The length of time a person takes to decide which shoes to purchase is normally distributed with a mean of 8.54 minutes and a standard deviation of 1.91. Find the probability that a randomly selected individual will take less than 6 minutes to select a shoe purchase. Is this. outcome unusual?

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4VA. Calculating normal probabilities (2:18)
4DF. Probabilities from Normal Distributions 3 (DOCX)
Probability is 0.91 , which is unusual as it is greater than $5 \%$
Probability is 0.09 , which is usual as it is not less than $5 \%$
Probability is 0.09 , which is unusual as it is less than $5 \%$
Probability is 0.91 , which is usual as it is greater than $5 \%$

Step 1 ：First, we need to calculate the Z-score. The Z-score is a measure of how many standard deviations an element is from the mean. The formula for calculating the Z-score is: \(Z = \frac{X - \mu}{\sigma}\)

Step 2 ：In this case, X is the value we are interested in, which is 6 minutes. \(\mu\) is the mean, which is 8.54 minutes, and \(\sigma\) is the standard deviation, which is 1.91 minutes.

Step 3 ：Substituting the given values into the formula, we get: \(Z = \frac{6 - 8.54}{1.91} = \frac{-2.54}{1.91} = -1.33\)

Step 4 ：Now, we need to find the probability that a Z-score is less than -1.33. We can use a Z-table or a calculator with a normal distribution function for this.

Step 5 ：Looking up -1.33 in the Z-table, we find that the probability is approximately 0.0925 or 9.25%.

Step 6 ：Therefore, the probability that a randomly selected individual will take less than 6 minutes to select a shoe purchase is 0.0925 or 9.25%. Since this probability is less than 5%, it is considered unusual.

Step 7 ：So, the final answer is \(\boxed{0.0925}\) or \(\boxed{9.25\%}\)