Simple Probability a. The probability of rolling a 6 is one out of 6 or 1/6. 0.1666666667 b. The probability of rolling the number 6 twice in 2 rolls using only one die is 1/6 * 1/6 0.0277777778 c. If the probability of seeing a male house finch at a birdfeeder is .45 and the probability of seeing a female house finch is .35; what is the probability of seeing, first a male house finch. then a female house finch, and finally another male house finch? 0.070875 a. There are only 2 types of Pokemon: rare and common. If the probability of finding a rare Pokemon at a Pokestop is .05, what is the probability that the first 8 Pokemon that you find are common, and then you find one rare Pokemon at a Pokestop that has a lure attached to it. Assume that you will find at least 9 Pokemon in the 30 minutes that the lure is attached. 0.0331710216 4. An apple juice bottling company maintains records concerning the number of unacceptable bottles of juice obtained from the filling and capping machines. Based on past data, the probability that a bottle came from machine I and was nonconforming is 0.05 and the probability that a bottle came from machine II and was nonconforming is 0.075. These probabilities represent the probability of one bottle out of the total sample having the specified characteristics. Half the bottles are filled on machine I and the other half are filled on machine II. A. If a filled bottle is selected at random, what is the probability that it is a nonconforming bottle? Addition rule for mutually exclusive events Either the bottles came from machine I or they came from machine II. Therefore, the events are mutually exclusive. The probability of A or B is equal to the probability of A plus the probability of B. Probability that a bottle is nonconforming = 0.05+ 0.075 = Machine I Machine II Conforming 900 850 Nonconforming 100 150 1000 1000 The probability is 250/2000 = 0.125 0.125 1750 250 2000 that a bottle is nonconforming. B. If a filled bottle is selected at random, what is the probability that it was filled on machine II? Simple probability: Since 50% of the bottles are filled on machine II, then the probability is 50%. C. If a filled bottle is selected at random, what is the probability that it was filled on machine I and is a conforming bottle? Conforming Nonconforming Machine I Machine II 900 850 100 150 1000 1000 The probability is 900/2000 and is conforming. 1750 250 2000 0.45 that a bottle was filled on machine I The table below contains probabilities of each event. The green cells contain simple probabilities. The red cells contain joint probabilities. Conforming Nonconforming Machine I Machine II 0.45 0.425 0.05 0.075 0.875 0.125 0.5 0.5 1 Suppose that there is a lottery where you select 3 different numbers betw A number cannot be selected twice so you can select 123 or 012 but not 112 a. How many different three digit numbers can you choose from? 720 b. If you purchase one ticket, what is the chance that you will win? 0.0013888889 c. If you purchase 3 tickets, what is the likelihood that you will lose? 0.99583 different numbers between 0 and 9. 123 or 012 but not 112 or 100. 1. According to Investment Digest ("Diversification and the Risk/Reward Relationship", Winter 1994, 1-3), the mean of the annual return for common stocks from 1926 to 1992 was 15.4%, and the standard deviation of the annual return was 21.5%. During the same 67-year time span, the mean of the annual return for long-term government bonds was 5.5%, and the standard deviation was 7.0%. The article claims that the distributions of annual returns for both common stocks and long-term government bonds are bell-shaped and approximately symmetric. Assume that these distributions are distributed as normal random variables with the means and standard deviations given previously. Transformational Equation is Z = ( X - mean ) / standard deviation Note: The Pink cells contain the answers. Part A Using the transformation formula to compute the Z value: (0%-15.4%) / 21.5% Z-value: -0.72 Now, we go to the Z table and find the value corresponding to Z=.72. The value that we find is 0.2642 This is the area between the mean (Z=0) and the actual value that we are looking for. Always draw the probability graph. This helps us find the area we are interested in. In our case in part a, we are asked for the probability that the returns are more than 0%. We know that 0% is left of the mean (mean is 15.4%) so we want the area between 0 and the mean + the total area on the right of the mean. The area on the right of the mean is 50%. To calculate the actual probability, we need to add 0.2642 to .5 or 26.4% to 50% 76.40% Remember, in every normal distribution mean=median=mode Remember, in every normal distribution mean=median=mode Part B We will solve part B in a similar way. First we calculate the transformation equation: (.20 - .154) / .215 This gives us .213 We lookup this value in the Z table and find it at .0832 So, we know that the area between the mean (15.4) and the value that we are looking for is .0832. We need the total value to the left of 20, so we need to add .5 which is the probability that the returns are under the mean. .50 + .0832 = .5832 58.32% Note that the information on mean returns and standard deviation of government bonds is given in the question but is not needed to compute the answer. Isn't it like decisions we make every day at our job? MGMT 650 Fall 2016 Problem Set 2 1. Simple Probability: Compute these probabilities: a. The probability of rolling the number 6 using one die. b. The probability of rolling the number 6 twice in 2 rolls using only one die. c. If the probability of seeing a male house finch at a birdfeeder is .45 and the probability of seeing a female house finch is .35; what is the probability of seeing, first a male house finch, then a female house finch, and finally another male house finch? d. There are only 2 type of Pokemon: rare and common. If the probability of finding a rare Pokemon at a Pokestop is .05, what is the probability that the first 8 Pokemon that you find are common, and then you find one rare Pokemon at a Pokestop that has a lure attached to it. Assume that you will find at least 9 Pokemon in the 30 minutes that the lure is attached. 2. Joint Probability: An apple juice bottling company maintains records concerning the number of unacceptable bottles of juice obtained from the filling and capping machines. Based on past data, the probability that a bottle came from machine I and was nonconforming is 0.05 and the probability that a bottle came from machine II and was nonconforming is 0.075. These probabilities represent the probability of one bottle out of the total sample having the specified characteristics. Half the bottles are filled on machine I and the other half are filled on machine II. Compute: a. If a filled bottle is selected at random, what is the probability that it is a nonconforming bottle? b. If a filled bottle is selected at random, what is the probability that it was filled on machine II? c. If a filled bottle is selected at random, what is the probability that it was filled on machine I and is a conforming bottle? 3. Lottery Probability: Suppose that there is a lottery where you select 3 different numbers between 0 and 9. A digit cannot be selected twice so you can select 123 but not 112. a. How many different three digit numbers are possible? b. If you buy one ticket, what is the likelihood you will win? c. If you buy 3 tickets, what is the likelihood you will lose? 4. Probability using the z test: According to Investment Digest ("Diversification and the Risk/Reward Relationship", Winter 1994, 1-3), the mean of the annual return for common stocks from 1926 to 1992 was 15.4%, and the standard deviation of the annual return was 21.5%. During the same 67-year time span, the mean of the annual return for long-term government bonds was 5.5%, and the standard deviation was 7.0%. The article claims that the distributions of annual returns for both common stocks and long-term government bonds are bell-shaped and approximately symmetric. Assume that these distributions are distributed as normal random variables with the means and standard deviations given previously. Here is a link to the z table: http://www.statsoft.com/textbook/sttable.html a. What is the probability that the stock returns are greater than 0%? b. What is the probability that the stock returns are less than 20%? MGMT 650 Fall 2016 Problem Set 3 (Last updated 9/19/2016) 1. Confidence Interval: Compute a 95% confidence interval for the population mean, based on the sample numbers 21, 22, 33, 34, 25, 26, and 139. Change the last value to 29 and re-compute the confidence interval. What is an outlier and how does it affect the confidence interval? 2. Confidence Interval: A sample from a normal population has size = 109 observations, mean = 7.99, and standard deviation = 3.67. What is the 95 percent confidence interval for the population mean? Use the Excel function, =CONFIDENCE.NORM() and the mean to find the two limits. 3. Confidence Limit: The director of admissions at the University of Maryland, University College is concerned about the high cost of textbooks for the students each semester. A sample of 25 students enrolled in the university indicates that x(bar) = $315.40 and s = $43.20. a. Using the .10 level of significance, is there evidence that the population mean is above $300? b. What is your answer if x(bar) = $315.40, s = $75.00, and the level of significance is 0.05? c. What is your answer if x(bar) = $305.11, s = $43.20, and the level of significance is 0.10? d. D. Based on the information in part a, what decision should the director make about the books used for the courses if the goal is to keep the cost below $300? 4. t-Test: Paired Two Sample for Means In week 7 we will study Hypothesis Testing in detail. For answering the following question, you need a brief idea of what hypothesis testing is. Please read the following introduction: http://www.socialresearchmethods.net/kb/hypothes.php The director for Weight Watchers International wants to determine if the changes in their program results in better weight loss. She selected 25 Weight Watcher members at random and compared their weight 6 months later to weight at the start of the program. Here are the results: (The weight in the column labeled \"After\" represents their weights six months later and \"Before\" represents their weight at the start of the sixmonth period.) The director used .05 as the significance level. Use Excel to test H0: After - Before 0 vs. HA: After - Before < 0. For each paired difference, compute After - Before. In Data Analysis, tTest: Paired Two Sample for means, select the After data for Variable 1 Range. Note that the critical value output by Data Analysis for this test is always positive. In this problem, the sign of the critical value is negative corresponding to a 1tailed test with lower reject region and negative lower critical value. State your conclusion. Perso n 1 2 Befor e 176 192 After 164 191 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 185 177 196 178 196 181 158 201 191 193 176 212 177 183 210 198 157 213 161 177 210 192 178 176 176 185 169 196 172 158 193 185 189 175 210 173 180 204 192 152 200 161 166 203 186 170