# There is currently a global pandemic and many researchers and scientists are working assiduously to find a vaccine. Assume that Advanced Researcher, a company that is currently testing a few vaccines confirms an 80% chance of effectiveness of one of their vaccines. Which probability distribution would you use to calculate the probability of success or failure of Advanced Researcher's vaccine if it were to be administered to people in society?

Question
Upper Level Math
There is currently a global pandemic and many researchers and scientists are working assiduously to find a vaccine. Assume that Advanced Researcher, a company that is currently testing a few vaccines confirms an 80% chance of effectiveness of one of their vaccines.
Which probability distribution would you use to calculate the probability of success or failure of Advanced Researcher's vaccine if it were to be administered to people in society?

2021-03-08
Step 1
Given,
There is currently a global pandemic and many researchers and scientists are working assiduously to find a vaccine. Assuming that Advanced Researcher, a company that is currently testing a few vaccines confirms an 80% chance of effectiveness of one of their vaccines.
$$\displaystyle\Rightarrow{P}{\left({S}{u}{\mathcal{{e}}}{s}{s}\right)}={0.8}$$
$$\displaystyle\Rightarrow{P}{\left({f}{a}{i}{l}{u}{r}{e}\right)}={1}-{0.8}={0.2}$$
Step 2
The probability distribution that we would use to calculate the probability of success or failure of Advanced Researcher's vaccine if it were to be administered to people in society is Binomial distribution.
Here, the assumptions are:
Each trial has only two possible outcomes, but only one occurs at a time
Each trial is mutually exclusive.
Each trial has equal probability.

