# Three alleles (alternative versions of a gene) A, B, and O determine the four blood types A (AA or AO), B (BB or BO), O (OO), and AB. The Hardy-Weinberg Law states that the proportion of Boas in a population who carry two different alleles is P = 2pq + 2pr + 2rq where p, g, and rare the proportions of A, B, and O in the population. Use the fact that p +q + r = 1 to show that P is at most frac{2}{3}.

Question
Comparing two groups
Three alleles (alternative versions of a gene) A, B, and O determine the four blood types A (AA or AO), B (BB or BO), O (OO), and AB. The Hardy-Weinberg Law states that the proportion of Boas in a population who carry two different alleles is
$$P = 2pq + 2pr + 2rq$$
where p, g, and rare the proportions of A, B, and O in the population.
Use the fact that $$p +q + r = 1$$ to show that P is at most
$$\frac{2}{3}.$$

2020-11-09
Maximum value of $$P = \frac{2}{3}$$
1) Concept:
Use the concept of absolute maximum and minimum.
To find the absolute maximum or minimum value of a continuous function f on the closed bonded set D:
1. Find the values of f at the critical points of f in D.
2. Find the extreme values of f on the boundary of D
3. The largest value of step 1 and step 2 is absolute maximum value and smallest value being the absolute minimum value.
2) Given:
Hardy-Weinberg Law:
The proportion of individuals in a population who carry two different allers is
$$P = 2pq + 2pr + 2rq$$
Where p, q and r be A, B and O blood group populations.
$$p + q r =1$$
3) Calculation:
Consider Hardy-Weinberg Law,
The proportion of individuals in a population who carry two different alleles is
$$P = 2pq + 2pr + 2rq$$
Where p, q and r be the A, B and O blood group populations.
By using fact, $$p + q + r = 1$$
write r in terms of p and q
$$r = 1 - p - q$$
Substitute this value of r in $$P = 2pq + 2pr + 2rq$$, then it becomes
$$P = P(p, q) = 2(1 - p - q)q + 2(1 - p - q) p + 2pq$$
$$P(p, q) = 2p - 2p^{2} + 2q - 2q^{2} - 2pq$$
As p, q, r is the proportion of species it ranges from 0 to 1.
Therefore, $$p \geq 0, q \geq 0, 1 - p - q \geq 0$$ this implies
$$p + q \leq 1$$
Therefore, domain of P is
$$D = {(p, q) 0 \leq p \leq 1, q \leq 1 - p}$$ which is closed set
bounded by lines $$p = 0, q = 0 and p + q = 1.$$
Now to find critical points, consider equation
$$P(p, q) = 2p - 2p^{2} + 2q - 2q^{2} - 2pq$$
Differentiating P partially with respect to p,
$$P_{p}(p, q) = 2 - 4p - 2q$$
Differentiating P partially with respect to q,
$$P_{q}(p, q) = 2 - 4q - 2p$$
Setting partial derivatives equal to 0, obtain the equations,
$$2 - 4p - 2q = 0 and 2 - 4q - 2p = 0$$
Therefore,
$$2p + q = 1 and p + 2q = 1$$
Solving these system of equations,
$$p + 2(1 - 2p) = 1$$
$$p = \frac{1}{3}$$
Substitute $$p = \frac{1}{3} in 2p + q = 1$$
$$q =1 - 2(\frac{1}{3}) = \frac{1}{3}$$
Thus the critical point is $$(\frac{1}{3}, \frac{1}{3})$$
Therefore,
$$p(\frac{1}{3}, \frac{1}{3}) = 2(\frac{1}{3}) - 2(\frac{1}{3})^{2} + 2(\frac{1}{3}) - 2(\frac{1}{3})^{2} - 2(\frac{1}{3})(\frac{1}{3})$$
$$=\frac{4}{1} - \frac{6}{9} = \frac{12-6}{9} = \frac{6}{9} = \frac{2}{3}$$
$$p(\frac{1}{3}, \frac{1}{3}) = \frac{2}{3}$$
Now find values of P on the boundary of D consisting three lines.
Along the line $$p = 0$$, q ranges between 0 and 1 that is
$$0 \leq q \leq 1.$$
Therefore, $$P(0, q) = 2q - 2q^{2}$$
Which represents downward parabola with maximum value at vertex $$(0, \frac{1}{2})$$
$$P(\frac{1}{2}, 0) = 2(\frac{1}{2}) - 2(\frac{1}{2})^{2} + 2(0) - 2(0)^2 - 2(0)(\frac{1}{2}) = \frac{1}{2}$$
Therefore,
$$P(0, \frac{1}{2}) = \frac{1}{2}$$
Similarly, along the line $$q = 0$$,
p ranges between 0 and 1 that is $$0 \leq p \leq 1.$$
Therefore, $$P(p, 0) = 2p - 2p^{2}$$
Which represents downward parabola with maximum value at vertex $$(\frac{1}{2}, 0)$$
$$P(\frac{1}{2}, 0) = 2(0) - 2(0)^{2} + 2(\frac{1}{2}) - 2(\frac{1}{2})^{2} - 2(0)(\frac{1}{2}) = \frac{1}{2}$$
Therefore,
$$P(\frac{1}{2}, 0) = \frac{1}{2}$$
Also along line $$p + q = 1$$
where $$0 \leq p \leq 1,$$
$$P(p, q) = P(p, 1 - p) = 2p - 2p^{2} + 2(1 - p) - 2(1 - p)^{2} - 2p(1 - p)$$
$$= 2p - 2p^{2} + 2 - 2p - 2 + 4p - 2p^{2} - 2p + 2p^{2}, 0 \leq p \leq 1$$
$$P(p, 1 - p) = 2p - 2p^{2}$$
Which is downward parabola whose vertex is at $$(\frac{1}{2}, \frac{1}{2})$$
Therefore, maximum value is
$$P(\frac{1}{2}, \frac{1}{2}) = 2(\frac{1}{2}) - 2(\frac{1}{2})^{2} = \frac{1}{2}$$
Therefore, comparing all values at boundary with value of P at the critical point
The absolute maximum value of P(p, q) on D is $$\frac{2}{3}.$$
Therefore, the maximum value of P is $$\frac{2}{3}.$$
Conclusion:
The maximum value of P is \frac{2}{3}.

