Answer :
To find the approximate value of [tex]\( P(z \geq -1.25) \)[/tex] for a standard normal distribution, follow these steps:
1. Understand the problem:
We are asked to find the probability that the standard normal variable [tex]\( z \)[/tex] is greater than or equal to [tex]\(-1.25\)[/tex]. We have a standard normal table which provides cumulative probabilities from the mean ([tex]\( z = 0 \)[/tex]) up to specified positive values of [tex]\( z \)[/tex].
2. Use symmetry of the normal distribution:
The standard normal distribution is symmetric around [tex]\( z = 0 \)[/tex]. This means that [tex]\( P(z \geq -1.25) \)[/tex] is equivalent to [tex]\( P(z \leq 1.25) \)[/tex], because the probability of the distribution to the right of [tex]\(-1.25\)[/tex] is the same as the probability to the left of [tex]\( 1.25 \)[/tex].
3. Look up the cumulative probability:
According to the table, the cumulative probability [tex]\( P(z \leq 1.25) \)[/tex] is given as [tex]\( 0.8944 \)[/tex]. This means that the probability that [tex]\( z \)[/tex] is less than or equal to [tex]\( 1.25 \)[/tex] is [tex]\( 0.8944 \)[/tex].
4. Interpret the result:
Since the cumulative probability [tex]\( P(z \leq 1.25) = 0.8944 \)[/tex], which is equivalent to [tex]\( P(z \geq -1.25) = 0.8944 \)[/tex], we can convert this probability into a percentage to match the answer choices.
5. Convert to percentage:
Multiply the probability by 100 to get the percentage:
[tex]\[ 0.8944 \times 100 = 89.44\% \][/tex]
The closest option provided is [tex]\( 89\% \)[/tex].
Therefore, the approximate value of [tex]\( P(z \geq -1.25) \)[/tex] is [tex]\( 89\% \)[/tex].
The correct answer is [tex]\( 89\% \)[/tex], which matches the option given in the problem.
1. Understand the problem:
We are asked to find the probability that the standard normal variable [tex]\( z \)[/tex] is greater than or equal to [tex]\(-1.25\)[/tex]. We have a standard normal table which provides cumulative probabilities from the mean ([tex]\( z = 0 \)[/tex]) up to specified positive values of [tex]\( z \)[/tex].
2. Use symmetry of the normal distribution:
The standard normal distribution is symmetric around [tex]\( z = 0 \)[/tex]. This means that [tex]\( P(z \geq -1.25) \)[/tex] is equivalent to [tex]\( P(z \leq 1.25) \)[/tex], because the probability of the distribution to the right of [tex]\(-1.25\)[/tex] is the same as the probability to the left of [tex]\( 1.25 \)[/tex].
3. Look up the cumulative probability:
According to the table, the cumulative probability [tex]\( P(z \leq 1.25) \)[/tex] is given as [tex]\( 0.8944 \)[/tex]. This means that the probability that [tex]\( z \)[/tex] is less than or equal to [tex]\( 1.25 \)[/tex] is [tex]\( 0.8944 \)[/tex].
4. Interpret the result:
Since the cumulative probability [tex]\( P(z \leq 1.25) = 0.8944 \)[/tex], which is equivalent to [tex]\( P(z \geq -1.25) = 0.8944 \)[/tex], we can convert this probability into a percentage to match the answer choices.
5. Convert to percentage:
Multiply the probability by 100 to get the percentage:
[tex]\[ 0.8944 \times 100 = 89.44\% \][/tex]
The closest option provided is [tex]\( 89\% \)[/tex].
Therefore, the approximate value of [tex]\( P(z \geq -1.25) \)[/tex] is [tex]\( 89\% \)[/tex].
The correct answer is [tex]\( 89\% \)[/tex], which matches the option given in the problem.
