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Answer:
a) B. The distribution is approximately normal.
b) 0.0322 = 3.22%
c) 0.0202 = 2.02%
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 establishes 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
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
Mean and standard deviation:
Sample of 49:
This means that
(a) Describe the sampling distribution of x.
By the Central Limit Theorem, approximately normal, and the correct answer is given by option B.
(b) What is P (x > 94.55) ?
This is 1 subtracted by the p-value of Z when X = 94.55, so:
By the Central Limit Theorem
has a p-value of 0.9678.
1 - 0.9678 = 0.0322
So 0.0322 = 3.22%
Question c:
This is the p-value of Z when X = 82.85. So
has a p-value of 0.0202.
So
0.0202 = 2.02%