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- Suppose that in a dial-up connection you may be connected through one of three types of channels with varying quality of transmission. In Type I the channel has an error probability of 0.01, in Type Il the channel has an error probability of 0.005, and in Type III the channel has an error probability of 0.001. Suppose we assume that for the service provider used, 20% of the channels are of Type I, 30% of the channels are of Type II and 50% of the channels are of Type II. a. What is the probability of error for an arbitrary connection? (Hint: This is similar to the resistors with the nominal value problem) b. Suppose we observe an error, what is the probability that we were connected via a channel Type I?A data contains 17 predictors with DS and DW as categorical variable. X1, X2, X3, X4, X5, DS, DW, X1*DS, X1*DW, X2* DS, X2*DW, X3*DS, X3*DW, X4*DS, X4*DW, X3-squared, X4- squared and X3-squared *DW. Predictor variable X5 is omitted from the model. Which of the following plots should be considered to check for Linearity, Independence, Normal Distribution, and Equal Variance Condition? Select all that apply: ☐ Scatterplot of residuals (vertical) and fitted values (horizontal) ☐ Scatterplot of residuals (vertical) and X₁ (horizontal) ☐ Scatterplot of residuals (vertical) and X2 (horizontal) Scatterplot of residuals (vertical) and X3 (horizontal) ☐ Scatterplot of residuals (vertical) and X4 (horizontal) ☐ Scatterplot of residuals (vertical) and X5 (horizontal) ☐ Scatterplot of residuals (vertical) and Y (horizontal) ☐ Normal probability plot of the residuals Normal probability plot of YA risk averse individual faces uncertainty with two outcomes: good, bad. Theindividual has income $1260 at good and $840 at bad outcome. The probability of good outcome is 5/7 (so the probability of bad outcome is 1 – 5/7 = 2/7). The individual can buy any nonnegative x units of insurance. Every unit of insurance has price $p and it pays $1 in the event of bad outcome. In this insurance market, the unit price of insurance is known to be p = 2/3. (a)Suppose the individual buys x units of insurance. Determine the individual's net income under good income, net income under bad income and the average net income. Draw these three in a diagram as functions of x. (b) For the individual: (i) compare full insurance with over insurance and (ii) compare full insurance with partial insurance. Then determine best choice of insurance for the individual.
- Information theory is concerned with the transmission of data, usually encoded as a stream of 0s and 1s, over communication channels. Because channels are “noisy,” there is a chance that some 0s sent through the channel are mistakenly received at the other end as 1s, and vice versa. The majority of digits sent, however, are not altered by the channel. Draw a tree diagram that depicts the type of bit sent (either 0 or 1) and the type of bit received at the end of the channel.The operations manager of a company that manufactures car mufflers wants to determine whether there are any differences in the quality of workmanship among the three daily shifts. The manager randomly selects 500 car mufflers and carefully inspects them. Each muffler is either classified as perfect, satisfactory, or defective, and the shift that produced it is also recorded. The two categorical variables of interest are: shift and condition of the muffler produced. The data can be summarized by the two-way cross tabulation table. DO these data provide sufficient evidence at the 5% Significance level to infer that there are differences in quality among the three shifts? Satisfactory Defective Total Shift 1 116 112 230 Shift 2 65 148 Shift 3 27 78 107 Total 208 269Provost Office conducted a study of independence between two binary variables, (X,Y). Variable X indicates a student's major (MATH or Computer Science) and Y characterizes individual performance (Above or Below Average). A group of n = 600 respondents has been summarized in the contingency table below. Observed Frequencies Y = ABOVE Y = BELOW Row Sum X = MATH 140 60 X = COMP Science 310 90 Column Sum 600 Expected Counts Y = ABOVEY = BELOW Row Sum X = MATH X = COMP Science Column Sum 600 Test independence of (X, Y) at the significance level a = 0.05 • ESTIMATE EXPECTED COUNTS under the null hypothesis of homogeneity / independence between X and Y. Then place them into the table provided above. • Evaluate the TEST STATISTIC for this study • Specify CRITICAL VALUE (or VALUES) required for hypothesis testing procedure. • Formulate REJECTION RULE and then state your decision about independence. 7
