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Answer:
97.74% probability that the sample mean would differ from the true mean by less than 1.9 dollars if a sample of 92 5-gallon pails is randomly selected
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 random variable X, with mean and standard deviation
, the sample means with size n of at least 30 can be approximated to a normal distribution with mean
and standard deviation
In this problem, we have that:
What is the probability that the sample mean would differ from the true mean by less than 1.9 dollars if a sample of 92 5-gallon pails is randomly selected?
This is the pvalue of Z when X = 29 + 1.9 = 30.9 subtracted by the pvalue of Z when X = 29 - 1.9 = 27.1. So
X = 30.9
By the Central Limit Theorem
has a pvalue of 0.9887
X = 27.1
has a pvalue of 0.0113
0.9887 - 0.0113 = 0.9774
97.74% probability that the sample mean would differ from the true mean by less than 1.9 dollars if a sample of 92 5-gallon pails is randomly selected
Answer:
How many units need to be sold to produce the maximum revenue? 1000 units
How many in dollars is the maximum revenue when the maximum of units are sold? $350,000
Step-by-step explanation:
We get max value of a function if we differentiate it and set it equal to 0.
We need to differentiate r(x) and set it equal to 0 and solve for x.
<u><em>That would be number of units sold to get max revenue.</em></u>
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<u>Then we take that "x" value and substitute into r(x) to get the max revenue amount.</u>
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Before differentiating, we see the rules shown below:
Where
f'(x) is the differentiated function
Now, let's do the process:
So, 1000 units need to be sold for max revenue
Now, substituting, we get:
The max revenue amount is $350,000