Final answer:
To determine the constants that make the probability statements correct, we can use the cumulative distribution function and the inverse of the cumulative distribution function for the standard normal distribution. For each case, we calculate the value of the constant c using the appropriate formula or equation.
Step-by-step explanation:
For each case, we need to find the value of the constant c that makes the probability statement correct.
(a) Φ(c) = 0.9842: To find c, we can use the inverse of the standard normal distribution function. Φ(c) represents the probability that a standard normal random variable is less than c. So, we need to find the value of c for which this probability is 0.9842. Using a standard normal distribution table or a calculator, we can find that c is approximately 2.18.
(b) P(0 ≤ Z ≤ c) = 0.3051: This represents the probability that a standard normal random variable is between 0 and c. We can use the cumulative distribution function of the standard normal distribution to find this probability. By calculating Φ(c) - Φ(0), where Φ refers to the cumulative distribution function, we can find that c is approximately 0.53.
(c) P(c ≤ Z) = 0.1190: This represents the probability that a standard normal random variable is greater than or equal to c. Again, we can use the cumulative distribution function to find the value of c. By calculating 1 - Φ(c), we can find that c is approximately -1.17.
(d) P(−c ≤ Z ≤ c) = 0.6528: This represents the probability that a standard normal random variable is between -c and c. To find the value of c, we can use the inverse of the cumulative distribution function. By solving the equation Φ(c) - Φ(-c) = 0.6528, we find that c is approximately 0.83.
(e) P(c ≤ |Z|) = 0.0128: This represents the probability that the absolute value of a standard normal random variable is greater than or equal to c. By calculating 2(1 - Φ(c)), where 2 represents the area in both tails of the standard normal distribution, we can find that c is approximately 2.36.