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:
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
Combining them:
The formula for the z-score is:
In this problem, we have that:
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
A sample size of 18.