Question from my human pop genetics class

Question

I apologize that this is a long one but I'm fearing I've done all of this wrong. I'm especially confused with graphing all of this for part C.

(25 pts) Some people do not recognize the bitter taste of phenylthiocarbamide (PTC). The inability to taste PTC is due to a single autosomal recessive gene (tt). Alimba and colleagues (2010) studied a population from Nigeria and found the frequency of the t allele was q = 0.542.

(a) Approximately what proportion of people in that population tastes PTC? Consider the experiment in which sets of 4 people are randomly selected from the population and assayed for how many in each group taste PTC.

(b) What kind of “experiment” is this (binomial or geometric)?

(c) Graph the distribution of tasters expected from this experiment.

(d) What is the probability that two or fewer people will be tasters?

(e) What is the average number of tasters expected?

(f) What is the standard deviation?

Answer 1

No worries, let's walk through this step by step once more.

(a) What proportion of people in that population tastes PTC?

Given:

• The frequency of the t allele (recessive) is q = 0.542 which means the frequency of the T allele (dominant) is p = 1 - q = 0.458

• People who taste PTC are either homozygous dominant (TT) or heterozygous (Tt).

To find the proportion of tasters in the population, use Hardy-Weinberg proportions:

• p² = frequency of TT (tasters) = (0.458)20.2097

• 2pq frequency of Tt (tasters) = 2 x 0.458 x 0.542 0.4968

Therefore, the total proportion of tasters is:

p ^ 2 + 2pq = 0.2097 + 0.4968 = 0.7065

(b) What kind of experiment is this (binomial or geometric)?

This is a binomial experiment because:

• There are two possible outcomes for each person: tasting or not tasting PTC.

• The probability of tasting is constant at 0.7065.

We are sampling a fixed number of people (4).

In a binomial experiment, we are counting how many "successes" (tasters) occur in a fixed number of trials (people sampled).

(c) Graph the distribution of tasters expected from this experiment

We can model the number of tasters in a group of 4 using a binomial distribution with:

n = 4 (number of people in a group),

p0.7065 (probability of tasting PTC).

The binomial probability formula is: P(X = k) = binomial(n,k) * p ^ k * (1 - p) ^ (n - k)

(d) What is the probability that two or fewer people will be tasters?

To find the probability of two or fewer tasters, sum the probabilities of getting 0, 1, or 2 tasters. This is calculated using the cumulative distribution function (CDF) of the binomial distribution.

For X as the number of tasters: P(X2)=P(X=(0)+ P(X = 1) P(X = 2)

(e) What is the average number of tasters expected?

The expected (mean) number of tasters in a group of 4 is: E(X) = np = 4 * 0.7065 = 2.83

(f) What is the standard deviation?

The standard deviation for a binomial distribution is:

sigma = sqrt(np(1 - p))

By plugging these values into the formula, you can calculate the standard devi

Answer 2

for (a) p=1-q= 0.458 . Tasters are TT and Tt, so calculate p2 and 2pq, add them and you get 0.705 or ~ 70% of the population tastes PTC

(b) Binomial

(c) Calculate the probabilities of having 0,1,2,3,4 tasters in each set of 4

Use the probability mass function for binomial distribution, calculate each provability, graph it