Final answer:
The probability of an iguana having a tail longer than 19.5 inches is approximately 0.93. The probability of an iguana's tail being between 18 and 25.5 inches is approximately 0.67. Both probabilities are calculated using z-scores and the standard normal distribution table.
Step-by-step explanation:
To find the probability that a randomly selected iguana has a tail longer than 19.5 inches, we need to calculate the z-score and then use the standard normal distribution to find the probability.
The z-score is calculated by taking the value in question, subtracting the mean, and dividing by the standard deviation. So for an iguana tail length of 19.5 inches:
z = (19.5 - 24) / 3
z = -4.5 / 3
z = -1.5
Using the standard normal distribution table, we find the area to the right of z = -1.5, which is approximately 0.9332. This means the probability of an iguana having a tail longer than 19.5 inches is 0.93 (rounded to two decimals).
To find the probability that an iguana's tail is between 18 and 25.5 inches long, we need to calculate z-scores for both values:
For 18 inches: z = (18 - 24) / 3 = -6 / 3 = -2
For 25.5 inches: z = (25.5 - 24) / 3 = 1.5 / 3 = 0.5
Using the standard normal distribution table, we find the areas corresponding to these z-scores and then calculate the difference:
Area to the left of z = 0.5 is approximately 0.6915
Area to the left of z = -2 is approximately 0.0228
Probability of being between 18 and 25.5 inches is 0.6915 - 0.0228 = 0.67 (rounded to two decimals).