Text Book: Introduction to Probability Models 11th edition, Sheldon M. Ross: Chapter 5, Question 42(b)

Can you please explain how the expression in b was derived and which theorem (or provide page # of the theorem) the question is referring to? I see the solution but I don't understand it. (like why T1+T2<=1)

Thank you!

You wish to test the following claim (HaHa) at a significance level of α=0.001α=0.001.

      Ho:μ=88.9
      Ha:μ>88.9

You believe the population is normally distributed, but you do not know the standard deviation. You obtain a sample of size n=79 with mean M=91.5 and a standard deviation of SD=16.3

What is the test statistic for this sample? (Report answer accurate to three decimal places.)
test statistic =

What is the p-value for this sample? (Report answer accurate to four decimal places.)
p-value =

The p-value is... (choose one)

• less than (or equal to) α
• greater than α

This test statistic leads to a decision to...(choose one)

• reject the null
• accept the null
• fail to reject the null

As such, the final conclusion is that...(choose one)

• There is sufficient evidence to warrant rejection of the claim that the population mean is greater than 88.9.
• There is not sufficient evidence to warrant rejection of the claim that the population mean is greater than 88.9.
• The sample data support the claim that the population mean is greater than 88.9.
• There is not sufficient sample evidence to support the claim that the population mean is greater than 88.9.

When do you use Chi-squared test with Yates correction versus without Yates correction?

A pediatrician wants to determine the relation that may exist between a​ child's height and head circumference. She randomly selects 5 children and measures their height and head circumference. The data are summarized below. A normal probability plot suggests that the residuals are normally distributed. Complete parts​ (a) and​ (b) below. Height​ (inches), x 27.75 25 27 26 26.5 Head Circumference​ (inches), y 17.6 16.9 17.5 17.3 17.3 ​(a) Use technology to determine s Subscript b 1. s Subscript b 1equals nothing   ​(Round to four decimal places as​ needed.) ​(b) Test whether a linear relation exists between height and head circumference at the alphaequals0.01 level of significance. State the null and alternative hypotheses for this test. Choose the correct answer below. A. Upper H 0​: beta 0equals0 Upper H 1​: beta 0not equals0 B. Upper H 0​: beta 0equals0 Upper H 1​: beta 0greater than0 C. Upper H 0​: beta 1equals0 Upper H 1​: beta 1not equals0 D. Upper H 0​: beta 1equals0 Upper H 1​: beta 1greater than0 Determine the​ P-value for this hypothesis test. ​P-valueequals nothing ​(Round to three decimal places as​ needed.) What is the conclusion that can be​ drawn? A. Reject Upper H 0 and conclude that a linear relation does not exist between a​ child's height and head circumference at the level of significance alphaequals0.01. B. Reject Upper H 0 and conclude that a linear relation exists between a​ child's height and head circumference at the level of significance alphaequals0.01. C. Do not reject Upper H 0 and conclude that a linear relation does not exist between a​ child's height and head circumference at the level of significance alphaequals0.01. D. Do not reject Upper H 0 and conclude that a linear relation exists between a​ child's height and head circumference at the level of significance alphaequals0.01.

In the study, the authors state the following: “Because ventilator-free days and days free of multiple organ dysfunction syndromes are known to have a bimodal distribution, the data were initially analyzed by means of Student’s t-test, with between-group differences presented as means and 95% confidence intervals. A secondary analysis of these outcome measures involving a bootstrapped t-test was also conducted to support the results of the primary analysis…”

Try to Justify the logic of these two sentences.

i. why did they initially use a method based on Student’s t-test and why did they do a secondary analysis with a “bootstrapped t-test”. Try to explain briefly what the latter means.

This is an open question, please feel free to answer this question. If you know something or even just some points, I will grade it as helpful. Thanks!
If you think more detail is needed, please also write a comment!

c. Taking every 20th passenger in your sorted table, create a new table that lists Passenger ID and Rating for the 20 data points you will now have (In excel please) with formula. I am stomped by this.

You measure 33 textbooks' weights, and find they have a mean weight of 53 ounces. Assume the population standard deviation is 9.1 ounces. Based on this, construct a 99% confidence interval for the true population mean textbook weight.

