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1 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 ! 6! 5 4! 6 5 4! C C = = = = 75 (5 1)!1! (6 )!! 4! 4! 11! ! C = = = = ! 3! 8! ( ) P = = = 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) = = = 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? = 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 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 Independent events: P ( B & R & = = = b. Selecting a white, a blue, and a red ball Independent events: P ( W & B & = W ) = = /0/007

2 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 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 Calculate mean: µ = = = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) σ = = = = = =.6667 round to one decimal position σ =.7 16 σ = = round to one decimal position σ = 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: a. Mean = ( )/1 = 163/1 = Mean = b. Median = (10+11)/ = 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 Data 8.f. Is the number 35 an outliner? Justify your answer: No Q IQR = (1.5)(18.5 7) = = = > 35 6/0/007

3 8.g. Calculate a rough estimate of the standard deviation: Rule of thumb Range s = = = = 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 Weighted average = = = = 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 ) / (total number of values) =.5/6 = 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 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 % = 6 = CVar salary Accountants age is more variable than their salary. = σ % = = 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 a. z = = = 0.93 σ 1.5 µ b. z = = = 0.85 σ 00 c. µ z = = = 1.4 σ 5 Answer: the score of 630 has the highest relative position. 3 6/0/007

4 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) = = 0.9 neither) = = The numbers of endangered species for several groups are listed here. Location Mammals Birds Reptiles Amphibians TOTALS USA Foreign TOTALS 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 51/663 = 561/663 or b) Bird USA) = Bird & USA) / USA) = (78/663) / (164/663) = 78/165 = c) Warm-blooded) = 314/ /663 = 567/663 or 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 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) = ( )/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: = 0 (or subtract the lower value from the mean: = 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 = = = or 93.75% Step 4: Multiply this percentage with the total values to find the number within the given interval: = /0/007

5 b. Step1: Use Chebyshev s theorem to find k (since we know the percentage): 0.80 = 1 1/k 1/k = 1 1/k = /k = = (0.0)k 1/0.0 = k 5 = k k = SQRT(5) k =.4 Step: Upper value: mean +.4(standard dev) = (5) = = 61. Lower Value: mean.4(standard dev) = 50.4(5) = = 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 = 15 Divide by standard deviation: 15/5 = % of values will lie between 35 and 65. Multiply this percentage with 500 to find total number of values in interval 35-65: 500 * = 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