Rose and Jack plan to study together for the AMS311 test. They decide to meet at the library between 8:00pm and 8:30pm. Assume that they each arrive (independently) at a random time (uniformly) in this interval. It is possible that someone has to wait up to 30 minutes for the other to arrive.

a). What is the probability that someone (Rose or Jack, whichever arrives first) must wait more than 20 minutes until the other one arrives? What is the probability that Jack waits for more than 20 minutes?

b). What is the expected amount of time that somebody (the first person to arrive) waits? Formulate the problem and solve. Make sure you carefully define the random variables you use!

Use this extract taken from the article, “L-Glutamine changes gut bacteria leading to weight loss,” (appeared in Preventdisese.com on November 2, 2019) to answer the questions that follow:

L-Glutamine is the most common amino acid found in your muscles and it plays a key role in protein metabolism, and the ability to secrete human growth hormone, which helps metabolize body fat and support new muscle growth. Researchers have now found that a daily L-glutamine dose of 30 grams per day was associated with a significant reduction in the ratio of specific biomarkers for obesity. The 30g dose studied was associated with a significant reduction in the ratio of Firmicutes to Bacteroidetes in obese and overweight people. “The finding that L-glutamine promotes changes in the gut microbiota composition provides support for the importance of some nutrients in modulating the intestinal bacterial profile,” wrote the researchers in Nutrition. “These changes resembled the weight loss programs established in the literature”. A new study, albeit small scale and of limited duration, suggested that the amino acid L-glutamine may also have weight management potential by changing the bacterial composition in the gut. The Brazilian researchers did not observe any changes in body weight during their 14 day study, but noted that a longer intervention period “may result in metabolic changes”. The researchers recruited 33 overweight and obese adults, aged between 23 and 59 and randomly assigned them to receive supplements of L-glutamine or L-alanine for two weeks. A reduction of 0.3 was observed in the ratio of Firmicutes to Bacteroidetes in the L-glutamine group, they added (from 0.85 to 0.57), while L-alanine was associated with an increase from 0.91 to 1.12. “Thus, these findings suggest that oral supplementation of L-glutamine have similar effects on gut microbiota as weight loss,” said the researchers. “We would like to highlight that although the age range of the volunteers was large (23-59 years) and aging may have an effect on intestinal microbiota, the results obtained in this study were statistically significant”.

(a) Is the study by Brazilian researchers that was cited in this article observational or experimental? In less than 50 words clearly explain your choice based on the extract given above.

(b) Identify the variable(s) of interest.

(c) Explain explicitly what a confounding variable is. Identify one plausible confounding variable in this study, explain why it is a confounding variable and suggest a possible way to overcome the impact of the confounding variable.

(d) Is the conclusion from this particular study reflected in the title of the article appropriate? Justify your response.

At Community Hospital, the burn center is experimenting with a new plasma compress treatment. A random sample of n₁ = 302 patients with minor burns received the plasma compress treatment. Of these patients, it was found that 250 had no visible scars after treatment. Another random sample of n₂ = 440patients with minor burns received no plasma compress treatment. For this group, it was found that 85 had no visible scars after treatment. Let p₁ be the population proportion of all patients with minor burns receiving the plasma compress treatment who have no visible scars. Let p₂ be the population proportion of all patients with minor burns not receiving the plasma compress treatment who have no visible scars.

(a) Find a 99% confidence interval for p₁ − p₂. (Round your answers to three decimal places.)

(b) Explain the meaning of the confidence interval found in part (a) in the context of the problem. Does the interval contain numbers that are all positive? all negative? both positive and negative? At the 99% level of confidence, does treatment with plasma compresses seem to make a difference in the proportion of patients with visible scars from minor burns?

Because the interval contains both positive and negative numbers, we can not say that there is a higher proportion of patients with no visible scars among those who received the treatment.We can not make any conclusions using this confidence interval.     Because the interval contains only negative numbers, we can say that there is a higher proportion of patients with no visible scars among those who did not receive the treatment.Because the interval contains only positive numbers, we can say that there is a higher proportion of patients with no visible scars among those who received the treatment.

