There are 12 paintings at an art show. Three of them are chosen randomly to
2 answers:
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Answer:
a) unknown
b) 1.43% probability that the average price for 30 gas stations is less than $4.55.
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:
a) What's the approximate probability that the average price for 16 gas stations is over $4.69?
Sample size less than 30 and unknow distribution. Therefore, the central limit theorem cannot be applied and the answer is unknown.
b) Find the probability that the average price for 30 gas stations is less than $4.55.
Now
This probability is the pvalue of Z when X = 4.55. So
By the Central Limit Theorem
has a pvalue of 0.0143
1.43% probability that the average price for 30 gas stations is less than $4.55.