1. On a lunch counter, there are 5 oranges and 6 apples. If 3 pieces of fruit are selected, find the probability that 1 orange and apples are selected. Order does not matter Combinations: 5C1 (1 ) 6C P orange and apples = C 5 1 6 11 3 5! 6! 5 4! 6 5 4! 5 6 5 C C = = = = 75 (5 1)!1! (6 )!! 4! 4! 11! 11 10 9 8! 11 10 9 C = = = = 165 11 3! 3! 8! 3 3 11 3 ( ) 75 5 5 P = = = 165 55 11. Place the word True or False in front of each statement (justify your answer):.1 TRUE In a combination, the arrangement ABC is the same as BCA. FALSE If the events A and B are independent, then A and = A) + Ans: A and = A).3 FALSE If A) = 0.6 and A and = 0.3, then B A) = 0.9 A and.3 Ans: P ( B A) = = = 0.5 0. 9 A).6.4 FALSE If an event is certain to occur, its probability E) = 0. Ans: E)=1.5 FALSE If E is an event, then E) + E) > 1 Ans: E) + E) = 1 3. When 7 dice are rolled, the sample space consists of how many events? 7 6 6 6 6 6 6 6 = 6 4. An inspector must select 3 tests to perform in a certain order on a manufactured part. He has a choice of 7 tests. How many ways can he perform the 3 different tests? Order is important Permutation: 7! 7 6 5 4! 7 P3 = = = 10 (7 3)! 4! 5. An urn contains 5 red balls, 5 blue balls, and 6 white balls. A ball is selected, its color is noted, and then it is placed back into the urn. A nd ball is selected, its color noted, and then placed back into the urn. Finally, a 3rd ball is selected and its color noted. Find the probability of 5.a. Selecting a blue, a red, and a red ball. 5 5 5 Independent events: P ( B & R & = = = 16 16 16 5.b. Selecting a white, a blue, and a red ball. 6 5 5 Independent events: P ( W & B & = W ) = = 16 16 16 15 4096 150 4096 1 6/0/007
6. Compute the mean for the grouped data shown below. Use the empty columns to organize your data. Show all formulas used. Classes Frequency: f Midpoint: m f * m 1-3 1 4-6 3 5 15 7-9 0 8 0 10-1 11 TOTALS 6 39 f 39 = m = = 6.5 n 6 7. Compute the population Variance ( σ ) and Standard Deviation (σ ) for the following population: 1, 0, 3, 1,, 5. Use the definition formula and show all steps and calculations. 1+ 0 + 3+ 1+ + 5 1 Calculate mean: µ = = = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 + 0 + 3 + 1 + + 5 1 + + 1 + 1 + 0 + 3 σ = = = 1+ 4 + 1+ 1+ 0 + 9 16 = = =.6667 round to one decimal position σ =.7 16 σ = = 1.639 round to one decimal position σ = 1.6 6 8. The lengths of service (in years) of the Chief Justice of the Supreme Court are: 7, 1, 5, 35, 8, 10, 15,, 11, 10, 1, 7 Arrange numbers in ascending order: 1 5 7 7 10 10 11 1 15 8 35 8.a. Mean = (1+5+7+7+10+10+11+1+15++8+35)/1 = 163/1 = 13.5833 Mean = 13.6 8.b. Median = (10+11)/ = 10.5 8.c. Mode = 7, 10 (7 occurs times. 10 occurs times) 8.d. Five-number summary (Low, Q1, m, Q3, High): Low = 1, Q1 = (7+7) = 7, m = 10.5, Q3 = (15+)/ = 18.5, High = 35 8.e. Construct a boxplot BoxPlot 1 7 10.5 18.5 35 0 5 10 15 0 5 30 35 40 Data 8.f. Is the number 35 an outliner? Justify your answer: No Q3 + 1.5 IQR = 18.5 + (1.5)(18.5 7) = 18.5 + 1.5 11.5 = 18.5 + 17.5 = 35.75 > 35 6/0/007
8.g. Calculate a rough estimate of the standard deviation: Rule of thumb Range 35 1 34 s = = = = 8.5 4 4 4 9. In a dental survey of third-grade students, the following distribution for the number of cavities was found. Find the average number of cavities using the weighted mean method. Number of Students Number of cavities 1 1 8 5 3 5 4 Weighted average = 1 1 + 8 + 5 3 + 5 4 1 + 8 + 5 + 5 = 1 + 16 + 15 + 0 30 = 63 30 =.1 10. Find the percentile rank for the test score 15 for the scores: 16, 1, 5, 15, 0, 1. Arrange scores in ascending order: 5, 1, 15, 16, 0, 1; P = (number values below 15 + 0.5) / (total number of values) =.5/6 =.41667 Percentile rank = 41.67% 11. What score corresponds to the 75% percentile for the scores in problem 10? 