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.
Excek spreadsheets are very helpful for these kinds of problem.
a) the probability of selecting a female is .3*.4 + .5*.5 + .2*.5 = .47. Ergo, the probability of picking a male is .53.
The probability of picking all males is .53^50 = something really small.
for n males it (50-choose-n)*.53^(50-n)*.47^n. So, for n=1 its 50*.53^(49) * .47 = 1.725E(-13). Repeat for n=2,n=3, and n=4. the sum of these 5 really small values is the probability of getting 46 or more male. I get 2.33E(-9). 1-minus this is the probability of getting at least 5 females.
b) The probability of NOT picking milk frog in a single pick is .8 The probability in 50 picks is .8^50. I get 1.427E(-5).
c) Largely repeat the steps in a)
To determine the probabilities in this scenario, we need to understand the concept of joint probability and use it to calculate the desired probabilities. Joint probability is the probability that two or more events occur together.
a) To determine the probability of at least five female bullfrogs in the sample, we need to calculate the probability of having exactly five, exactly six, and so on, up to fifty. Then we sum up these individual probabilities.
To calculate the probability of having exactly x female bullfrogs in a sample, we need to consider the following three probabilities:
1. The probability of selecting a bullfrog: 30% (from the total population percentage).
2. The probability of selecting a female bullfrog given it's a bullfrog: 40% (from the gender ratio of bullfrogs).
3. The probability of having exactly x female bullfrogs in a sample of 50 frogs: This can be calculated using the binomial probability formula.
Using the binomial probability formula, the probability of having exactly x successes (in this case, female bullfrogs) in n independent trials (in this case, sample of 50 frogs) can be calculated as:
P(x) = (nCx) * (p^x) * ((1-p)^(n-x))
where:
- nCx is the number of combinations of n items taken x at a time.
- p is the probability of success (in this case, the probability of selecting a female bullfrog given it's a bullfrog).
For each value of x (from 5 to 50), calculate P(x) using the formula above and sum up these probabilities to find the probability of at least five female bullfrogs in the sample.
b) To determine the probability that there will not be any mink frogs in the sample, we need to calculate the probability of not selecting a mink frog for each of the 50 frogs.
The probability of not selecting a mink frog can be calculated as:
1 - (Probability of selecting a mink frog)
To calculate the probability of selecting a mink frog, we need to consider the following two probabilities:
1. The probability of selecting a mink frog: (20% * 50%) + (20% * 50%) = 20% (from the species and gender ratios).
2. The probability of not selecting a mink frog: 1 - (Probability of selecting a mink frog).
For each of the 50 frogs, calculate the probability of not selecting a mink frog and multiply these probabilities together to find the probability that there will not be any mink frogs in the sample.
c) To determine whether having 30 spring peepers in the sample is unusual enough to reconsider the original estimate of their proportion, we need to conduct hypothesis testing.
The null hypothesis (H₀) would be that the proportion of spring peepers in the frog population is the same as the estimated proportion. The alternative hypothesis (H₁) would be that the proportion is significantly different.
To test the hypothesis, you can use a binomial test or a chi-square test. A binomial test would compare the observed number of spring peepers (30) to the expected number based on the estimated proportion and sample size (50). The chi-square test would involve comparing the observed frequencies of the different species to the expected frequencies based on the estimated proportions and sample size.
Using either test, you can calculate the p-value, which represents the probability of observing the data (or more extreme data) under the null hypothesis. If the p-value is below a chosen significance level (e.g., α = 0.05), you can reject the null hypothesis and conclude that the observed proportion of spring peepers in the sample is significantly different from the estimated proportion.
Keep in mind that this is just a general explanation of how to approach the problem. To obtain the actual probability values and test the hypothesis, you would need to perform the calculations using the specific values given in the table.