Mathematics, 24.02.2020 18:52 carolelai08

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?

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Answer from: marvinc5603

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 $\mu$ and standard deviation $\sigma$ , the zscore of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

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 $\mu$ and standard deviation $\sigma$ , a large sample size can be approximated to a normal distribution with mean $\mu$ and standard deviation $s = \frac{\sigma}{\sqrt{n}}$

In this problem, we have that:

$\mu = 207, \sigma = 17$

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

Pvalue of Z when X = 201.

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{201 - 207}{17}$

Z = -0.35

Z = -0.35 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 $n = 16, s = \frac{17}{\sqrt{16}} = 3.25$

Again the pvalue of Z when X = 201

$Z = \frac{X - \mu}{\sigma}$

Due to the Central Limit Theorem

$Z = \frac{X - \mu}{s}$

$Z = \frac{201 - 207}{3.25}$

Z = -1.85

Z = -1.85 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.

Now $n = 16, s = \frac{17}{\sqrt{37}} = 2.79$

$Z = \frac{X - \mu}{s}$

$Z = \frac{201 - 207}{2.79}$

Z = -2.15

Z = -2.15 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

Answer from: Quest

step-by-step explanation:

1. practice, practice & more practice

2. review errors

3. master the key concepts

do not try to memorise the processes.

4. understand your doubts.

5. create a distraction free study environment

6. create a mathematical dictionary

7. apply maths to real world problems.

Answer from: Quest

if x = 1, then the answer is 3.5

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