Answer:
0.31% probability that the average number of bikes rented per week, during the next 30 weeks, will be at least 510.
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:
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
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 random variable X, with mean
and standard deviation
, the sample means with size n of at least 30 can be approximated to a normal distribution with mean
and standard deviation ![s = \frac{\sigma}{\sqrt{n}}](https://tex.z-dn.net/?f=s%20%3D%20%5Cfrac%7B%5Csigma%7D%7B%5Csqrt%7Bn%7D%7D)
In this problem, we have that:
![\mu = 500, \sigma = 20, n = 30, s = \frac{20}{\sqrt{30}} = 3.65](https://tex.z-dn.net/?f=%5Cmu%20%3D%20500%2C%20%5Csigma%20%3D%2020%2C%20n%20%3D%2030%2C%20s%20%3D%20%5Cfrac%7B20%7D%7B%5Csqrt%7B30%7D%7D%20%3D%203.65)
Find the probability that the average number of bikes rented per week, during the next 30 weeks, will be at least 510.
This is 1 subtracted by the pvalue of Z when X = 510. So
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
By the Central Limit Theorem
![Z = \frac{X - \mu}{s}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7Bs%7D)
![Z = \frac{510 - 500}{3.65}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7B510%20-%20500%7D%7B3.65%7D)
![Z = 2.74](https://tex.z-dn.net/?f=Z%20%3D%202.74)
has a pvalue of 0.9969
1 - 0.9969 = 0.0031
0.31% probability that the average number of bikes rented per week, during the next 30 weeks, will be at least 510.