To solve this problem, we need to determine which of the given probabilities is approximately equal to 0.2957. We'll use the probabilities given in the standard normal table and compare the values for the different ranges.
Here are the steps:
1. Understand the ranges of [tex]\(z\)[/tex] values given:
- [tex]\(P(-1.25 \leq z \leq 0.25)\)[/tex]
- [tex]\(P(-1.25 \leq z \leq 0.75)\)[/tex]
- [tex]\(P(0.25 \leq z \leq 1.25)\)[/tex]
2. Convert these [tex]\(z\)[/tex] ranges to probabilities using the values from the standard normal table:
- For [tex]\(P(0.25 \leq z \leq 1.25)\)[/tex]:
- From the table: [tex]\(P(z \leq 0.25) = 0.5987\)[/tex]
- From the table: [tex]\(P(z \leq 1.25) = 0.8944\)[/tex]
- Therefore, [tex]\(P(0.25 \leq z \leq 1.25) = P(z \leq 1.25) - P(z \leq 0.25) = 0.8944 - 0.5987 = 0.2957\)[/tex]
- For [tex]\(P(-1.25 \leq z \leq 0.75)\)[/tex]:
- From the table: [tex]\(P(z \leq -1.25) = 1 - P(z \leq 1.25) = 1 - 0.1056 = 0.1056\)[/tex] (since [tex]\(P(z \leq -1.25)\)[/tex] is same as 1 minus the positive half due to the symmetry of the normal distribution)
- From the table: [tex]\(P(z \leq 0.75) = 0.7734\)[/tex]
- Therefore, [tex]\(P(-1.25 \leq z \leq 0.75) = P(z \leq 0.75) - P(z \leq -1.25) = 0.7734 - 0.1056 = 0.6678\)[/tex]
- For [tex]\(P(-1.25 \leq z \leq 0.25)\)[/tex]:
- From the table: [tex]\(P(z \leq -1.25) = 1 - 0.1056 = 0.1056\)[/tex]
- From the table: [tex]\(P(z \leq 0.25) = 0.5987\)[/tex]
- Therefore, [tex]\(P(-1.25 \leq z \leq 0.25) = P(z \leq 0.25) - P(z \leq -1.25) = 0.5987 - 0.1056 = 0.4931\)[/tex]
3. Compare the calculated probabilities:
- [tex]\(P(0.25 \leq z \leq 1.25) = 0.2957\)[/tex]
- [tex]\(P(-1.25 \leq z \leq 0.75) = 0.6678\)[/tex]
- [tex]\(P(-1.25 \leq z \leq 0.25) = 0.4931\)[/tex]
By comparing these probabilities, we can see that:
- The probability for the range [tex]\(0.25 \leq z \leq 1.25\)[/tex] is exactly 0.2957.
Thus, the answer is:
[tex]\[ P(0.25 \leq z \leq 1.25) \][/tex]
Therefore, the correct option to mark is:
[tex]\[ P(0.25 \leq z \leq 1.25) \][/tex]
