Final answer:
To determine the value of the constant c in different probability statements involving a standard normal distribution, we use the properties of the distribution and find the corresponding z-scores. For each case, we use a standard normal distribution table or calculator to find the z-score that matches the given probability. Then, we determine the value of c based on the z-score.
Step-by-step explanation:
(a) ?(c) = 0.9821:
To determine the value of c, we need to find the z-score that corresponds to a probability of 0.9821 in a standard normal distribution. Using a standard normal distribution table or calculator, we find that the z-score is approximately 2.20. Therefore, c = 2.20.
(b) P(0 ≤ Z ≤ c) = 0.2939:
To find the value of c, we need to find the z-score that corresponds to a probability of 0.2939 when z is between 0 and c. Using a standard normal distribution table or calculator, we find that the z-score is approximately 0.58. Therefore, c = 0.58.
(c) P(c ≤ Z) = 0.1335:
To find the value of c, we need to find the z-score that corresponds to a probability of 0.1335 when z is greater than or equal to c. Using a standard normal distribution table or calculator, we find that the z-score is approximately -1.10. Therefore, c = -1.10.
(d) P(-c ≤ Z ≤ c) = 0.6476:
To find the value of c, we need to find the z-score that corresponds to a probability of 0.6476 when z is between -c and c. Using a standard normal distribution table or calculator, we find that the z-score is approximately 0.4. Therefore, c = 0.4.
(e) P(c ≤ |Z|) = 0.0128:
To find the value of c, we need to find the z-score that corresponds to a probability of 0.0128 when |z| is greater than or equal to c. Using a standard normal distribution table or calculator, we find that the z-score is approximately 2.71. Therefore, c = 2.71.