Olive-Oil producers have expanded to approximately 120 producers statewide. Analysts estimate the average price consumers pay for a bottle of olive oil is $21, with a standard deviation of $2.40. What is the probability that the average price paid by a random sample of 36 olive-oil buying consumers is less than $22?

To find the probability that the average price paid by a random sample of 36 olive-oil buying consumers is less than $22, we need to use the Central Limit Theorem.

The Central Limit Theorem states that if a random sample is taken from a population with any distribution, as the sample size increases, the distribution of the sample means approaches a normal distribution.

In this case, we are given the average price consumers pay for a bottle of olive oil, which has a known population mean of $21 and a standard deviation of $2.40. Since the sample size is large (n=36), we can assume that the distribution of the sample means will follow a normal distribution.

To calculate the probability, we need to standardize the value of $22 using the formula for the standard error of the mean:

Standard Error = Standard Deviation / sqrt(Sample Size)
Standard Error = $2.40 / sqrt(36)
Standard Error = $2.40 / 6
Standard Error = $0.40

Next, we need to calculate the z-score, which represents the number of standard deviations a value is from the mean. We can use the formula:

z = (X - Mean) / Standard Error
z = ($22 - $21) / $0.40
z = $1 / $0.40
z = 2.5

Now, we can use the standard normal distribution table or a calculator to find the probability associated with a z-score of 2.5. Looking up the z-score of 2.5 in the table, we find that the probability is approximately 0.9938.

Therefore, the probability that the average price paid by a random sample of 36 olive-oil buying consumers is less than $22 is approximately 0.9938, or 99.38%.