### Relevant Questions

The table below shows the number of people for three different race groups who were shot by police that were either armed or unarmed. These values are very close to the exact numbers. They have been changed slightly for each student to get a unique problem.
Suspect was Armed:
Black - 543
White - 1176
Hispanic - 378
Total - 2097
Suspect was unarmed:
Black - 60
White - 67
Hispanic - 38
Total - 165
Total:
Black - 603
White - 1243
Hispanic - 416
Total - 2262
Give your answer as a decimal to at least three decimal places.
a) What percent are Black?
b) What percent are Unarmed?
c) In order for two variables to be Independent of each other, the P $$(A and B) = P(A) \cdot P(B) P(A and B) = P(A) \cdot P(B).$$
This just means that the percentage of times that both things happen equals the individual percentages multiplied together (Only if they are Independent of each other).
Therefore, if a person's race is independent of whether they were killed being unarmed then the percentage of black people that are killed while being unarmed should equal the percentage of blacks times the percentage of Unarmed. Let's check this. Multiply your answer to part a (percentage of blacks) by your answer to part b (percentage of unarmed).
Remember, the previous answer is only correct if the variables are Independent.
d) Now let's get the real percent that are Black and Unarmed by using the table?
If answer c is "significantly different" than answer d, then that means that there could be a different percentage of unarmed people being shot based on race. We will check this out later in the course.
Let's compare the percentage of unarmed shot for each race.
e) What percent are White and Unarmed?
f) What percent are Hispanic and Unarmed?
If you compare answers d, e and f it shows the highest percentage of unarmed people being shot is most likely white.
Why is that?
This is because there are more white people in the United States than any other race and therefore there are likely to be more white people in the table. Since there are more white people in the table, there most likely would be more white and unarmed people shot by police than any other race. This pulls the percentage of white and unarmed up. In addition, there most likely would be more white and armed shot by police. All the percentages for white people would be higher, because there are more white people. For example, the table contains very few Hispanic people, and the percentage of people in the table that were Hispanic and unarmed is the lowest percentage.
Think of it this way. If you went to a college that was 90% female and 10% male, then females would most likely have the highest percentage of A grades. They would also most likely have the highest percentage of B, C, D and F grades
The correct way to compare is "conditional probability". Conditional probability is getting the probability of something happening, given we are dealing with just the people in a particular group.
g) What percent of blacks shot and killed by police were unarmed?
h) What percent of whites shot and killed by police were unarmed?
i) What percent of Hispanics shot and killed by police were unarmed?
You can see by the answers to part g and h, that the percentage of blacks that were unarmed and killed by police is approximately twice that of whites that were unarmed and killed by police.
j) Why do you believe this is happening?
Do a search on the internet for reasons why blacks are more likely to be killed by police. Read a few articles on the topic. Write your response using the articles as references. Give the websites used in your response. Your answer should be several sentences long with at least one website listed. This part of this problem will be graded after the due date.
Is statistical inference intuitive to babies? In other words, are babies able to generalize from sample to population? In this study,1 8-month-old infants watched someone draw a sample of five balls from an opaque box. Each sample consisted of four balls of one color (red or white) and one ball of the other color. After observing the sample, the side of the box was lifted so the infants could see all of the balls inside (the population). Some boxes had an “expected” population, with balls in the same color proportions as the sample, while other boxes had an “unexpected” population, with balls in the opposite color proportion from the sample. Babies looked at the unexpected populations for an average of 9.9 seconds (sd = 4.5 seconds) and the expected populations for an average of 7.5 seconds (sd = 4.2 seconds). The sample size in each group was 20, and you may assume the data in each group are reasonably normally distributed. Is this convincing evidence that babies look longer at the unexpected population, suggesting that they make inferences about the population from the sample? Let group 1 and group 2 be the time spent looking at the unexpected and expected populations, respectively. A) Calculate the relevant sample statistic. Enter the exact answer. Sample statistic: _____ B) Calculate the t-statistic. Round your answer to two decimal places. t-statistic = ___________ C) Find the p-value. Round your answer to three decimal places. p-value =
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a. Are the groups to be compared independent samples or depentend samples? Why?
b. Show all speps of a test equality of the two population means for a two-sided alternative hypotesis. Report the P-value and interpret.
c. What assumptions must you make the inference in part b to be valid?
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a. Construct a $$95\%$$ confidence interval for the difference between the proportions of customers purchasing tractors with and without warranties. Be sure to check all necessary assumptions and interpret the interval.
b. Test the hypothesis that offering the warranty increases the proportion of customers who eventually purchase a tractor. Be sure to check all necessary assumptions, state the null and alternative hypotheses, obtain the p-value, and state your conclusion. Should a manager offer a warranty based on this test?
Using the Minitab statistical analysis program to enter the data and perform the analysis, complete the following: Construct a one-sided $$\displaystyle{95}\%$$ confidence interval for the true difference in population means. Test the null hypothesis that the population means are identical at the 0.05 level of significance.