Derive a general expression for the average failure rate of a Weibull distribution between two point in time, t1 and t2.

Question 1

A camera is located 3km away from the entrance gate of a port to count the number of incoming trucks. Trucks that have passed the camera but have not passed the gate are considered to be in the queueing system, in which some trucks are waiting to be serviced by the gate and the other trucks are being serviced by the gate. If truck A passes the camera before truck B, it will pass the gate no later than truck B (first-come-first serve). If truck A passes the camera later than truck B, it will pass the gate no earlier than truck B. Suppose the port does not work at mid-night, and the gate opens at 7:00am. The cumulative numbers of trucks that have passed the camera and the gate at different times of a day are shown in the table below. For brevity, assume trucks pass the camera and the gate only at 6:00, 6:10, 6:20, etc. That is, suppose that in the table below 5 trucks passed the camera exactly at 6:10, and 10 trucks pass the gate exactly at 7:10.

Table Q1-1. Cumulative numbers of trucks that have passed the camera and the gate

Q1.1: Calculate the average flow rate per hour of the system for the three hours from 6:00am to 8:59:59am.

Q1.2: (You can use software tools, e.g., Microsoft Excel, to help answer this question.) Calculate the average flow time of all flow units.

Q1.3: (You can use software tools, e.g., Microsoft Excel, to help answer this question.) Calculate the average inventory of the system from 6:00am to 8:59:59am, that is, the average number of trucks in the system at 6:00:01, 6:10:01, 6:20:01, 6:30:01, 6:40:01, … 8:40:01, 8:50:01.

The following table shows the beta carotene and chlorophyll contents of curry (spice) leaves under different postharvest treatments.

Were there significant differences in carotene and chlorophyll contents under the different postharvest treatments?                                                     

Authors in the New England Journal of Medicine investigated a possible link between Zika virus and microcephaly (or other fetus developmental abnormalities) and researchers collected the following data from Rio de Janeiro state in Brazil (data from March 2016):

Test for independence between maternal Zika virus infection and a baby’s microcephaly. Report here the test you are using, the p-value you get, and the conclusion you come to

The following table shows the beta carotene and chlorophyll contents of curry (spice)

leaves under different postharvest treatments.

Were there significant differences in carotene and chlorophyll contents under the

different post harvest treatments?                   

Three machines M1, M2 and M3 produce nominally identical items. The evidence of the engineers past experience is that 5 % of the output from machine M1 is faulty, 3.5 % of the output from machine M2 is faulty and 2.5 % of the output from machine M3 is faulty. On a given day, M1 has produced 15 % of the total output, M2 has produced 30 % and M3 the remainder.

An item selected at random is found to be faulty.

1. Use Bayes' theorem to find the probability that it was produced by M1.
2. Use Bayes' theorem to find the probability that it was produced by M2.
3. Use Bayes' theorem to find the probability that it was produced by M3.

A researcher believes that there is a positive relationship between the time spent in studying and the score a student gets at the end of the semester. She selected a sample of students and recorded the following data:

After analyzing the data using Microsoft Excel, the researcher got the outputs below:

Answer the following (Please, use at least 2 decimals when you type numbers ):

a.    The correlation coefficient (r) = Answer

b.    The relationship between the time spent in studying and the score a student gets is: Answerweak and positivestrong and negativeweak and negativestrong and positive

c.    The y-intercept equals: Answer

d.    The regression (time spent) coefficient equals = Answer

e.    The regression model will be: AnswerScore = 70.704 + 0.123 MinuteMinute = 0.123 + 70.704 ScoreScore = 0.123 + 70.704 MinuteMinute = 70.704 + 0.123 Score

f.     From the regression model, the researcher can predict that if a student spends 2.5 hours, the score he gets will be: Answer

g.    From the regression model, if a student did not study at all, his score would be expected to be: Answer

h.    The coefficient of determination for this model is:  Answer95708890%

Consider the following Markov chain:

1. What is the probability that the first passage time from 2 to 1 is 3?
2. What is the expected first passage time from 2 to 1?
3. What is the expected first passage time from 2 to 2 (recurrence time for 2)? What is the relation between this expectation and the steady-state probability of being at state 2?

Calculate the​ 95% prediction intervals for the four different investments included in the following table. Small Stocks ​S&P 500 Corporate Bonds ​T-Bills Average Return 18.37​% 11.84​% 6.47​% 3.46​% Standard Deviation of returns 38.79​% 20.01​% 6.98​% 3.14​% A.The​ 95% prediction interval of small stocks is between ?​% and ?​%. (Round to two decimal places and put the lower number​ first.) B. The​ 95% prediction interval of the​ S&P500 is between ?​% and ?%.(Round to two decimal places and put the lower number​ first.) C. The​ 95% prediction interval of corporate bonds is between ?​% and ?%. (Round to two decimal places and put the lower number​ first.) D. The​ 95% prediction interval of​ T-bills is between ?% and ?% (Round to two decimal places and put the lower number​ first.)

A producer of soap dispensers claims its machine dispense liquid with a standard deviation of 3.1 ounces. A random sample of 20 machines found the standard deviation to be 4.6 ounces. Is there enough evidence, at the α = 0.05 level of significance, to suggest the standard deviation is not 3.1 ounces?

A manufacturer of 10cm marbles claims the standard deviation of it’s marble's diameter is less than 0.21cm. A random sample of 12 marbles found the standard deviation of the diameter to be 0.14cm. Is there enough evidence, at the α = 0.01 level of significance, to back up the manufacturers claim?

A powerpack is manufactured with a mean lifetime of 4 years and a standard deviation of 0.6 years. What percent of the powerpacks will have a lifetime greater than 4.1 years?

If a company makes LED lights with a mean lifetime of 2000 hours and a standard deviation of 120 hours. What percent of the lights have a lifetime between 1840 and 2100 hours?

Discrete Math: 7) How many 4-element DNA sequences..

a) Do not contain the base T?

b) Contain the sequence ACG?

c) Contain all four bases A, T, C, and G?

d) Contain exactly three of the four bases?

Please show all work

A bottling company uses a filling machine to fill plastic bottles with a popular cola. The bottles are supposed to contain 300 milliliters (ml). In fact, the contents vary according to a normal distribution with mean µ=303 ml and standard deviation σ= 3 ml

What is the probability that an individual bottle contains less than 300 ml?

What is the probability that the sample mean of 80 bottles is less than 300 ml?

The results indicated no difference between the means when the statstics-only condition was compared to the combined statistics+victim condition. Compute the t_LSD value here.

donations differed significantly by request type, F(2, 55) = 3.35, p < .05,

η² = .109, MSE. 2.98. In planned comparisons based on Small et al.’s

findings (2007, Study 3), donations were higher when a personalized victim

appeared alone (M = $3.18, n =17) than when statistics were added to the

victim request (M = $2.00, n = 22, p = .04) or when statistics were presented

alone (M = $1.79, n = 19, p = .02). The latter two means did not differ.

Five hundred consumers were surveyed about a new brand of snack food, Crunchicles. Their age groups and preferences are given in the table.

One consumer from the survey is selected at random. Use reduced fractions for your responses to each of the following questions.

What is the probability that the consumer is 18–24 years of age, given that he/she dislikes Crunchicles?



What is the probability that the selected consumer dislikes Crunchicles?



What is the probability that the selected consumer is 35–55 years old or likes Crunchicles?



If the selected consumer is 70 years old, what is the probability that he/she likes Crunchicles?

what are the strength and weaknesses of formal theory and statistical analysis? how can these methods be combined with qualitative methods?
(word limit:700 words)