### Relevant Questions

Give a full and correct answer Why is it important that a sample be random and representative when conducting hypothesis testing? Representative Sample vs. Random Sample: An Overview Economists and researchers seek to reduce sampling bias to near negligible levels when employing statistical analysis. Three basic characteristics in a sample reduce the chances of sampling bias and allow economists to make more confident inferences about a general population from the results obtained from the sample analysis or study: * Such samples must be representative of the chosen population studied. * They must be randomly chosen, meaning that each member of the larger population has an equal chance of being chosen. * They must be large enough so as not to skew the results. The optimal size of the sample group depends on the precise degree of confidence required for making an inference. Representative sampling and random sampling are two techniques used to help ensure data is free of bias. These sampling techniques are not mutually exclusive and, in fact, they are often used in tandem to reduce the degree of sampling error in an analysis and allow for greater confidence in making statistical inferences from the sample in regard to the larger group. Representative Sample A representative sample is a group or set chosen from a larger statistical population or group of factors or instances that adequately replicates the larger group according to whatever characteristic or quality is under study. A representative sample parallels key variables and characteristics of the large society under examination. Some examples include sex, age, education level, socioeconomic status (SES), or marital status. A larger sample size reduced sampling error and increases the likelihood that the sample accurately reflects the target population. Random Sample A random sample is a group or set chosen from a larger population or group of factors of instances in a random manner that allows for each member of the larger group to have an equal chance of being chosen. A random sample is meant to be an unbiased representation of the larger population. It is considered a fair way to select a sample from a larger population since every member of the population has an equal chance of getting selected. Special Considerations: People collecting samples need to ensure that bias is minimized. Representative sampling is one of the key methods of achieving this because such samples replicate as closely as possible elements of the larger population under study. This alone, however, is not enough to make the sampling bias negligible. Combining the random sampling technique with the representative sampling method reduces bias further because no specific member of the representative population has a greater chance of selection into the sample than any other. Summarize this article in 250 words.
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.
State whether the investigation in question is an observational study or a designed experiment. Justify your answer in each case.
The Salk Vaccine. In the 1940s and early 1950s, the public was greatly concerned about polio. In an attempt to prevent this disease, Jonas Salk of the University of Pittsburgh developed a polio vaccine. In a test of the vaccine’s efficacy, involving nearly 2 million grade-school children, half of the children received the Salk vaccine, the other half received a placebo, in this case an injection of salt dissolved in water. Neither the children nor the doctors performing the diagnoses knew which children belonged to which group, but an evaluation center did. The center found that the incidence of polio was far less among the children inoculated with the Salk vaccine. From that information, the researchers concluded that the vaccine would be effective in preventing polio for all U.S. school children, consequently, it was made available for general use.
Potential buyers for a new car were randomly divided into two groups. One group was shown the "A" version of an ad for the car, while the other group was shown the "B" version of the ad. All were then tested on their recall of key points made in the ad. The researcher should run a hypothesis test based upon a comparison of means for ?
In another study, a healthcare insurance company took measures of subscribers’ cardiac (heart) health. The people were then provided an app for their phones which provided "nudges" and reminders about heart-healthy behaviors, such as eating more vegetables and less fried or fatty food, taking walks and breaks from sitting too long, and getting enough sleep. After 4 months of having the app, the cardiac health measures were taken again, with the objective of seeing if nudges from the app would result in decreased cardiac risk. The researcher should run a hypothesis test based on a comparison of means for?
A company is marketing a new product they say works better than the traditional test tube. There is so much interest in the product that 30 different labs around the world are testing the claim that this product is actually better. If each study uses an alpha level (alpha) of .10, and if the null hypothesis is true (that the test tube isn't any better that the traditional one), how many of the hypothesis tests would we expect to incorrectly find statistical significance (that is, conclude that the new test tube is better, when it actually isn't)?
Scientists are working with a sample of cobalt-56 in their laboratory. They begin with a sample that has 60 mg of cobalt-56, and they measure that after 31 days, the mass of cobalt-56 sample is 45.43 mg. Recall that the differential equation which models exponential decay is $$\frac{dm}{dt}=-km$$ and the solution of that differential equation if $$m(t)=m_0e^{-kt}$$, where $$m_0$$ is the initial mass and k is the relative decay rate.
a) Use the information provided to compute the relative decay rate k. Show your calculation (do not just cit a formula).
b) Use the information provided to determine the half-life of cobalt-56. Give your answer in days and round to the second decimal place. Show your calculation (do not just cite a formula).
c) To the nearest day, how many days will it take for the initial sample of 60mg of cobalt-56 to decay to just 10mg of cobalt-56?
d) What will be the rate at which the mass is decaying when the sample has 50mg of cobalt-56? Make sure to indicate the appropriate units and round your answer to three decimal places.
Scientists are working with a sample of cobalt-56 in their laboratory. They begin with a sample that has 60 mg of cobalt-56, and they measure that after 31 days, the mass of cobalt-56 sample is 45.43 mg. Recall that the differential equation which models exponential decay is $$\displaystyle{\frac{{{d}{m}}}{{{\left.{d}{t}\right.}}}}=-{k}{m}$$ and the solution of that differential equation if $$\displaystyle{m}{\left({t}\right)}={m}_{{0}}{e}^{{-{k}{t}}}$$, where $$\displaystyle{m}_{{0}}$$ is the initial mass and k is the relative decay rate.
a) Use the information provided to compute the relative decay rate k. Show your calculation (do not just cit a formula).
b) Use the information provided to determine the half-life of cobalt-56. Give your answer in days and round to the second decimal place. Show your calculation (do not just cite a formula).
c) To the nearest day, how many days will it take for the initial sample of 60mg of cobalt-56 to decay to just 10mg of cobalt-56?
d) What will be the rate at which the mass is decaying when the sample has 50mg of cobalt-56? Make sure to indicate the appropriate units and round your answer to three decimal places.
A bipolar alkaline water electrolyzer stack module comprises 160 electrolytic cells that have an effective cell area of $$\displaystyle{2}{m}^{{2}}$$. At nominal operation, the current density for a single cell of the electrolyzer stack is 0.40 $$\displaystyle\frac{{A}}{{c}}{m}^{{2}}$$. The nominal operating temperature of the water electrolyzer stack is $$\displaystyle{70}^{\circ}$$ C and pressure 1 bar. The voltage over a single electrolytic cell is 1.96 V at nominal load and 1.78 V at 50% of nominal load. The Faraday efficiency of the water electrolyzer stack is 95% at nominal current density, but at 50% of nominal load, the Faraday efficiency decreases to 80%.
Calculate the nominal stack voltage:
1. A researcher is interested in finding a 98% confidence interval for the mean number of times per day that college students text. The study included 144 students who averaged 44.7 texts per day. The standard deviation was 16.5 texts. a. To compute the confidence interval use a ? z t distribution. b. With 98% confidence the population mean number of texts per day is between and texts. c. If many groups of 144 randomly selected members are studied, then a different confidence interval would be produced from each group. About percent of these confidence intervals will contain the true population number of texts per day and about percent will not contain the true population mean number of texts per day. 2. You want to obtain a sample to estimate how much parents spend on their kids birthday parties. Based on previous study, you believe the population standard deviation is approximately $$\displaystyle\sigma={40.4}$$ dollars. You would like to be 90% confident that your estimate is within 1.5 dollar(s) of average spending on the birthday parties. How many parents do you have to sample? n = 3. You want to obtain a sample to estimate a population mean. Based on previous evidence, you believe the population standard deviation is approximately $$\displaystyle\sigma={57.5}$$. You would like to be 95% confident that your estimate is within 0.1 of the true population mean. How large of a sample size is required?
$$1) Use Euclidean algorithm to find \(d= GCD(X, Y).$$
2) Find the integers s and tsuch that $$d = sX + tY$$
3) With the answer obtained from a, what is the ratio of subjects selected from population $$X\ and\ Y, PX : PY.$$
4) Find Least Common Multiple for $$Xand Y, LCM(X, Y).$$