Question

Each contestant in the Hunger Games must be trained to compete. Suppose that the time it takes to train a contestant has mean 5 days and standard deviation 4 days, independent of the time it takes other contestants to train. If the Hunger Games has 100 contestants to train, approx imate the probability that it will take less than 450 days to train all the contestants. Leave your answer in terms of the standard normal distribution phi(a).

Answer 1

Answer:

11.51% probability that it will take less than 450 days to train all the contestants.

Step-by-step explanation:

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

Normal probability distribution

When the distribution is normal, we use 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 the sum of n variables, the mean is $\mu*n$ and the standard deviation is $s = \sigma\sqrt{n}$

In this question:

$n = 100, \mu = 100*5 = 500, s = 4\sqrt{100} = 40$

Approximate the probability that it will take less than 450 days to train all the contestants.

This is the pvalue of Z when X = 450.

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

By the Central Limit Theorem

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

$Z = \frac{450 - 500}{40}$

$Z = -1.2$

$Z = -1.2$ has a pvalue of 0.1151

11.51% probability that it will take less than 450 days to train all the contestants.