A random sample of 100 suspension helmets used by motorcycle riders and automobile race-car drivers was subjected to an impact test, and on 36 of these helmets some damage was observed. Using the traditional method (not plus 4) find a 95% confidence interval for the true proportion of helmets that would sustain damage in this testing. Be sure to assess assumptions and interpret the interval in context.

An article presents results of a survey of adults with diabetes. The average body mass index (BMI) in a sample of 15 men was 30.4, with a standard deviation of 0.6. The average BMI in a sample of 19 women was 31.1 with a standard deviation of 0.2. Assuming BMI is normally distributed, find a 95% confidence bound for the difference in mean BMI between men and women with diabetes. Be sure to interpret the interval in context.

Based on a​ poll, among adults who regret getting​ tattoos, 23​% say that they were too young when they got their tattoos. Assume that ten adults who regret getting tattoos are randomly​ selected, and find the indicated probability. Complete parts​ (a) through​ (d) below.

a. Find the probability that none of the selected adults say that they were too young to get tattoos.

b. Find the probability that exactly one of the selected adults says that he or she was too young to get tattoos.

c. Find the probability that the number of selected adults saying they were too young is 0 or 1.

d. If we randomly select ten ​adults, is 1 a significantly low number who say that they were too young to get​ tattoos?

▼No, Yes,

because the probability that

▼at least 1, more than 1, exactly 1, less than 1, at most 1

of the selected adults say that they were too young is

▼greater than less than or equal to

0.05.

Exhibit 2:

Current statistics show that, about 5% of the patients who are infected with the Novel Coronavirus are in serious or critical condition, and need ventilators and oxygen facilities. Suppose this is the population proportion. (The questions related to this Exhibit are designed so that you can see how statistical analyses can be used to fight against pandemics.)

1. Refer to Exhibit 2. Assume that in the City of Gotham, in the first week of outbreak, 256 citizens are tested positive for COVID-19. At least how many ventilators should be prepared to meet the possible demand. (Round up to the nearest integer that is larger than the result.)

2. Refer to Exhibit 2. Suppose the No. 1 district of the City of Gotham has 8100 residents living in it. If you were the head of the health department of the No. 1 district. To cope with the possible all-infection outbreak of COVID-19 in your district, you prepared 600 ventilators. Assuming in the worst case scenario, what is the probability that your prepared medical equipments are overwhelmed by the serious conditioned patients who are in need of the ventilators? (Round up to nearest four decimal place.)

3. Refer to Exhibit 2. Suppose the No. 1 district of the City of Gotham has 8100 residents living in it. If you were the head of the health department of the No. 1 district. To cope with the possible all-infection outbreak of COVID-19 in your district, you prepared 600 ventilators. Assuming in the worst case scenario, what is the probability that your prepared medical equipments are overwhelmed by the serious conditioned patients who are in need of the ventilators? (Round up to nearest four decimal place.)

Let X be the random variable for the sum obtained by rolling two fair dice.

(1) What is the probability density function?

(2) What is the cumulative probability density function?

(3) What is the expected value?

(4) What is the variance?

A Rollercoaster’s auditors estimate that the average daily loss from those illegally riding without tickets is at least $200, but wants to determine the accuracy of this statistic. The company researcher takes a random sample of losses over 64 days and finds that X = $198 and s = $15. Test at α = 0.01.

MODEL

X Y Z

NUMBER OF DEFECTIVE CARS SOLD 50 100 350

TOTAL NUMBER OF CARS SOLD 150 250 600

Suppose that we randomly select 2 different (First and Second) consumers each of whom purchased a new MERCEDES car in 2020. Given this experiment answer all of the following 10 questions.

Q1) What is the probability of the first consumer’s car to be MODEL X?

Q2)What is the probability of the first consumer’s car to be either MODEL Y or MODEL Z?

Q3)What is the probability of the second consumer’s car to be either MODEL X or MODEL Z?

Q4) What is the probability of the first consumer’s car to be either DEFECTIVE or MODEL Y?

Q5) What is the probability of the second consumer’s car to be either NON-DEFECTIVE or MODEL Z?

Q6)If the second consumer’s car is MODEL Y, what is the probability that İt is NON-DEFECTIVE?

Q7) If the first consumer’s car is NON-DEFECTIVE what is the probability that it is MODEL Z?

Q8) What is the probability of the cars of both of these 2 consumers to be DEFECTIVE?

Q9)If the car of the first consumer is MODEL Z what is the probability of the car of the second consumer to be MODEL X?

Q10) If the car of the second consumer is DEFECTIVE, what is the probability of the car of the first consumer to be MODEL Y?

Recall in our discussion of the binomial distribution the research study that examined schoolchildren developing nausea and vomiting following holiday parties. The intent of this study was to calculate probabilities corresponding to a specified number of children becoming sick out of a given sample size. Recall also that the probability, i.e. the binomial parameter "p" defined as the probability of "success" for any individual, of a randomly selected schoolchild becoming sick was given.

