Suppose the gpa scores of the students in a class are normally distributed with a mean of 3.4 and a standard deviation of 0.2 .
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Answer:
0.0869 = 8.69% probability of getting more than 61% green balls.
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 normal distributions can be solved using the z-score formula.
In a set with mean and standard deviation
, the z-score 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 p-value, 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.
For a proportion p in a sample of size n, the sampling distribution of the sample proportion will be approximately normal with mean and standard deviation
The machine is known to have 99 green balls and 78 red balls.
This means that
Mean and standard deviation:
a. Calculate the probability of getting more than 61% green balls.
This is 1 subtracted by the pvalue of Z when X = 0.61. So
By the Central Limit Theorem
has a pvalue of 0.9131
1 - 0.9131 = 0.0869
0.0869 = 8.69% probability of getting more than 61% green balls.