Gretchen is refinishing her gazebo that is in the shape of a regular octagon. The length of one side is 3 feet and the apothem i
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Answer:
a) The mean is $55 and the standard deviation is $0.6957
b) The shape of the sampling distribution of x is approximately normal.
c) 0.0749 = 7.49% probability that the average cell phone service paid by the sample of cell phone users will exceed $56.
Step-by-step explanation:
Normal probability distribution
When the distribution is normal, we use 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.
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
In this question:
Mean of the distribution: 55
Standard deviation of the distribution: 22
Sample of 1000.
This means that:
Part A: What are the mean and standard deviation of the sample distribution of X?
By the Central Limit Theorem, the mean is 55 and the standard deviation is .
Part B: What is the shape of the sampling distribution of x?
By the Central Limit Theorem, the shape of the sampling distribution of x is approximately normal.
Part C: What is the probability that the average cell phone service paid by the sample of cell phone users will exceed $56?
This is 1 subtracted by the pvalue of Z when . So:
By the Central Limit Theorem
has a pvalue of 0.9251
1 - 0.9251 = 0.0749
0.0749 = 7.49% probability that the average cell phone service paid by the sample of cell phone users will exceed $56.