text: Question
  
  # A: Let z be a random variable with a standard normal distribution. Find the indicated probability....
  
  A: Let z be a random variable with a standard normal distribution. Find the indicated probability. (Round your answer to four decimal places.)
  P(z ≤ 1.11) =
  
  B: Let z be a random variable with a standard normal distribution. Find the indicated probability. (Round your answer to four decimal places.)
  
  P(z ≥ −1.24) =
  
  C: Let z be a random variable with a standard normal distribution. Find the indicated probability. (Round your answer to four decimal places.)
  
  P(−1.78 ≤ z ≤ −1.14) =
  
  D: Assume that x has a normal distribution with the specified mean and standard deviation. Find the indicated probability. (Round your answer to four decimal places.)
  
  μ = 40; σ = 16
  
  P(50 ≤ x ≤ 70) =
  
  E: Assume that x has a normal distribution with the specified mean and standard deviation. Find the indicated probability. (Round your answer to four decimal places.)
  
  μ = 97; σ = 12
  
  P(x ≥ 90) =
  
  We need to use standard normal z table to solve this problem
  
  A)
  
  From z table, P(z<1.11) = 0.8665
  
  B)
  
  From z table, P(z>-1.24) = 0.8925
  
  C)
  
  P(-1.78<z<-1.14) = p(z<-1.14) - p(z<-1.78)
  
  P(z<-1.14) = 0.1271
  
  P(z<-1.78) = 0.0375
  
  So answer is 0.1271 - 0.0375 = 0.0896
  
  D)
  
  As the data is normally distributed we can use standard normal z table to estimate the answers
  
  Z = (x-mean)/s.d
  
  Given mean = 40
  
  S.d = 16
  
  P(50 < x< 70)
  
  P(x<70)
  
  Z = (70-40)/16 = 1.88
  
  From z table, P(z<1.88) = 0.9699
  
  P(x<50)
  
  Z = (50 - 40)/16 = 0.63
  
  From z table, P(z<0.63) = 0.7357
  
  Required probability is 0.9699 - 0.7357 = 0.2342
  
  E)
  
  P(x>90)
  
  Z = (90 - 97)/12 = -0.58
  
  From.z.table, P(z>-0.58) = 0.7190
  
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explanation: The document contains multiple questions related to finding probabilities for standard normal distributions and normal distributions with specified means and standard deviations. Each question is followed by a detailed solution using z-tables.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: Question
  
  # A: Let z be a random variable with a standard normal distribution. Find the indicated probability....
  
  A: Let z be a random variable with a standard normal distribution. Find the indicated probability. (Round your answer to four decimal places.)
  P(z ≤ 1.11) =
  
  B: Let z be a random variable with a standard normal distribution. Find the indicated probability. (Round your answer to four decimal places.)
  
  P(z ≥ −1.24) =
  
  C: Let z be a random variable with a standard normal distribution. Find the indicated probability. (Round your answer to four decimal places.)
  
  P(−1.78 ≤ z ≤ −1.14) =
  
  D: Assume that x has a normal distribution with the specified mean and standard deviation. Find the indicated probability. (Round your answer to four decimal places.)
  
  μ = 40; σ = 16
  
  P(50 ≤ x ≤ 70) =
  
  E: Assume that x has a normal distribution with the specified mean and standard deviation. Find the indicated probability. (Round your answer to four decimal places.)
  
  μ = 97; σ = 12
  
  P(x ≥ 90) =
  
  We need to use standard normal z table to solve this problem
  
  A)
  
  From z table, P(z<1.11) = 0.8665
  
  B)
  
  From z table, P(z>-1.24) = 0.8925
  
  C)
  
  P(-1.78<z<-1.14) = p(z<-1.14) - p(z<-1.78)
  
  P(z<-1.14) = 0.1271
  
  P(z<-1.78) = 0.0375
  
  So answer is 0.1271 - 0.0375 = 0.0896
  
  D)
  
  As the data is normally distributed we can use standard normal z table to estimate the answers
  
  Z = (x-mean)/s.d
  
  Given mean = 40
  
  S.d = 16
  
  P(50 < x< 70)
  
  P(x<70)
  
  Z = (70-40)/16 = 1.88
  
  From z table, P(z<1.88) = 0.9699
  
  P(x<50)
  
  Z = (50 - 40)/16 = 0.63
  
  From z table, P(z<0.63) = 0.7357
  
  Required probability is 0.9699 - 0.7357 = 0.2342
  
  E)
  
  P(x>90)
  
  Z = (90 - 97)/12 = -0.58
  
  From.z.table, P(z>-0.58) = 0.7190
  
  #### Earn Coins
  
  Coins can be redeemed for fabulous gifts.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains multiple questions related to finding probabilities for standard normal distributions and normal distributions with specified means and standard deviations. Each question is followed by a detailed solution using z-tables.
  
  Label: multiple-question-and-answer