Question

(1 point)

Answer 1

Answer:

0.9988 = 99.88% probability that the mean number of eggs laid would differ from 790 by less than 30.

Step-by-step explanation:

To solve this question, we need to understand the normal probability distribution and the central limit theorem.

Normal Probability Distribution:

Problems of normal distributions can be solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$, the z-score 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 p-value, 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 normally distributed random variable X, with mean $\mu$ and standard deviation $\sigma$, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean $\mu$ and standard deviation $s = \frac{\sigma}{\sqrt{n}}$.

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

Mean 790 and standard deviation 92.

This means that $\mu = 790, \sigma = 92$

Samples of 98

This means that $n = 98, s = \frac{92}{\sqrt{98}}$

What is the probability that the mean number of eggs laid would differ from 790 by less than 30?

This is the pvalue of Z when X = 790 + 30 = 820 subtracted by the pvalue of Z when X = 790 - 30 = 760. So

X = 820

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

By the Central Limit Theorem

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

$Z = \frac{820 - 790}{\frac{92}{\sqrt{98}}}$

$Z = 3.23$

$Z = 3.23$ has a pvalue of 0.9994

X = 760

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

$Z = \frac{760 - 790}{\frac{92}{\sqrt{98}}}$

$Z = -3.23$

$Z = -3.23$ has a pvalue of 0.0006

0.9994 - 0.0006 = 0.9988

0.9988 = 99.88% probability that the mean number of eggs laid would differ from 790 by less than 30.