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Math 3A Name: Date: Probability and Statistics Test Review Key Facts Expected Value – multiply outcome by its probability and add them all together (expected value is a synonym for mean) Mean ( x ) – average of numbers in data set Deviation ( x - x ) – how far a data value is from the mean Variance ( s 2 ) – average of squared deviations Standard Deviation ( s ) – square root of variance When you add c to each number in the data set: o Add c to the mean o Variance and standard deviation stay the same When you multiply each number in the data set by k: o Multiply the mean and standard deviation by k o Multiply the variance by k2 When a data set includes frequencies, multiply data values by frequencies before averaging (this includes when averaging squared deviations to find variance!) If the results of two spinners (or number cubes, etc.) are added together: o The two original means add to give you the new mean o The two original variances add to give you the new variance If you do n trials of an experiment (with mean x and variance s 2 ) and sum the results: o Mean = x × n , Variance = s 2 × n , Standard Deviation = s 2 × n Practice 1. The spinner shown to the right is spun 2 times and the results are added together. a. Write out the entire sample space. 6 1 4 2 4 b. What is the most likely sum? c. What is the least likely sum? d. What is the mean and standard deviation of this spinner? 3 2. A raffle is conducted and the payouts for the tickets are shown below. a. Calculate the expected value of the payout on a ticket. b. If it costs $4 to buy a ticket, would you want to buy one? Why? For problems 3&4: Calculate the mean, variance, and standard deviation using the tables. 3. x 4 6 8 9 4. x 3 6 7 8 5. Looking at the data sets from #3 and 4, which has the larger spread? How do you know? 6. Calculate the mean, variance, and standard deviation for the data set below using the table. You need to enter the headings yourself! x Frequency 0 2 1 2 3 4 5 3 7 1 8 2 10 2 Total s= s2 = x= 7. A certain data set has the following statistics: x=3 s 2 = 20 s = 4.472 a. Find the mean, variance and standard deviation if you add 6 to each number in the data set. x= s2 = s= b. Find the mean, variance and standard deviation if you multiply each number in the data set by 2. x= s2 = s= 8. Consider two number cubes with different numbers on their faces: Cube 1: 2, 3, 3, 5, 6, 8 Cube 2: 2, 4, 5, 6, 7, 9 Find the mean, variance, and standard deviation of the sum when rolling the two number cubes. 9. Recall the statistics for a regular number cube: x = 3.5 a. s 2 = 2.917 s =1.708 Find the mean, variance, and standard deviation for the sum of 20 number cubes. b. Find the mean, variance, and standard deviation for the sum of 100 number cubes. 10. Recall the statistics for a multiple-choice question with 5 options: x = 0.2 a. s 2 = 0.16 s = 0.4 Find the mean, variance, and standard deviation for the number of questions correct on a 25-question test. b. Find the mean, variance, and standard deviation for the number of questions correct on a 50-question test. Answers 1. a. 2,3,3,4,4,4,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,10,10,10,10,12 b. 5,7,and 8 are all equally likely since they all occur 6 times in the sample space c. either 2 or 12 d. x = 6.6 s » 2.26 2. Expected value is the same as the mean. The mean is $1.75 b. No, you only win $1.75 on average, so you lose $2.25 if it costs you $4 for the ticket 3. x = 274 = 6.75 s 2 = 14.75 s = 3.6875 =1.920 4 = 3.6875 4. x = 244 = 6 s 2 = 144 = 3.5 s = 3.5 =1.871 5. #4 has larger spread because it has a higher variance and standard deviation 72 6. x = 16 = 4.5 s 2 = 166 s = 10.375 = 3.221 16 =10.375 7. a. x = 3+ 6 = 9 s 2 = 20 s = 4.472 2 2 b. x = 3× 2 = 6 s = 20 × 2 = 80 s = 4.472× 2 = 8.944 8. Cube 1: x = 4.5 s = 2.062 Cube 2: x = 5.5 s = 2.217 2 Sum: x = 4.5+ 5.5 =10 s = 2.0622 + 2.2172 = 9.167 s = 9.167 = 3.028 2 9. a. x = 3.5× 20 = 70 s = 58.34 = 7.638 s = 2.917× 20 = 58.34 2 b. x = 3.5×100 = 350 s = 291.7 =17.079 s = 2.917×100 = 291.7 2 10. a. x = 0.2 × 25 = 5 s = 4 =2 s = 0.16 × 25 = 4 2 b. x = 0.2 × 50 =10 s = 8 = 2.828 s = 0.16 × 50 = 8