7 Binomial Test
Satellite forest mapping
Rapid ʻŌhiʻa Death (ROD) is a devastating pathogen affecting native ʻōhiʻa forests in Hawaiʻi. Much work has been done to try to document its spread. Vaughn et al. (2018) used remote sensing images from 2017 to detect areas of potential ROD. Remote sensing uses fancy cameras attached to low-flying airplanes to capture many different wavelengths of light. The wavelengths are then analyzed to classify areas as likely ROD or not. The image above is from the Kohala Mountains on the Island of Hawaiʻi and shows 28.67% ROD in forested areas. Suppose Vaughn et al. were able to re-survey this same area now in 2024 and calculate the percentage of ROD as 34.78%.
Questions 1–2 refer to the prompt above.
True or false: given the percentage data, would a binomial test be an appropriate method to test the hypothesis that ROD area has increased over time? [1 point]
- TRUE
- FALSE
Instead, imagine you did a “boots on the ground” survey in 2024 of individual trees in the Kohala Mountains and recorded whether each tree was infected with ROD or not. Imagine you find that out of 1000 trees you surveyed, 34.8% are infected. In this situation, would a binomial test be an appropriate method to test the hypothesis that ROD prevalence has increased compared to the 28.67% value from 2017 (true or false)? [1 point]
- TRUE
- FALSE
Hybrid viability
Hawaiian Drosophila (yes that famous genus of fruit fly) are super diverse and super awesome. They speciate like crazy due, in part, to reproductive isolation. When studying reproductive isolation between species, scientists will often allow different species to mate then count the viability of hybrid offspring. Suppose that you are measuring the viability of hybrid eggs laid by a female fruit fly. This female fruit fly will lay eggs over a period of one week, but she needs to be transferred to a fresh vial of food (yeast) every day. After you transfer her to a new vial, you count the number of eggs she laid and record the number of viable eggs. Here are your data:
| Day | Eggs laid | Viable offspring |
|---|---|---|
| 1 | 6 | 3 |
| 2 | 6 | 3 |
| 3 | 8 | 4 |
| 4 | 13 | 6 |
| 5 | 9 | 4 |
| 6 | 8 | 5 |
| 7 | 9 | 6 |
Questions 3–6 refer to the prompt and table above
If you wish to perform a binomial test on total offspring viability for this fly, what would the number of trials be in these data? [1 point]
- 1
- 2
- 7
- 31
- 59
If you wish to perform a binomial test on total offspring viability for this fly, what would the value for the test statistic be in these data? [1 point]
- 1
- 2
- 7
- 31
- 59
Matings within the same species of Drosophila usually yield eggs that are 90% viable. If you wish to test whether matings between species are less viable than matings within species, a form of reproductive isolation, what would an appropriate null hypothesis be? [1 point]
- \(H_0: p = 0\)
- \(H_0: p = 0.5\)
- \(H_0: p = 0.9\)
- \(H_0: p = 1\)
You estimate that the proportion of viable hybrid offspring from this female is \(\hat{p} = 0.53\) and the 95% confidence interval for \(\hat{p}\) is 0.40–0.65. Based on these results, would you conclude that hybrid offspring from this fly are less viable than offspring from typical crosses within species? [1 point]
- No, because \(P > 0.05\)
- Yes, because \(P < 0.05\)
- No, because \(p = 0.9\), the typical proportion of viable offspring for within species crosses, is well outside the 95% confidence intervals
- Yes, because \(p = 0.9\), the typical proportion of viable offspring for within species crosses, is well outside the 95% confidence intervals
Binomial distribution
In questions 7–10 you’ll work directly with the binomial distribution
- Which of the following figures correctly shows the sampling distribution for \(n = 20\) trials with \(p = 0.20\)? [1 point]
Use R code to calculate the probability of observing exactly 10 success from the binomial distribution for \(n = 20\) trials with \(p = 0.20\) [1 point] Report both your R code and the answer. For example if the question was “Use R code to calculate the sum of 1 plus 1” you would report:
1 + 1 # 2Use R code to find the probability of 10 or more success from the binomial distribution for \(n = 20\) trials with \(p = 0.20\) [1 point] Report your answer in the same format as for question (8.)
Imagine I flipped a coin 20 times and got 6 heads versus 14 tails. Use R code to test the null hypothesis that this is a fair coin using the binomial test (hint: \(p = 0.5\) is the correct value of \(p\) for the null hypothesis of a fair coin). [1 point] Again report your answer in the format of questions (8.–9.)