pregnancies of a certain animal are approximately normally distributed with mean u=207 and standard deviation 17 days.

(a) what is the probability that a randomly selected pregnancy lasts less than 201 days?
(b) what is the probability that a random sample of 16 preganacies has a mean geatation period of 201 days or less?
(c) what is the probability that a random sample of 37 preganacies has a mean geatation period of 201 days or less?

a) 36.32% probability that a randomly selected pregnancy lasts less than 201 days.

b) 3.22% probability that a random sample of 16 preganacies has a mean geatation period of 201 days or less.

c) 1.58% probability that a random sample of 37 preganacies has a mean geatation period of 201 days or less

Step-by-step explanation:

To solve this problem, we have to know the normal probability distribution and the central limit theorem.

Normal probability distribution

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean and standard deviation , the zscore of a measure X is given by:

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

Central Limit theorem

The Central Limit Theorem estabilishes that, for a random variable X, with mean and standard deviation , a large sample size can be approximated to a normal distribution with mean and standard deviation

In this problem, we have that:

(a) what is the probability that a randomly selected pregnancy lasts less than 201 days?

Pvalue of Z when X = 201.

has a pvalue of 0.3632.

So 36.32% probability that a randomly selected pregnancy lasts less than 201 days.

(b) what is the probability that a random sample of 16 preganacies has a mean geatation period of 201 days or less?

Now

Again the pvalue of Z when X = 201

Due to the Central Limit Theorem

has a pvalue of 0.0322.

So 3.22% probability that a random sample of 16 preganacies has a mean geatation period of 201 days or less.

(c) what is the probability that a random sample of 37 preganacies has a mean geatation period of 201 days or less?

Now

has a pvalue of 0.0158.

1.58% probability that a random sample of 37 preganacies has a mean geatation period of 201 days or less

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step-by-step explanation:

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step-by-step explanation: