Suppose a box contains tickets, each labeled by an integer. Let X,Y , and Z be the results of draws at random with replacement from the box. Show that, no matter what the distribution of numbers in the box, a) P(X + Y is even) ≥1/2; b) P(X + Y + Z is a multiple of 3) ≥1/4.
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Suppose a box contains tickets, each labeled by an integer. Let X,Y , and Z be the results of draws at random with replacement from the box. Show that, no matter what the distribution of numbers in the box,
a) P(X + Y is even) ≥1/2;
b) P(X + Y + Z is a multiple of 3) ≥1/4.
(a) We can obtain an even number if both ticket numbers are even or both are odd
Let
probability of even number of tickets
Probability of odd number of tickles
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- Ex 1: A production facility employs 20 workers on the day shift, 15 workers on the swing shift, and 10 workers on the graveyard shift. A quality control consultant is to select 6 of these workers for in-depth interviews. Suppose the selection is made in such a way that any particular group of 6 workers has the same chance of being selected as does any other group (drawing 6 slips without replacement from among 45). a) How many selections result in all 6 workers coming from the day shift? b) What is the probability that all 6 selected workers will be from the day shift? c) What is the probability that all 6 selected workers will be from the same shift?Buu has 6 different types (named 1 to 6) of chocolates (in large quantity). Now he plays a game to eat those chocolates. He throws a die (of six sides, non-biased, if the die shows 1, he eats chocolate 1. If the die shows 2, he eats chocolate 2 and so on. Now let X be the number of chocolates of type 1 are eaten and Y be the number of chocolates of type 2 are eaten. What is joint PMF of X and Y ?(Total number of times he roll the die is 4.)A large bakery has many different products for sale. Suppose that 200 come in between 6am and 10am. Of the 200 customers 140 order donuts, 100 order cinnamon rolls, and 80 order both. Make a Venn Diagram to model the situation.
- Suppose X= {1,2,3,4,5,6,7}. Which of the following are partitions of X? A. X1= {1,2,5}, X2= {0,3}, X3= {-4,6,7} B. X1= {1,2,3,4,5,6,7} C. X1= {1,2,5}, X2= {3}, X3= {6,7}, X4= {4} D. X1= {1,2,5}, X2= {3,4,5}, X3= {2,6,7} E. X1= {1,2,5}, X2= {3}, X3= {6,7} F. X1= {1,2,5}, X2= {3,4}, X3= {6,7} G. X1= {1,2,5}, X2= {3,6}, X3= {6,7}, X4= {4,5}Classify each of the following as either quantitative or Qualitative (first blank) and either nominal, ordinal, interval or ratio (second blank): 1)The time an accountant spends at work during tax season. 2) The ordered preference that investors have for different types of investment funds. 3) The country where an individual is born. there are 2 for each one Plzz explainb) The Ekatahuna Widget factory is to replace its old machines, and must choose among the three types now available on the market. One of each type (A1, A2 and A3) is borrowed for testing. The 21 workers are randomly allocated, seven to each new machine. Each worker is timed for assembling 10 widgets using the old type of machine, and x (in seconds) is the average time taken. Then, after practice, the worker is timed for 10 widgets on the new machine, with Y being the average time in seconds. A1 A2 Аз y y y 23 22 21 23 24 25 22 23 24 26 28 25 24 22 24 26 26 26 24 26 28 27 25 28 30 28 26 27 30 29 32 30 28 29 32 29 33 33 198 31 31 34 32 Sum 197 188 178 191 184 Note: Ey, = 22' + 24° +.+ 32' = 15327; E = 23' +.+32' = 15893 and Exy=15572 Conduct the analysis of covariance, and test the significance of the covariate and adjusted treatment means.
- A researcher is interested in understanding if there is a difference in the proportion of undergrad and grad students at UCI who prefer online teaching to in person teaching, at the a = 0.05 level. They take 2 samples, first, a sample of 300 undergrad students. The second, is a sample of 172 grad students. Of the undergrads, 186 said they preferred online lectures, and of the graduate students, 104 said that they prefer online lectures. Let p₁ = the proportion of undergrad students who prefer online class and p2 = the proportion of grad students who prefer online lectures. (a) Set up the null and alternative hypothesis (using mathematical notation/numbers AND interpret them in context of the problem). (b) Calculate the test statistic. (c) Calculate the critical value. (d) Draw a picture of the distribution of the test statistic under Ho. Label and provide values for the critical value and the test statistic, and shade the critical region. (e) Make and justify a statistical decision at…Spam Email Filters. A study by Forbes indicated that the five most common words appearing in spam emails are shipping!, today!, here!, available, and fingertips!. Many spam filters separate spam from ham (email not considered to be spam) through application of Bayes' theorem. Suppose that for one email account, 1 in every 10 messages is spam and the proportions of spam messages that have the five most common words in spam email are given below. shipping! 0.051 today! 0.045 here! 0.034 available 0.014 fingertips! 0.014 Also suppose that the proportions of ham messages that have these words are: shipping! 0.0015 today! 0.0022 here! 0.0022 available 0.0041 fingertips! 0.0011 a. If a message includes the word shipping!, what is the probability the message is spam? If a message includes the word shipping!, what is the probability the message is ham? Should messages that include the word shipping! be flagged as spam? b. If a message includes the word…Which of these settings does not allow use of a matched pairs ?t procedure? 1)You interview 64 female students in their freshman year and again in their senior year and ask each about their ideal number of children. 2)You interview 64 female students and their mothers and ask each about their ideal number of children. 3)You interview both the husband and the wife in 64 married couples and ask each person about their ideal number of children. 4)You interview a sample of 64 unmarried male students and another sample of 64 unmarried female students and ask each about their ideal number of children.
- Buu has 6 different types (named 1 to 6) of chocolates (in large quantity). Now he plays a game to eat those chocolates. He throws a die (of six sides, non-biased, if the die shows 1, he eats chocolate 1. If the die shows 2, he eats chocolate 2 and so on. Now let X be the number of chocolates of type 1 are eaten and Y be the number of chocolates of type 2 are eaten. What is joint PMF of X and Y?only B pleaseA researcher randomly selected 158 personal vehicles and noted the type of vehicle and its color. The two-way table displays the data. A 5-column table with 4 rows. Column 1 has entries sedan, S U V, convertible, total. Column 2 is labeled Red with entries 14, 9, 20, 43. Column 3 is labeled Blue/Green with entries 31, 19, 6, 56. Column 4 is labeled Black/Gray with entries 22, 34, 3, 59. Column 5 is labeled Total with entries 67, 62, 29, 158. The columns are titled Vehicle color and the rows are titled vehicle type. Suppose a vehicle is randomly selected. Let event C = convertible and R = red. What is the value of P(R|C)? StartFraction 29 Over 158 EndFraction StartFraction 43 Over 158 EndFraction StartFraction 20 Over 43 EndFraction StartFraction 20 Over 29 EndFraction
