Answer:
1) 0.1587 = 15.15% probability that x is less than 2
2) 7.64% probability that there are fewer than 100 accidents at the roundabout in a year
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 and standard deviation
, the zscore of a measure X is given by:
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 and standard deviation
, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
and standard deviation
.
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:
1)
Here we have
What is the probability that x is less than 2?
This is the pvalue of Z when X = 2.
By the Central Limit Theorem
has a pvalue of 0.1515
0.1587 = 15.15% probability that x is less than 2
2)
100/52 = 1.9231
This is the pvalue of Z when X = 1.9231
has a pvalue of 0.0764
7.64% probability that there are fewer than 100 accidents at the roundabout in a year