Suppose you are now in a different reality, in which this binomial probability parameter p is now unknown to you but you are still interested in carrying out the original study described above, though you must first estimate p with a certain level of confidence. Furthermore, you would also like to collect data from adults to examine the difference between the proportion with nausea and vomiting following holiday parties of schoolchildren and adults, which will reflect any possible age differences in becoming sick. You obtain research funding to randomly sample 49 schoolchildren and 42 adults with an inclusion criterion that a given participant must have recently attended a holiday party, and conduct a medical evaluation by a certified pediatrician and general practitioner for the schoolchildren and adults, respectively. After anxiously awaiting your medical colleagues to complete their medical assessments, they email you data contained in the following tables.

What is the estimated 95% confidence interval (CI) of the difference in proportions between schoolchildren and adults developing nausea and vomiting following holiday parties? Assign groups 1 and 2 to be schoolchildren and adults, respectively.

Please note the following: 1) in practice, you as the analyst decide how to assign groups 1 and 2 and subsequently interpret the results appropriately in the context of the data, though for the purposes of this exercise the groups are assigned for you; 2) 0 and 1 are defined as no and yes, respectively, which is a typical coding scheme in Public Health; 3) you might calculate a CI that is different from any of the multiple choice options listed below due to rounding differences, therefore select the closest match; and 4) you may copy and paste the data into Excel to facilitate analysis.

Select one:

a. -0.1543 to 0.2244

b. -0.1365 to 0.2522

c. -0.1208 to 0.2837

d. -0.1529 to 0.2900

Suppose x has a distribution with μ = 12 and σ = 5.

(a) If a random sample of size n = 31 is drawn, find μ_x, σ_x and P(12 ≤ x ≤ 14). (Round σ_x to two decimal places and the probability to four decimal places.)

(b) If a random sample of size n = 67 is drawn, find μ_x, σ_x and P(12 ≤ x ≤ 14). (Round σ_x to two decimal places and the probability to four decimal places.)

(c) Why should you expect the probability of part (b) to be higher than that of part (a)? (Hint: Consider the standard deviations in parts (a) and (b).)
The standard deviation of part (b) is  ---Select--- smaller than the same as larger than part (a) because of the  ---Select--- smaller larger same sample size. Therefore, the distribution about μ_x is  ---Select--- the same wider narrower .

Given a data set with 100 observations, a goodness of fit test to see if a sample follows a uniform distribution or a poisson distribution or a normal distribution will have the same number of degrees of freedom. true or false

and

When a contingency table of expected frequencies is constructed, the null hypothesis is that all of the cells in the table are equally likely. true or false

thank you :)

Question 2

In a study of the relationship between X = mean daily temperature for the month and Y = monthly charges on the electrical bill the following data was gathered:

We are interested in fitting the following simple linear regression model: Y = Xβ + ε

a) Calculate X′X, (X′X)^-1 and X′Y and then calculate the least squares estimates of β₀ and β_1.

b)   Calculate the variance-covariance matrix of b, and use it to perform a t-test to test the null hypothesis that β₁ = 0. Use α = 0.05.

Use the matrix approach to calculate the following:

c)   Calculate a 95% confidence interval for the mean of Y when X = 20.

d)   Calculate a 95% prediction interval for an individual new value of Y when X = 20.

4.11 Consider the hardness testing experiment described in Section 4.1. Suppose that the experiment was conducted as described and that the following Rockwell C‐scale data (coded by subtracting 40 units) obtained:

1. Analyze the data from this experiment.
2. Use the Fisher LSD method to make comparisons among the four tips to determine specifically which tips differ in mean hardness readings.
3. Analyze the residuals from this experiment.

Problem 2 The following observations are stopping distances (ft) of a bus at 25 mph, where the population stopping distance is normally distributed

32.1

30.6

31.4

30.4

31.0

31.9

a) Does the data suggest that true average stopping distance exceeds 30’ with α = .01?

b) In order to minimize Type I and Type II errors, what sample size would be necessary (α = .01 and β = .10) when μ’ = 31 and σ = .65?

One college class had a total of 80 students. The average score for the class on the last exam was 83.9 with a standard deviation of 5.8. A random sample of 32 students was selected. a. Calculate the standard error of the mean. b. What is the probability that the sample mean will be less than 85​? c. What is the probability that the sample mean will be more than 84​? d. What is the probability that the sample mean will be between 82.5 and 84.5​? a. The standard error of the mean is nothing. ​(Round to two decimal places as​ needed.) b. The probability that the sample mean will be less than 85 is nothing. ​(Round to four decimal places as​ needed.) c. The probability that the sample mean will be more than 84 is nothing. ​(Round to four decimal places as​ needed.) d. The probability that the sample mean will be between 82.5 and 84.5 is nothing. ​(Round to four decimal places as​ needed.)

Researchers are interested in the effect of a certain nutrient on the growth rate of plant seedlings. Using a hydroponics grow procedure that utilized water containing the nutrient, they planted six tomato plants and recorded the heights of each plant 14 days after germination. Those heights, measured in millimeters, were

55.2 ,

59.9 ,

61.7,

62.4 ,

64.2,

and

67.2 .

Using​ technology,

find

the​ 95% confidence interval for the population mean

muμ.

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