A group of statistics students decided to conduct a survey at Loyola Marymount University to find the mean number of hours students spent studying per week. a. They sampled 576 students and found the mean to be 20 hours. Assuming we know the population standard deviation is 7 hours, what is the confidence interval at the 95% level of confidence? b. They sampled 576 students and found the mean to be 20 hours. Assuming we know the population standard deviation is 7 hours, what is the confidence interval at the 99% level of confidence? What do you notice as the difference between the solution for (a) and (b)? c. They sampled 100 students and found the mean to be 21 hours. Assuming we don’t know the population standard deviation, but we estimate the sample standard deviation to be 6 hours, what is the confidence interval at the 95% level of confidence? d. Assuming we know the population standard deviation is 7 hours, what is the required sample size if the error should be less than 15 minutes with a 95% level of confidence?

1. Economists often track employment trends by measuring the proportion of people who are “underemployed,” meaning they are either unemployed or would like to work full time but are only working part-time. In the summer of 2019, 18.5% of Americans were “underemployed.” The mayor of Detroit wants to show the voters that the situation is not as bad in his city as it is in the rest of the country. His staff takes a simple random sample of 400 Detroit residents and finds that 60 of them are underemployed.  

(a) Does the data give convincing evidence that the proportion of underemployed in Detroit is lower than elsewhere in the country? Perform the appropriate statistical test.

(b) The mayor’s political rival claims that the same poll actually fails to provide sufficient evidence that the underemployment rate in Detroit is any different from the rest of the country. Explain how it is possible for him to come to this conclusion.

(c) Suppose the true underemployment rate in Detroit is actually only 14%. If the mayor were to perform the exact same test again, what is the probability that the mayor’s test results in a Type II Error at the 5% level?

Find the area under the standard normal curve. Round your answer to four decimal places.

(a) Find the area under the standard normal curve to the right of z= −1.97.

(b) Find the area under the standard normal curve that lies between z= 1.26 and z=2.32.

(c) Find the area under the standard normal curve that lies outside the interval between z=0.46 and z=1.75.

(d) Find the area under the standard normal curve to the left of z= −0.94.

4. Here is a fact about permutations: (**) nPk = n!/(n-k)!, for all k € ≤ n. Let’s prove this via mathematical induction for the fixed case k=3.

(i) Write clearly the statement (**) we wish to prove. Be sure your statement includes the phrase “for all n” .

(ii) State explicitly the assumption in (**) we will thus automatically make about k=2.

(iii) Now recall that to prove by induction means to show that If mPk = m!/(m-k)! is true for all € k ≤ m then m+1Pk = (m+1)!/((m+1)-k )! for all € k ≤ m +1 must also be true. State what we must prove in the case k=3. Include the relevant statement about k=2 here, as you will need to use it in (iv).

(iv) OK so now prove (**) for the case k=3.

(a) Verify the theorem is true for the “base case” n=3 (I.E) that (**) is true for k=0,1,2,3 when n=3. You can do these four verifications by elementary means. Just remember what we mean by permutations, and thus convince us these four statements are true.

(b) Now use your cleverness to prove the underlined statement (iii) is true.

(c) Now state the fact that you have proven (**) to be true for k=3 and all n.

You get called for Jury duty. There are 100 potential jurors and the judge needs to select twelve of them. Assume the jury is chosen at random.

(a) How many different juries can the judge make from that group?

(b) How many different juries can the judge make which include you?

(c) What is the probability that you will end up on the jury?

Question 1

For smartphone, the battery performance is one of the important technical specifications. A company has designed a version of smartphone and has done some tests on the battery life. Let X denotes the amount of time that the fully charged battery can last for Internet surfing. The unit of X is hour and it is omitted in the following discussion.

(a) Suppose we know that X is normally distributed with mean μ=10 and standard deviation σ=1.2, i.e., X~N(10,〖1.2〗^2). For a randomly picked phone, what is the probability that its battery can last for longer than 9.3 hours for internet surfing?

(b) Suppose we know that X~N(10,〖1.2〗^2). If we randomly pick 36 phones and calculate the average of battery time for internet surfing, which is denoted as X ̅. Then, what is the value of P(X ̅>9.3)?

