Question

RADIATION ON MARS

Answer 1

Answer:

0.71% probability that the daily average amount of radiation received by a rover will exceed 102

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.

In this problem, we have that:

$\mu = 100, \sigma = 20, n = 600, s = \frac{20}{\sqrt{600}} = 0.8165$

What is the probability that the daily average amount of radiation received by a rover will exceed 102?

This is 1 subtracted by the pvalue of Z when X = 102. So

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

By the Central Limit Theorem

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

$Z = \frac{102 - 100}{0.8165}$

$Z = 2.45$

$Z = 2.45$ has a pvalue of 0.9929

1 - 0.9929 = 0.0071

0.71% probability that the daily average amount of radiation received by a rover will exceed 102