0 c = (np)/100 = (6)(75)/100 = 4.5 Round up to next whole number: c = 5; Count to the 5 th position from the left, which corresponds to the score 0. 1. The average age of the accountants at YZ Corp. is 6 years, with a standard deviation of 6 years; the average salary of the accountants is $31,000, with a standard deviation of $4,000. Compare the variation of age and income. 6 CVar = σ age 100 100 3.1% = 6 = CVar salary Accountants age is more variable than their salary. = σ 4000 100 100 1.9% = 31000 = 13. Which score indicates the highest relative position? a. A score of 3. on a test with mean of 4.6 and σ = 1.5 b. A score of 630 on a test with mean of 800 and σ = 00 c. A score of 43 on a test with mean of 50 and σ = 5 Calculate the z-score for each case and choose the highest value: µ 3. 4.6 a. z = = = 0.93 σ 1.5 µ 630 800 b. z = = = 0.85 σ 00 c. µ 43 50 z = = = 1.4 σ 5 Answer: the score of 630 has the highest relative position. 3 6/0/007
14. The probability that a student owns a car is 0.65, and the probability that a student owns a computer is 0.8. If the probability that a student owns both is 0.55, what is the probability that a given student owns neither a car nor a computer? car or computer) = car) + computer) - both) = 0.65 + 0.8 0.55 = 0.9 neither) = 1 0.9 =.08 15. The numbers of endangered species for several groups are listed here. Location Mammals Birds Reptiles Amphibians TOTALS USA 63 78 14 10 165 Foreign 51 175 64 8 498 TOTALS 314 53 78 18 663 If one endangered species is selected at random, find the probability that it is a) Foreign or a mammal b) It is a bird given that it is found in the USA c) Warm-blooded Add the TOTALS entries to the table a) Foreign or a Mammal) = 498/663 + 314/663 51/663 = 561/663 or 0.846 b) Bird USA) = Bird & USA) / USA) = (78/663) / (164/663) = 78/165 = 0.473 c) Warm-blooded) = 314/663 + 53/663 = 567/663 or 0.855 16. A sales representative who visits customers at home finds that she sells 0, 1,, 3, or 4 items according to the following distribution: Items Sold Frequency 0 8 1 10 3 3 4 1 4 Find the probability that she sells the following: a) Exactly one item: Exactly 1) = 10/4 = 5/1 b) At least one item: At Least 1) = (10+3++1)/4 = 16/4 = /3 17. In a distribution of 00 values, the mean is 50 and the standard deviation is 5. Use Chebyshev s theorem to answer the following questions: a. At least how many values will fall between 30 and 70? b. Find the range of values that at least 80% of the values lie a. Step 1: Subtract the mean from the larger value: 50 30 = 0 (or subtract the lower value from the mean: 50 30 = 0) Step : Divide the difference by the standard deviation to get k: k = 0/5 = 4 Step 3: Use Chebyshev s theorem to find the percentage: 1-1/k = 1 1/4 = 1 1/16 = = 1 0.065 = 0.9375 or 93.75% Step 4: Multiply this percentage with the total values to find the number within the given interval: 00 0.9375 = 187.5 4 6/0/007
b. Step1: Use Chebyshev s theorem to find k (since we know the percentage): 0.80 = 1 1/k 1/k + 0.80 = 1 1/k = 1 0.80 1/k = 0.0 1 = (0.0)k 1/0.0 = k 5 = k k = SQRT(5) k =.4 Step: Upper value: mean +.4(standard dev) = 50 +.4(5) = 50 + 11. = 61. Lower Value: mean.4(standard dev) = 50.4(5) = 50 11. = 38.8 18. In a distribution of 500 values with a Bell-shaped distribution, the mean is 50 and the standard deviation is 5. Use the Empirical rule to answer the following: a. How many values will fall in the interval 35 to 65? b. Find the interval that contains 95% of the value a. 65 50 = 15 Divide by standard deviation: 15/5 = 3 99.7% of values will lie between 35 and 65. Multiply this percentage with 500 to find total number of values in interval 35-65: 500 * 0.997 = 498.5 b. 95% of the values lie within standard deviations of the mean: * 5 = 10 Interval: 50 ± 10 [40, 60] 99.7% 95% 68% 3s s 1s + 1s + s + 3s Empirical Rule for Bell-Shaped Distributions 5 6/0/007