A national park is famous for its beautiful desert landscape and its many natural rock formations. The following table is based on information gathered by a park ranger of all rock formations of at least 3 feet. The height of the rock formation is rounded to the nearest foot.
Height of rock formation, feet 3–9 10–29 30–49 50–74 75 and higher
Number of rock formations in park 117 89 32 38 15
For a rock formation chosen at random from this park, use the preceding information to estimate the probability that the height of the rock formation is as follows. (Round your answers to two decimal places.)
(a) 3 to 9 feet
(b) 30 feet
(c) 3to 49 feet
(d) 10 to 74 feet
(e) 75 feet or taller
Divide the number in the desired category by the total number of observations in each case.
To estimate the probability for each scenario, we need to calculate the relative frequency or proportion for each height range.
First, let's calculate the total number of rock formations in the park:
Total number of rock formations = 117 + 89 + 32 + 38 + 15 = 291
Now, let's calculate the probability for each scenario:
(a) For heights from 3 to 9 feet:
Number of rock formations in this range = 117
Probability = Number of rock formations in this range / Total number of rock formations
Probability = 117 / 291
Probability ≈ 0.40
(b) For heights of exactly 30 feet:
Number of rock formations = 32
Probability = Number of rock formations / Total number of rock formations
Probability = 32 / 291
Probability ≈ 0.11
(c) For heights from 3 to 49 feet:
Number of rock formations in this range = 117 + 89 + 32 = 238
Probability = Number of rock formations in this range / Total number of rock formations
Probability = 238 / 291
Probability ≈ 0.82
(d) For heights from 10 to 74 feet:
Number of rock formations in this range = 89 + 32 + 38 = 159
Probability = Number of rock formations in this range / Total number of rock formations
Probability = 159 / 291
Probability ≈ 0.55
(e) For heights of 75 or taller:
Number of rock formations = 15
Probability = Number of rock formations / Total number of rock formations
Probability = 15 / 291
Probability ≈ 0.05
Therefore, the estimated probabilities are:
(a) 0.40
(b) 0.11
(c) 0.82
(d) 0.55
(e) 0.05
To estimate the probabilities for each scenario, we need to calculate the relative frequency of each category in the distribution.
First, we calculate the total number of rock formations in the park:
Total Rock Formations = 117 + 89 + 32 + 38 + 15 = 291
Now we can calculate the relative frequency of each category by dividing the number of rock formations in that category by the total number of rock formations:
(a) Probability of 3 to 9 feet = Number of rock formations in 3-9 feet category / Total Rock Formations
Probability of 3 to 9 feet = 117 / 291
Probability of 3 to 9 feet ≈ 0.40 (rounded to two decimal places)
(b) Probability of 30 feet = Number of rock formations in 30 feet category / Total Rock Formations
Probability of 30 feet = 32 / 291
Probability of 30 feet ≈ 0.11 (rounded to two decimal places)
(c) Probability of 3 to 49 feet = (Number of rock formations in 3-9 feet category + Number of rock formations in 10-29 feet category + Number of rock formations in 30-49 feet category) / Total Rock Formations
Probability of 3 to 49 feet = (117 + 89 + 32) / 291
Probability of 3 to 49 feet ≈ 0.84 (rounded to two decimal places)
(d) Probability of 10 to 74 feet = (Number of rock formations in 10-29 feet category + Number of rock formations in 30-49 feet category + Number of rock formations in 50-74 feet category) / Total Rock Formations
Probability of 10 to 74 feet = (89 + 32 + 38) / 291
Probability of 10 to 74 feet ≈ 0.60 (rounded to two decimal places)
(e) Probability of 75 feet or taller = Number of rock formations in 75 and higher category / Total Rock Formations
Probability of 75 feet or taller = 15 / 291
Probability of 75 feet or taller ≈ 0.05 (rounded to two decimal places)
To summarize:
(a) Probability of 3 to 9 feet ≈ 0.40
(b) Probability of 30 feet ≈ 0.11
(c) Probability of 3 to 49 feet ≈ 0.84
(d) Probability of 10 to 74 feet ≈ 0.60
(e) Probability of 75 feet or taller ≈ 0.05