- Consider the following simple linear regression. iid Y = Po + B₁X + e, e~ N(0,02), i 1,..., n.The following table was generated from the sample data of 10 junior high students regarding the average number of hours they are unsupervised per nigh, the average number of hours they play video games per night, and their final grades in their math class. The dependent variable is the final grade, the first independent variable (x1) is the number of hours unsupervised each night, and the second independent variable (x2) is the number of hours of video games each night. Coefficients Standard Error t-Stat p-value Intercept 50.375000 7.719473 6.525704 0.000326 Hours Unsupervised 6.682500 1.484194 4.502444 0.002790 Hours Playing Video Games 1.535000 1.145568 1.339947 0.222124 Step 1: Write the multiple regression equation for the computer output given. y=_+_x1+_x2 Step 2: Indicate if any of the independent variables could be eliminated at the 0.05 level of significance. A) x1 B) x2 C) Keep all variablesA researcher is interested in examining whether the location that a person lives is related to the number of hours that they spend on the internet each week. The researcher collected data from a sample of 30 participants who were classified in one of three groups: (1) 10 people who live in an urban setting, (2) 10 people who live in a suburban setting, and (3) 10 people who live in a rural setting. Each participant reported the number of hours they spend on the internet in a typical week (the dependent variable). The researcher found the following descriptive statistics: Urban participants reported an average of 8.9 hours of internet use per week with a standard deviation of 2.77. Suburban participants reported an average of 12.7 hours of internet use per week with a standard deviation of 4.88. Rural participants reported an average of 9.8 hours of internet use per week with a standard deviation of 2.82. Using the data that was collected, the researcher found the following: The…
- Suppose that a new Internet company Mumble.com requires all employees to take a drug test. Mumble.com can afford only the inexpensive drug test—the onewith a 5% false-positive rate and a 10% false-negativerate. (That means that 5% of those who are not usingdrugs will incorrectly test positive and that 10% of thosewho are actually using drugs will test negative.) Supposethat 10% of those who work for Mumble.com are usingthe drugs for which Mumble is checking. (Hint: It maybe helpful to draw a tree diagram to answer the questionsthat follow.)a. If one employee is chosen at random, what is theprobability that the employee both uses drugs and testspositive?b. If one employee is chosen at random, what is theprobability that the employee does not use drugs buttests positive anyway?c. If one employee is chosen at random, what is the probability that the employee tests positive?d. If we know that a randomly chosen employee has testedpositive, what is the probability that he or she usesdrugs?A researcher is interested in examining whether the location that a person lives is related to the number of hours that they spend on the internet each week. The researcher collected data from a sample of 30 participants who were classified in one of three groups: (1) 10 people who live in an urban setting, (2) 10 people who live in a suburban setting, and (3) 10 people who live in a rural setting. Each participant reported the number of hours they spend on the internet in a typical week (the dependent variable). The researcher found the following descriptive statistics: Urban participants reported an average of 8.9 hours of internet use per week with a standard deviation of 2.77. Suburban participants reported an average of 12.7 hours of internet use per week with a standard deviation of 4.88. Rural participants reported an average of 9.8 hours of internet use per week with a standard deviation of 2.82. Using the data that was collected, the researcher found the following: The…Q1. An engineering statistics class has 265 students; 60% are electrical engineering majors, 10% are industrial engineering majors, and 30% are mechanical engineering majors. A sample of four students is selected randomly without replacement for a project team. Let X and Y denote the number of industrial engineering and mechanical engineering majors, respectively (Hint: range for both variables is 0..4). Determine the following: (a) fXY (x, y) (b) fX (x ) (c) E (X ) (d) fY|3 (y) (e) E (Y|X = 3) (f) V (Y|X = 3) (g) Are X and Y independent? Why?