Give your answers as decimals, to two places

< μμ <

Box 1: Enter your answer as an integer or decimal number. Examples: 3, -4, 5.5172
Enter DNE for Does Not Exist, oo for Infinity

Box 2: Enter your answer as an integer or decimal number. Examples: 3, -4, 5.5172
Enter DNE for Does Not Exist, oo for Infinity

Consider the following sequence of numbers:

Test whether the 3rd, 8th, 13th, and so on, numbers in the sequence given above auto correlated. (Use α = 0.05)

5 In a certain northern Wisconsin town (cough...cough...Oliver), there are 100 families. 30 families have 1 child, 50 families have 2 children, and 20 families have 3 children. The children are then ranked according to age: the oldest has rank 1, second oldest rank 2, third oldest rank 3. Suppose you select a random child from the town. Find the probability mass function, expected value, and variance of the childs rank.

Tom is trying to enter the used car business. He knows that Jean-Ralphio will sell him a car that needs repairs. Once repaired, Tom can sell it for $100 more than he spent to purchase it. Further, he knows that each car has an 80% chance of being a good car, and a 20% chance of being a bad car. Good cars only cost $20 to repair, but bad cars cost $200 to repair.

Mona-Lisa decides to sweeten the deal by offering Tom a warranty. Tom can pay her $40 and in return, she will pay half of the repair costs, up to $80. Therefore, Tom’s choices are to buy the car, buy the car and the warranty, or not buy anything and stick to his day job wih the Parks Department.

1. (a) Draw the decision making chart to help Tom. Be sure to include his actions, the states of the car, and his resulting payouts.

2. (b) Based on the Expected Monetary Value criterion, what should Tom do?

3. (c) Now assume that you are uncertain of the probability that the car is good. De- termine how much the probability can change before the optimal option changes under the Expected Monetary Value criterion.

4. (d) Return to the original probabilities of 0.8 and 0.2 from the original problem statement. Assume the repair cost for a good car is unknown. How much could this value change before the optimal option changes under the Expected Monetary Value criterion?

18 teams, 5 of which are professional, take part in a football tournament. At the beginning of the tournament the 18 teams are randomly divided into two groups (A and B) of 9 teams each. Determine the probabilities of the following events:

a) all professional teams end up in one group

b) 2 of the professional teams are in one group and the other 3 professional teams are in the other group

Random samples of employees in fast-food restaurants where the employer provides a training program were drawn. Of a sample of 67 employees who had not completed high school, 11 had participated in a training program provided by their current employer. Of an independent random sample of 113 employees who had completed high school but had not attended college, 27 had participated. Test at the 1% significance level the null hypothesis that the participation rates are the same for the two groups against the alternative that the rate is lower for those who have not completed high school.

Conduct the appropriate hypothesis test and report the p-value. Do not round intermediate calculations. Round your answer to four decimal places. Include the leading zero. Format: 0.0000

A total of 8 neutrinos, presumably from Supernova 1987A, as observed in an underground detector located in a salt mine near Cleveland.

(a) If the average number of "background" neutrinos observed per day is know to be 2, calculate the probability that 8 or more such background events will be detected in one day.

(b) If the average number of "background" neutrinos observed per day is know to be 2, calculate the probability that 8 or more such background events will be detected in a 10-minute period.

(c) Based on your answers to parts (a) and (b): Should the experimenters (call them Team A) who observe 8 or more events distributed over a one-day period publish their results as a "discovery," or simply attribute these "events" to a fluctuation in the background rate? If Team B observes 8 or more events within a 10-minute period, is this an important discovery, or likely statistical fluctuation?

Can you please explain how s could e true or false?

It is okay to violate the assumption of independent errors as long as the data are normally distributed.

Thank you! ?

The manufacturer of hardness testing equipment uses​ steel-ball indenters to penetrate metal that is being tested.​ However, the manufacturer thinks it would be better to use a diamond indenter so that all types of metal can be tested. Because of differences between the two types of​ indenters, it is suspected that the two methods will produce different hardness readings. The metal specimens to be tested are large enough so that two indentions can be made.​ Therefore, the manufacturer uses both indenters on each specimen and compares the hardness readings. Construct a​ 95% confidence interval to judge whether the two indenters result in different measurements.

​Note: A normal probability plot and boxplot of the data indicate that the differences are approximately normally distributed with no outliers.

Specimen   Steel ball   Diamond
1   51   53
2   57   55
3   61   63
4   71   74
5   68   69
6   54   55
7   65   68
8   51   51
9   53   56

Construct a​ 95% confidence interval to judge whether the two indenters result in different​ measurements, where the differences are computed as​ 'diamond minus steel​ ball'.

The lower bound is __?__ .

The upper bound is __?__. ​

(Round to the nearest tenth as​ needed.)