Suppose that the populations of the three species are distributed as shown in the following table. You capture a frog, note its gender and species, and then release it. This process is repeated until you have captured and recorded 50 frogs.

Species gen.ratio
Species %total pop. Males Females
bullfrog 30% 60% 40%
spring peeper 50% 50% 50%
mink frog 20% 50% 50%

a) Determine the probability that there will be at least five female bullfrogs in sample.
b)Determine the probability that there will not be any mink frogs in sample.
c) 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.

To determine the probability of certain outcomes, we need to calculate the probabilities of individual events and then combine them using the rules of probability.

a) To determine the probability that there will be at least five female bullfrogs in the sample, we need to consider all possible combinations of male and female bullfrogs.

The probability of selecting a female bullfrog is 40% of the total bullfrog population, which is 0.30 * 0.40 = 0.12.

The probability of selecting a male bullfrog is 60% of the total bullfrog population, which is 0.30 * 0.60 = 0.18.

Now, let's use these probabilities to calculate the probability of having at least five female bullfrogs in the sample.

We can use the binomial distribution formula to calculate the probability:

P(X >= 5) = P(X = 5) + P(X = 6) + ... + P(X = 50), where X is the number of female bullfrogs in the sample.

To calculate individual probabilities, we use the binomial coefficient formula:

P(X = x) = (nCx) * p^x * (1 - p)^(n - x),

where n is the number of trials (50 frogs), x is the number of successful outcomes (female bullfrogs), p is the probability of success (0.12), and (nCx) represents the binomial coefficient (the number of ways to choose x items from n items).

Using this formula, we calculate the probabilities for each value of X from 5 to 50 and sum them up. You can use a programming language or a statistical software package like R or Python to perform the calculations.

b) To determine the probability that there will not be any mink frogs in the sample, we need to consider the probability of not selecting a mink frog in each capture.

The probability of not selecting a mink frog is 50% of the total mink frog population, which is 0.20 * 0.50 = 0.10.

To calculate the probability of not having any mink frogs in the sample, we can raise this probability to the power of the number of frogs captured (50 in this case):

P(no mink frogs) = (1 - 0.10)^50

Calculate this expression to find the probability.

c) To determine whether the sample of 30 spring peepers is unusual enough to reconsider the original estimate of their proportion of the frog population, we need to perform a hypothesis test.

First, let's define the null hypothesis and the alternative hypothesis:

Null hypothesis (H0): The proportion of spring peepers in the frog population is equal to the original estimate of 50% (0.50).
Alternative hypothesis (H1): The proportion of spring peepers in the frog population is different from the original estimate of 50% (0.50).

Next, we can perform a proportion test using a statistical software package like R or Python. The test will compare the observed proportion in the sample (30 out of 50 captured frogs) with the hypothesized proportion of 50%.

The result of the test will give us a p-value, which represents the probability of observing a sample proportion as extreme as the one we observed (30/50) if the null hypothesis is true.

If the p-value is low (typically below a predetermined significance level such as 0.05), we reject the null hypothesis and conclude that the observed proportion is significantly different from the original estimate. If the p-value is high, we fail to reject the null hypothesis and conclude that the observed proportion is not significantly different from the original estimate.