(c) Suppose we don’t know the distribution of X, but only know that its mean μ=10 and its standard deviation σ=1.2. Will the result in (b) be affected seriously? Why?

(d) Suppose we know nothing about X and take a random sample to infer the mean of X. If the sample size is n=36, and after calculation we find the sample mean is X ̅=9.8 and the sample standard deviation is s=1.3. Construct a 95% confidence interval estimate of the mean of X.

(e) Suppose we know that the standard deviation of X is σ=1.2. Based on the same sample in (d), what is the 95% confidence interval estimate of the mean of X? What is the sampling error in this case? If we want to reduce the sampling error by half, how many additional phones are required?

A section of an Introduction to Psychology class took an exam under a set of unusual circumstances. The class took the exam in the usual classroom, but heavy construction noise was present throughout the exam. For all previous exams using the same format and same questions, student scores were normally distributed with a mean of µ = 75.00 and a population standard deviation (sigma) = 10.50. To understand the possible effects of the construction noise, you have been asked to perform a number of statistical procedures for the following sample of exam scores obtained during the construction noise:

Construction Noise Exam Scores

57 58 59 60 61 64 65 66 66 67 67 68

68 69 69 70 70 70 70 71 72 72 72 72

72 73 75 77 78 81 82 83 84 88 96 100

What is the sum of squared deviations (SS) of the sample?  Variance of the sample? Standard deviation of the sample?What is the z score for a raw score of 90 in the SAMPLE (as well as POPULATION)? Be sure to use the sample calculation for a z-score.

PLEASE ANSWER THIS THOROUGHLY LABLED THANK U <33

For this assignment you will choose a topic and create a data collection instrument. When selecting a topic, remember you need to collect both quantitative and qualitative data. Data collection could be done in the form of a survey via a medium such as Survey Monkey or Facebook or through available workplace content such as sales orders. You will want to collect data from approximately 30 observations. This means you will want to survey at least 30 people or select data from 30 different sales orders.

In the following problem, check that it is appropriate to use the normal approximation to the binomial. Then use the normal distribution to estimate the requested probabilities.

Do you take the free samples offered in supermarkets? About 64% of all customers will take free samples. Furthermore, of those who take the free samples, about 41% will buy what they have sampled. Suppose you set up a counter in a supermarket offering free samples of a new product. The day you were offering free samples, 327 customers passed by your counter. (Round your answers to four decimal places.)

(a) What is the probability that more than 180 will take your free sample?


(b) What is the probability that fewer than 200 will take your free sample?


(c) What is the probability that a customer will take a free sample and buy the product? Hint: Use the multiplication rule for dependent events. Notice that we are given the conditional probability P(buy|sample) = 0.41, while P(sample) = 0.64.


(d) What is the probability that between 60 and 80 customers will take the free sample and buy the product? Hint: Use the probability of success calculated in part (c).

150 subjects in a psychological study had a mean score of 36 on a test instrument designed to measure anger. The standard deviation of the 150 scores was 12. Find a 98% confidence interval for the mean anger score of the populations from which the sample was selected.

Select one: a. (34.1, 37.9) b. (33.5, 38.5) c. (33.7, 38.3) d. None of other answer is necessary true.

Which of the following factors can possibly make the observed linear correlation from a sample in a different direction or pattern from the true correlation in the population? Can be many answers.

1. Presence of a non-linear relationship

2. Outliers that deviate from the main cluster of data

3. Different correlations for different subgroups within the data

4. Range restriction on the predictor and/or criterion variables

5. Unreliable measures of the predictor and/or the criterion variables

6. Standardizing the scores on the predictor and/or the criterion

Included in a classification or prediction model, highly correlated values, or variables that are unrelated to the outcome of interest can lead to overfitting, and reliability can suffer. T/F

It is best to normalize when the units of measurement are common for the variables and when their scale reflects their importance. T/F

This is a useful procedure for reducing the number of predictors in the model by analyzing the input variables. It is intended to be used with quantitative variables. A.correlation Analysis , B Correspoding Anaylysis, C Principle components analysis D, ABC

This provides the sense of how dispersed the data are relative to the mean. A.Standard Deviation, B.Variance, C.Mean D.Average