Final answer:
a) The probability is 0.8413. b) The probability is 0.1587. c) The probability is 0.2417.
Step-by-step explanation:
a) To find the probability that a randomly selected person spent less than $17, we need to calculate the z-score for $17 using the formula: z = (x - μ) / σ where x is the value we are interested in, μ is the mean, and σ is the standard deviation. In this case, x = $17, μ = $15, and σ = $2. Substituting these values into the formula, we get: z = (17 - 15) / 2 = 1. We can then use a z-table to find the probability that corresponds to a z-score of 1, which is 0.8413. Therefore, the probability that a randomly selected person spent less than $17 is 0.8413.
b) To find the probability that a randomly selected person spent more than $17, we can subtract the probability of spending less than $17 from 1. From part (a), we know that the probability of spending less than $17 is 0.8413. Therefore, the probability of spending more than $17 is 1 - 0.8413 = 0.1587.
c) To find the probability that a randomly selected person spent between $12 and $14, we need to calculate the z-scores for $12 and $14 using the formula from part (a). For $12: z = (12 - 15) / 2 = -1.5, and for $14: z = (14 - 15) / 2 = -0.5. We can then use the z-table to find the probabilities that correspond to these z-scores. The probability for -1.5 is 0.0668, and the probability for -0.5 is 0.3085. To find the probability between $12 and $14, we subtract the probability for $12 from the probability for $14: 0.3085 - 0.0668 = 0.2417. Therefore, the probability that a randomly selected person spent between $12 and $14 is 0.2417.