Question

6. (4.11.2) A 500-page book contains 250 sheets of paper. The thickness of the paper used to manufacture the book has mean 0.08 mm and standard deviation 0.01 mm. a. What is the probability that a randomly chosen book is more than 20.2 mm thick (not including the covers)

Answer 1

Answer:

10.38% probability that a randomly chosen book is more than 20.2 mm thick

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 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 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.

For sums of n elements, the mean is $n\mu$ and the standard deviation is $s = \sqrt{n}\sigma$

In this problem, we have that:

$\mu = 250*0.08 = 20, \sigma = \sqrt{250}*0.01 = 0.1581$

What is the probability that a randomly chosen book is more than 20.2 mm thick (not including the covers)

This is 1 subtracted by the pvalue of Z when X = 20.2. So

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

By the Central Limit Theorem

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

$Z = \frac{20.2 - 20}{0.1581}$

$Z = 1.26$

$Z = 1.26$ has a pvalue of 0.8962

1 - 0.8962 = 0.1038

10.38% probability that a randomly chosen book is more than 20.2 mm thick