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Normal Distribution Starters

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Normal Distribution Starters

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  1. Binomial ASolnsBinomial BSolnsBinomial CSolnsPoisson ASolnsPoisson BSolnsPoisson CSolnsPoisson DSolnsNormal Dist ASolns (z values)Normal Dist BStarter B Solns AStarter B Solns BNormal Dist CStarter C: Solns 1Starter C: Solns 2Normal Dist DStarter D: Solns 1Starter D: Solns 2Normal Dist EStarter E: Solns 1Starter E: Solns 2Normal Dist FStarter F: Solns 1Starter F: Solns 2Normal Dist GStarter G: Solns 1Starter G: Solns 2Normal Dist HStarter H Solns (inverse z values) Normal Dist IStarter I: Solns 1Starter I: Solns 2Mixed problems ASolnMixed problems BSolnMixed problems CSoln Normal Distribution Starters

  2. 1st Page Binomial A 1) On average one bowl in every 4 has lumpy porridge. If big daddy B has 6 bowls of porridge, find the probability: (a) There are exactly 5 bowls with lumpy porridge. (b) There are at most 1 bowl with lumpy porridge. 2) Goldilocks has nightmares on 4 nights each week on average. a) Find the probability of her having more than 2 nights with nightmares in a week. b) Given that she had less than 5 nightmare nights in a week, find the probability that she had only 1 nightmare night.

  3. 1st Page Binomial A Soln 1) On average one bowl in every 4 has lumpy porridge. If big daddy B has 6 bowls of porridge, find the probability:   (a) There are exactly 5 bowls with lumpy porridge. (b) There are at most 1 bowl with lumpy porridge. 2) Goldilocks has nightmares on 4 nights each week on average.   a) Find the probability of her having more than 2 nights with nightmares in a week. b) Given that she had less than 5 nightmare nights in a week, find the probability that she had only 1 nightmare night.

  4. 1st Page Binomial B 1) It is known that 60% of candidates will achieve in an examination. If 5 people sit the examination find the probability: (a) Exactly 3 candidates will achieve. (b) At least 3 candidates will achieve. 2) A drug is known to be 90% effective when it is used to cure a disease. If 20 people are given the drug then X is binomial with n = 20,  = 0.9. a) Find the mean on the binomial distribution b) Find the standard deviation on the binomial distribution

  5. 1st Page Binomial B Soln 1) It is known that 60% of candidates will achieve in an examination. If 5 people sit the examination find the probability: (a) Exactly 3 candidates will achieve. (b) At least 3 candidates will achieve. 2) A drug is known to be 90% effective when it is used to cure a disease. If 20 people are given the drug then X is binomial with n = 20,  = 0.9. a) Find the mean on the binomial distribution b) Find the standard deviation on the binomial distribution

  6. 1st Page Binomial C 1) List the conditions of the binomial distribution. 2) It is known that 18.5 % of people can turn their eyelids inside out. In a group of 20 people what is the probability that more than 1 person can turn their eyelids inside out?

  7. 1st Page Binomial C Soln 1)  Binomial distribution occurs when: (a) There is a fixed number (n) of trials. (b) The result of any trial can be classified as a “success” or a “failure” (c) The probability of a success ( or p) is constant from trial to trial. (d) Trials are independent. P(X =x) = nCxx(1 - )n-x 2) It is known that 18.5 % of people can turn their eyelids inside out. In a group of 20 people what is the probability that more than 1 person can turn their eyelids inside out?

  8. 1st Page Poisson A 1) List the conditions of the Poisson distribution. 2) In a particular marine reserve there are on average 1.25 crayfish per m2 of seafloor. In a 20m2 area what is the probability that there are 10 crayfish?

  9. 1st Page Poisson A Soln 1) Poisson distribution occurs when: (a) Trials are independent. (b) The events cannot occur simultaneously (c) Events are random and unpredictable (d) The probability of an event occurring is proportional to the interval length (for small intervals) 2) In a particular marine reserve there are on average 1.25 crayfish per m2 of seafloor. In a 20m2 area what is the probability that there are 10 crayfish?

