A small business owner estimates his mean daily profit as $970 with a standard deviation of $129. His shop is open 102 days a year. What is the probability that his annual profit will not exceed $100,000?

Carry your intermediate computations to at least four decimal places. Report your result to at least three decimal places.

The probability that the shopkeeper's annual profit will not exceed $100,000 is 0.2090.

Step-by-step explanation:

According to the Central Limit Theorem if we have a population with mean μ and standard deviation σ and we select appropriately huge random samples (n ≥ 30) from the population with replacement, then the distribution of the sum of values of X, i.e ∑X, will be approximately normally distributed.  

Then, the mean of the distribution of the sum of values of X is given by,  

And the standard deviation of the distribution of the sum of values of X is given by,  

The information provided is:

μ = $970

σ = $129

n = 102

Since the sample size is quite large, i.e. n = 102 > 30, the Central Limit Theorem can be used to approximate the distribution of the shopkeeper's annual profit.

Then,

Compute the probability that the shopkeeper's annual profit will not exceed $100,000 as follows:

*Use a z-table for the probability.

Thus, the probability that the shopkeeper's annual profit will not exceed $100,000 is 0.2090.

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step-by-step explanation: