Calculus Online: Lab 5 Solutions

Question 1 (2 marks)

Thinking about area and probability is probably not natural for you at first. This question makes sure you understand the relationship. After 100 trials, what is the area of the yellow region in the diagram above. (Hint: you don't need to do any computation.)

This was a bit of a trick question. The area of the yellow region is always 1. The area of each bar is the proportion of trials (die rolls) with a particular outcome. For example, if there are two trials with results of 5 and 9, then there will be a bar above 5 with an area of 1/2 (1 trial out of 2) and a bar above 9 with the same area. Since the proportions of trials always adds to one, the area is always 1.

Question 2 (2 marks)

What is the variance of the distribution for the die with only 3's and 4's on it? Since this distribution is less spread out than for a normal die, you should get an answer smaller than 2.92.

Okay, here we need to apply the formula for the variance, which was developed just earlier in the lab. The mean for this die is 3.5. So the varience is (3-3.5)2/2+(4-3.5)2/2, which is 1/4.

Question 3 (2 marks)

As we increase the number of bins, the sum

\[ \mu=\sum_i x_i f(x_i) \Delta x \]

tends to which of the following integrals?.

a) \[ \int_{0}^{\infty} x ~dx \] b) \[ \int_{0}^{\infty} x^2f(x) ~dx \] c) \[ \int_{0}^{\infty} f(x) ~dx \] d) \[ \int_{0}^{1} f(x) ~dx \] e) \[ \int_{0}^{\infty} x f(x) ~dx \]

We know that answer a) can't be right since it doesn't involve f. Answers b), c), and d) can't be right since the factor of xi appears in the sum once, and this corresponds to a factor of x in the integrad. So, the correct answer is e).

Using the tool above you can also see the p.d.f. and c.d.f. for diameters of fish aged 1, 2, and 3 years old. Suppose we want to catch older fish, which are larger and less vital to the survival of the species. The following set of questions shows how the c.d.f. can be used to analyse problems like this.

Question 4 (4 marks)

In the appropriate area below, answer the following questions:
  1. What is the largest diameter of gill net that will catch 80% of all three-year old fish? (Hint: 20% of three-year old fish are smaller than what diameter?)
  2. What percent of two-year old fish will be caught using a net of this size?
  3. What percent of one-year old fish will be caught using a net of this size?
  4. What percent of all fish will be caught using a net of this size?

The trick here is to see that the CDF gives you exactly the wrong information. We want to catch 80% of all three year old fish. If we make a net, the smallest fish will go free. So we want to make a net such that 20% of the fish will go free. So we want a net with a diameter such that 20% of the (three-year old) fish have a smaller diameter. The problem then is to find a diameter such that the CDF at that point is 0.2. This diameter is around 9.2. The rest of the questions can then be answered by looking at the other CDF's at 9.2.

Supplemental Question

Using the diagram below, figure out how many fish you need catch in a single sample to be 80% certain that the average diameter of your catch is within 1cm of the true diameter.

To answer this question you need to recognize that the area between the dots under the curve is the probability that the sample mean will lay between the dots. Since we want our mean to be correct to within 1cm, we move the dots to 1 and -1. Then we increase n until the shaded area is 0.8. This happens at n=20