  10. 1st Page Poisson B 1) Goldilocks breaks three chairs per hour at school because she is over weight. What is the probability that she breaks no more than four chairs in an hour. (Assume chair breakages are independent ) 2) Goldilocks has on average five tantrums per hour. What is the probability that she has at least two tantrums in a given fifteen minute interval. (Assume tantrums are independent )

  11. 1st Page Poisson B Soln 1) Goldilocks breaks three chairs per hour at school because she is over weight. What is the probability that she breaks no more than four chairs and an hour. (Assume chair breakages are independent ) 2) Goldilocks has on average five tantrums per hour. What is the probability that she has at least two tantrums in a given fifteen minute interval. (Assume tantrums are independent )

  12. 1st Page Poisson C • 1) Baby bear cries four times a week on average. What is the mean and variance of the number of times he cries and a given day. • The probability that baby bears chair is not broken in a given week is 0.44 • What is the average number of times the chair is broken in a week. • What is the standard deviation of the number of times the chair is broken in a week. • What is the probability that baby bears chair is broken twice in a week.

  13. 1st Page Poisson C Solns • 1) Baby bear cries four times a week on average. What is the mean and variance of the number of times he cries and a given day. • The probability that baby bears chair is not broken in a given week is 0.44 • What is the average number of times the chair is broken in a week. • What is the standard deviation of the number of times the chair is broken in a week. • What is the probability that baby bears chair is broken twice in a week.

  14. 1st Page Poisson D • The bear hunting season is five months long. On average the bear family get angry four times a month in the bear hunting season and three times a month in the off-season. • What is the probability that the bear family get angry twice in the next month? • If the bear family did not get angry last month, what is the probability that it is the bear hunting season?

  15. 1st Page Poisson D Soln • The bear hunting season is five months long. On average the bear family get angry four times a month in the bear hunting season and three times a month in the off-season. • What is the probability that the bear family get angry twice in the next month? • If the bear family did not get angry last month, what is the probability that it is the bear hunting season?

  16. 1st Page 0 z ‘Z’ values Look up these ‘z’ values to find the corresponding probabilities 1) P(0 < z < 1.4) = 2) P(0 < z < 2.04) = 3) P(0 < z < 1.55) = 4) P(0 < z < 2.125) = 5) P(-0.844 < z < 0) = 6) P(-2.44 < z < 2.44) = 7) P(-0.85 < z < 1.646) = 8) P( z < 2.048) = 9) P(1.955 < z < 2.044) = 10) P( z < -2.111) =

  17. 1st Page 0 z ‘Z’ value Solutions Look up these ‘z’ values to find the corresponding probabilities 1) P(0 < z < 1.4) = 2) P(0 < z < 2.04) = 3) P(0 < z < 1.55) = 4) P(0 < z < 2.125) = 5) P(-0.844 < z < 0) = 6) P(-2.44 < z < 2.44) = 7) P(-0.85 < z < 1.646) = 8) P( z < 2.048) = 9) P(1.955 < z < 2.044) = 10) P( z < -2.111) =

  18. 1st Page 240mm 240mm 240mm 240mm 240mm Starter B A salmon farm water tank contains fish with a Mean length of 240mm Calculate the probability of the following (Std dev = 15mm) 1) P(A fish is between 240 and 250mm long) = 2) P(A fish is between 210 and 260mm long) = 3) P(A fish is less than 254mm long) = 4) P(A fish is less than 220mm long) = 5) P(A fish is between 255 and 265mm long) =

  19. 1st Page 240mm 240mm 240mm Starter B Solns 1 A salmon farm water tank contains fish with a Mean length of 240mm Calculate the probability of the following (Std dev = 15mm) 1) P(A fish is between 240 and 250mm long) = 2) P(A fish is between 210 and 260mm long) = 3) P(A fish is less than 254mm long) =

  20. 1st Page 240mm 240mm Starter B Solns 2 A salmon farm water tank contains fish with a Mean length of 240mm Calculate the probability of the following (Std dev = 15mm) 4) P(A fish is less than 220mm long) = 5) P(A fish is between 255 and 265mm long) =

  21. 1st Page 4.8kg Starter C A west coast population of mosquitoes have a Mean weight of 4.8kg Calculate the probability of the following (Std dev = 0.6kg) 1) What is the probability a mosquito is between 4.8kg and 5.8kg? 2) What percentage of mosquitoes are between 4kg and 5kg? 3) Out of a sample of 120 mosquitoes, how many would be over 6kg? 4) What percentage of mosquitoes are between 3kg and 4kg? 5) What percentage of mosquitoes are under 5.5kg?

  22. 1st Page 4.8kg 4.8kg Starter C: Solns 1 A west coast population of mosquitoes have a Mean weight of 4.8kg Calculate the probability of the following (Std dev = 0.6kg) 1) What is the probability a mosquito is between 4.8kg and 5.8kg? 2) What percentage of mosquitoes are between 4kg and 5kg?

  23. 1st Page 4.8kg 4.8kg 4.8kg Starter C: Solns 2 A west coast population of mosquitoes have a Mean weight of 4.8kg Calculate the probability of the following (Std dev = 0.6kg) 3) Out of a sample of 120 mosquitoes, how many would be over 6kg? 4) What percentage of mosquitoes are between 3kg and 4kg? 5) What percentage of mosquitoes are under 5.5kg?

  24. 1st Page 3.6g Starter D A room contains flies with a Mean weight of 3.6g and a Standard Deviation of 0.64kg 1) What is the probability a fly is between 3.6g and 5.8g? 2) What percentage of flies are between 3g and 5g? 3) Out of a sample of 40 flies, how many would be under 4g? 4) What percentage of flies are between 2g and 3g? 5) What percentage of flies are under 2.5g

  25. 1st Page 3.6g 3.6g Starter D: Solns 1 A room contains flies with a Mean weight of 3.6g and a Standard Deviation of 0.64kg 1) What is the probability a fly is between 3.6g and 5.8g? 2) What percentage of flies are between 3g and 5g?

  26. 1st Page 3.6g 3.6g 3.6g Starter D: Solns 2 A room contains flies with a Mean weight of 3.6g and a Standard Deviation of 0.64kg 3) Out of a sample of 40 flies, how many would be under 4g? 4) What percentage of flies are between 2g and 3g? 5) What percentage of flies are under 2.5g

  27. 1st Page 1.25t Starter E The mean weight of a loader scoop of coal is 1.25 tonnes and a standard deviation of 280 kg 1) What percentage of scoops are between 1.3 and 1.5 tonnes? 2) What percentage of scoops are less than 1 tonne? 3) Out of a sample of 500 scoops, how many would be over 1.4 tonnes? 4) What percentage of scoops are more than 1.6 tonnes? 5) What percentage of scoops are between 1 tonne and 2 tonnes

  28. 1st Page 1.25t 1.25t Starter E: Solns 1 The mean weight of a loader scoop of coal is 1.25 tonnes and a standard deviation of 280 kg 1) What percentage of scoops are between 1.3 and 1.5 tonnes? 2) What percentage of scoops are less than 1 tonne?

  29. 1st Page 1.25t 1.25t 1.25t Starter E: Solns 2 The mean weight of a loader scoop of coal is 1.25 tonnes and a standard deviation of 280 kg 3) Out of a sample of 500 scoops, how many would be over 1.4 tonnes? 4) What percentage of scoops are more than 1.6 tonnes? 5) What percentage of scoops are between 1 tonne and 2 tonnes

  30. 1st Page 4.6kg Starter F The weight of glue paste Ralph eats in a day is normally distributed with a mean of 4.6kg & standard deviation of 1.3kg 1) How many days in November will Ralph eat less than 5.5kg of glue paste? 2) What percentage of days does he eat less than 4kg of glue paste? 3) Ralph vomits when he eats more than 6kg of glue in a day. What is the chance of this happening? 4) What percentage of days does he eat between 4.2kg and 5kg of glue? 5) What is the probability he does not eat between 3.5 & 5 kg of glue paste?

  31. 1st Page 4.6kg 4.6kg Starter F: Solns 1 The weight of glue paste Ralph eats in a day is normally distributed with a mean of 4.6kg & standard deviation of 1.3kg 1) How many days in November will Ralph eat less than 5.5kg of glue paste? 2) What percentage of days does he eat less than 4kg of glue paste?

  32. 1st Page 4.6kg 4.6kg 4.6kg Starter F: Solns 2 The weight of glue paste Ralph eats in a day is normally distributed with a mean of 4.6kg & standard deviation of 1.3kg 3) Ralph vomits when he eats more than 6kg of glue in a day. What is the chance of this happening? 4) What percentage of days does he eat between 4.2kg and 5kg of glue? 5) What is the probability he does not eat between 3.5 & 5 kg of glue paste?

  33. 1st Page 10.4kg Starter G Itchy & Scratchy have a hammer collection which is normally distributed with a mean of 10.4 kg & standard deviation of 2.3kg 1) What percentage of the hammers weigh less than 8kg? 2) What is the probability a hammer weighs between 11kg & 14kg? 3) Scratchy’s head splits open if the hammer is more than 15kg. What is the chance of this happening? 4) A truck is loaded with 200 hammers. How many of these would be 12kg or less? 5) 90% of hammers weigh more than what weight?

  34. 1st Page 10.4kg 10.4kg Starter G: Solns 1 Itchy & Scratchy have a hammer collection which is normally distributed with a mean of 10.4 kg & standard deviation of 2.3kg) 1) What percentage of the hammers weigh less than 8kg? 2) What is the probability a hammer weighs between 11kg & 14kg?

  35. 1st Page 10.4kg 10.4kg 10.4kg Starter G: Solns 2 Itchy & Scratchy have a hammer collection which is normally distributed with a mean of 10.4 kg & standard deviation of 2.3kg) 3) Scratchy’s head splits open if the hammer is more than 15kg. What is the chance of this happening? 4) A truck is loaded with 200 hammers. How many of these would be 12kg or less? 5) 90% of hammers weigh more than what weight?

  36. 1st Page 0 z Inverse ‘Z’ values Look up these probabilities to find the corresponding ‘z’ values 1) 2) 0.4 0.3 3) 4) 0.45 0.85 5) 6) 0.65 0.08 7) 8) 0.12 0.02

  37. 1st Page 0 z Inverse ‘Z’ values: Solns Look up these probabilities to find the corresponding ‘z’ values 1) 2) 0.4 0.3 3) 4) 0.45 0.85 5) 6) 0.65 0.08 7) 8) 0.12 0.02

  38. 1st Page 120mL Starter I Kenny is practicing to be in a William Tell play. He suffers some blood loss which is normally distributed with a mean of 120mL & standard deviation of 14mL 1) What is the probability his blood loss is less than 100mL? 2) 80% of the time his blood loss is more then ‘M’ mL. Find the value of ‘M’ 3) Kenny passes out when his blood loss is too much. This happens 5% of the time. What is the maximum amount of blood loss Kenny can sustain? 4) 30% of the time Kenny is not concerned by his blood loss? What is his blood loss when he starts to be concerned? 5) The middle 80% of blood losses are between what two amounts?

  39. 1st Page 120mL 120mL 120mL Starter I: Solns 1 Kenny is practicing to be in a William Tell play. He suffers some blood loss which is normally distributed with a mean of 120mL & standard deviation of 14mL 1) What is the probability his blood loss is less than 100mL? 2) 80% of the time his blood loss is more then ‘M’ mL. Find the value of ‘M’ 3) Kenny passes out when his blood loss is too much. This happens 5% of the time. What is the maximum amount of blood loss Kenny can sustain?

  40. 1st Page 120mL 120mL Starter I: Solns 2 Kenny is practicing to be in a William Tell play. He suffers some blood loss which is normally distributed with a mean of 120mL & standard deviation of 14mL 4) 30% of the time Kenny is not concerned by his blood loss? What is his blood loss when he starts to be concerned? 5) The middle 80% of blood losses are between what two amounts? Lucky Kenny is not involved in the knife catching competition!

  41. 1st Page Mixed Problems A 1) Mean maximum temperature is 26°C Standard deviation = 5°C Temp measured to nearest degree. a) P(Temperature at least 30°C) b) P(Temperature between 20 and 30°C) c) P(Temperature between 20 and 24°C inclusive) 2) On average there are 4 frogs per litre of swamp water. What is the probability there are less than 4 frogs in a 2 litre bucket of swamp water

  42. 1st Page Mixed Problems A Soln 1) Mean maximum temperature is 26°C Standard deviation = 5°C Temp measured to nearest degree. a) P(Temperature at least 30°C) b) P(Temperature between 20 and 30°C) c) P(Temperature between 20 and 24°C inclusive) 2) On average there are 4 frogs per litre of swamp water. What is the probability there are less than 4 frogs in a 2 litre bucket of swamp water

  43. 1st Page Mixed Problems B The mean weight of a cake is 1.5kg with standard deviation of 0.34kg Homer weighs 120kg. 1) If Homer ate six cakes what is the mean and standard deviation of his total weight . 2) What is the probability Homer weighs over 130kg?

  44. 1st Page Mixed Problems B Soln The mean weight of a cake is 1.5kg with standard deviation of 0.34kg Homer weighs 120kg. 1) If Homer ate six cakes what is the mean and standard deviation of his total weight . 2) What is the probability Homer weighs over 130kg?

  45. 1st Page Mixed Problems C • Each day Homer is exposed to radiation for six minutes on average (variance of 1.5 minutes) • Radiation exposure is considered dangerous if it is for more than five minutes per day. • What is the probability that Homer is exposed to dangerous levels of radiation on two consecutive days • What is the probability that Homer is exposed to dangerous levels of radiation for at least three days in a five day week

  46. 1st Page Mixed Problems C Soln • Each day Homer is exposed to radiation for six minutes on average (variance of 1.5 minutes) • Radiation exposure is considered dangerous if it is for more than five minutes per day. • What is the probability that Homer is exposed to dangerous levels of radiation on two consecutive days • What is the probability that Homer is exposed to dangerous levels of radiation for at least three days in a five day week

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