Recent Question/Assignment

I only need help to answer questios Q6-Q8 in r studio
Some context:
Imagine a male insect that carries a gene sitting on the Y that is distorting the “fair” meiosis (a fair meiosis of an XY male should produce 50% Y sperm and 50% X sperm). These so called “gene drive systems” are real (some males carrying these gene drive are found in natural populations) or can be engineered by using molecular genetics constructs.
There are various projects around the world that aim to use gene drive as a very radical form of biological control: if you can engineer an extreme drive you generate in the long run a population consisting of males that cannot reproduce. See for instance the proposal to “eradicate” a mosquito species that is the vector for malaria:
The way we detect males carrying such a “drive” gene is by looking at the sex ratio (number of males versus female offspring) in the progeny of females mated with a male suspected to contain such a drive in his genome.
Under “the gene-drive scenario”“, a distorter gene sits on the Y and”cheats- during the male meiosis: a given sperm cell has 3/4 chances of carrying a Y and only 1/4 chances of carrying a X. So we expect 1/4 only of females (and 3/4 of males) in the progeny of females mated with males that carry the drive gene on their Y chromosome.
here is our data: In one progeny of an Aedes mosquito (spread dengue and yellow fever), we observed 14 males out of 20 offspring :
We have the “boring” null (H0H0) hypothesis (nothing extraordinary happens): “Mendelian inheritance” where we expect on average equal amounts of males and females.
And then we have the more exciting alternative hypothesis (HAHA): A gene-drive scenario described above where we expect 3/4 of males and 1/4 of females.
The first first step is to specify what is the null hypothesis distribution for our observation.
Q6: What is the probability distribution of the number of males in a progeny of 20 offsprings?
• Explain why you can use a binomial distribution to model the number of males in a progeny.
• Specify what binomial distribution you can use for the data under H0H0.
• Specify what binomial distribution you use can for data under HAHA?
Q7: Calculate the p-value given the data and H0H0 specified above
• Note here, for once, we are going to do a one-tailed test… Yes… The motivation is that if you reject the null, it is because there is gene drive that biologically is expected to result in an excess of males.
• Do you reject H0H0 at a=0.05a=0.05?
Q8: What is the minimum number of males you need to observe among 20 offspring before you reject H0H0 at a=0.005a=0.005?
Hint: try, different values to see how many males are needed (14, 15 etc) before the tail has a probability that is smaller than a=0.005a=0.005.