An instructor for a college biology lab has each of her 12 students (you are one of those 12 students) take their own SRS of 25 plants from an experimental field. She lets you know that the heights (measured in cm) of the plants in the field follow a N(50,4) distribution.
a. Find the probability that your SRS has an avg. height of at least 52 cm
b. What percentage of the 12 studnets would be expected to have a SRS with average height of at least 52 cm? Explain
c. What is the probability that one or more students choose a SRS with avg. height of at least 52 cm?
Please explain how to do this!
To solve these questions, we need to use the properties of a normal distribution and some basic probability calculations.
a. Find the probability that your SRS has an average height of at least 52 cm:
The average height of your SRS follows a normal distribution with mean 50 cm and standard deviation (σ) of 4 cm. We need to find the probability that the average height of your SRS is at least 52 cm.
To calculate this probability, we first need to standardize the value 52 cm using the formula: z = (x - μ) / σ, where z is the standardized score, x is the value we want to find the probability for, μ is the mean, and σ is the standard deviation.
In this case, let's calculate the z-score for 52 cm:
z = (52 - 50) / 4 = 0.5
Now we need to find the cumulative probability of z or find P(Z ≥ 0.5), where Z is the standard normal distribution.
You can use a standard normal distribution table or use a statistical software package to find this probability. For example, if you refer to a standard normal distribution table, you would find that P(Z ≥ 0.5) is approximately 0.3085.
So the probability that your SRS has an average height of at least 52 cm is approximately 0.3085.
b. What percentage of the 12 students would be expected to have an SRS with an average height of at least 52 cm:
Since each student takes their own SRS of 25 plants, their average heights also follow a normal distribution with mean 50 cm and standard deviation (σ) of 4 cm.
To find the percentage of students who would be expected to have an SRS with an average height of at least 52 cm, we can calculate the probability for each student.
From part a, we found that the probability for one student to have an average height of at least 52 cm is approximately 0.3085.
Since the students are randomly sampling from the same population, we can assume their samples are independent. Thus, we can multiply the probabilities together: (0.3085)^12 ≈ 0.00002509, or approximately 0.0025%.
Therefore, the percentage of the 12 students that would be expected to have an SRS with an average height of at least 52 cm is approximately 0.0025%.
c. What is the probability that one or more students choose an SRS with an average height of at least 52 cm:
To find the probability that one or more students choose an SRS with an average height of at least 52 cm, we can use the complement rule.
The complement rule states that the probability of an event happening is 1 minus the probability of the event not happening.
Since the probability of a student not having an average height of at least 52 cm is 1 - 0.3085 (from part a), the probability of all 12 students not having an average height of at least 52 cm is (1 - 0.3085)^12 ≈ 0.3035.
Therefore, the probability that one or more students choose an SRS with an average height of at least 52 cm is approximately 1 - 0.3035 ≈ 0.6965, or approximately 69.65%.