Question

please answer all questions

Environmental scientists believe that there is a decline in the population of frogs around the world. They speculate that declining frog populations are a symptom of deteriorating environmental conditions. How would these scientists gather accurate data about existing frog populations? How would they use these data to speculate on previous or future population levels? Imagine that a wetland you are studying has a population of 5000 frogs, including bullfrogs, spring peepers, and mink frogs. The populations of the three species are shown below Species Gender Ratio Percent of Total Marsh Population Species Females Males 60% 55% 52% 40% Bullfrog Spring peeper Mink frog 30% 50% 25% 48% You capture a frog, note its species and gender, and then release it. This process is repeated until you have captured and recorded 50 frogs 2. Discuss why you could model the number of a given type of frog with a binomial distribution. Create a probability distribution table for the number of mink frogs in the sample Determine the probability that there will be at least five female bullfrogs in the sample Determine the probability that there will not be any mink frogs in the sample. Suppose that there were 30 spring peepers in the sample.. Determine whether this is unusual enough to cause you to reconsider your original estimate of their proportion of the frog population. a. b. c. d. e. 3. Could you use a normal approximation for the number of frogs in question 2? Why or why not? 4. If you have a sample of 100 frogs, calculate and interpret the mean and the standard deviation of the number of female frogs. What is the probability that you will capture between 30 and 50 females? Construct a 90% confidence interval for the proportion of females captured. a. b. 5. What is a limitation of using the binomial distribution to estimate the frog population? 6. Suppose now you have caught and marked frogs with a band, then released them. A total of 54 male and 36 bullfrogs, 76 male and 74 spring peepers, and 29 male and 31 female mink frogs were marked with the band. A month later, you go out and capture 100 frogs. Of these, 5 have bands. a. What kind of distribution will you use to model this, and why? b. Discuss what has happened to the frog population.

Answer 1

2(a) Since frogs are chosen with replacement , the probability of choosing a particular type of frog remains constant.

Number of trials are finite (50)

Trials are independent

So we can use Binomial model.

(b) Let X be the number of mink frogs

X follows Binomial with n= 50 , p = 0.20

$P(X=x)= \binom{50}{x}0.2^{x}(1-0.2)^{50-x}$

(c)

Let X be the number of bull frogs

X follows Binomial with n= 50 , p = 0.3*0.4=0.12

$P(X\geq 5)= \sum_{5}^{50}\binom{50}{x}0.12^{x}(1-0.12)^{50-x}$

   $=1- \sum_{0}^{4}\binom{50}{x}0.12^{x}(1-0.12)^{50-x}$

= 1-0.2679

= 0.7320

(d)

$P(X=0) =\binom{50}{0}0.2^{0}(1-0.2)^{50}$

= 0.00001427

(e)

Let X be the number of bull frogs

X follows Binomial with n= 50 , p = 0.5

$P(X=30) =\binom{50}{30}0.5^{30}(1-0.5)^{50-30}$

= 0.0419

The probability we get is very low

which is unusual with p =0.5

So 30 spring peppers in the sample is unusual enough to reconsider the original proportion