Answer:
A sample size of 18.
Step-by-step explanation:
To solve this question, we have 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 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)
Combining them:
The formula for the z-score is:
![Z = \frac{X - \mu}{\frac{\sigma}{\sqrt{n}}}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Cfrac%7B%5Csigma%7D%7B%5Csqrt%7Bn%7D%7D%7D)
In this problem, we have that:
![\mu = 500, \sigma = 100](https://tex.z-dn.net/?f=%5Cmu%20%3D%20500%2C%20%5Csigma%20%3D%20100)
For what sample size would you expect a sample mean of 489 to be at the 33rd percentile?
This is n as such Z has a pvalue of 0.33 when X = 489. So when X = 489, Z = -0.47.
So
![Z = \frac{X - \mu}{\frac{\sigma}{\sqrt{n}}}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Cfrac%7B%5Csigma%7D%7B%5Csqrt%7Bn%7D%7D%7D)
![-0.47 = \frac{489 - 500}{\frac{100}{\sqrt{n}}}](https://tex.z-dn.net/?f=-0.47%20%3D%20%5Cfrac%7B489%20-%20500%7D%7B%5Cfrac%7B100%7D%7B%5Csqrt%7Bn%7D%7D%7D)
![-0.47*100 = -11\sqrt{n}](https://tex.z-dn.net/?f=-0.47%2A100%20%3D%20-11%5Csqrt%7Bn%7D)
![11\sqrt{n} = 47](https://tex.z-dn.net/?f=11%5Csqrt%7Bn%7D%20%3D%2047)
![\sqrt{n} = \frac{47}{11}](https://tex.z-dn.net/?f=%5Csqrt%7Bn%7D%20%3D%20%5Cfrac%7B47%7D%7B11%7D)
![\sqrt{n} = 4.27](https://tex.z-dn.net/?f=%5Csqrt%7Bn%7D%20%3D%204.27)
![\sqrt{n}^{2} = (4.27)^{2}](https://tex.z-dn.net/?f=%5Csqrt%7Bn%7D%5E%7B2%7D%20%3D%20%284.27%29%5E%7B2%7D)
![n = 18](https://tex.z-dn.net/?f=n%20%3D%2018)
A sample size of 18.