Answer :
To determine the probability for the given range [tex]\(P(0.6 \leq z \leq 2.12)\)[/tex], we'll follow these steps:
1. Find the cumulative probability for [tex]\(z = 0.6\)[/tex].
2. Find the cumulative probability for [tex]\(z = 2.12\)[/tex].
3. Subtract the cumulative probability for [tex]\(z = 0.6\)[/tex] from that for [tex]\(z = 2.12\)[/tex] to get the probability that [tex]\(z\)[/tex] lies between these two values.
### Step-by-Step Solution:
1. Finding [tex]\(P(z \leq 0.6)\)[/tex]:
- The cumulative probability [tex]\(P(z \leq 0.6)\)[/tex] is approximately [tex]\(0.7257\)[/tex].
2. Finding [tex]\(P(z \leq 2.12)\)[/tex]:
- The cumulative probability [tex]\(P(z \leq 2.12)\)[/tex] is approximately [tex]\(0.9830\)[/tex].
3. Calculate [tex]\(P(0.6 \leq z \leq 2.12)\)[/tex]:
- We subtract the cumulative probability at [tex]\(z = 0.6\)[/tex] from the cumulative probability at [tex]\(z = 2.12\)[/tex]:
[tex]\[ P(0.6 \leq z \leq 2.12) = P(z \leq 2.12) - P(z \leq 0.6) \][/tex]
[tex]\[ P(0.6 \leq z \leq 2.12) = 0.9830 - 0.7257 = 0.2573 \][/tex]
4. Convert the probability to a percentage:
- [tex]\(0.2573\)[/tex] in percentage terms is [tex]\(25.73\%\)[/tex].
Therefore,
[tex]\[ P(0.6 \leq z \leq 2.12) \approx 25.73\% \][/tex]
Given the choices:
- [tex]\(16 \%\)[/tex]
- [tex]\(26 \%\)[/tex]
- [tex]\(73 \%\)[/tex]
- [tex]\(98 \%\)[/tex]
The closest answer is:
[tex]\[ \boxed{26 \%} \][/tex]
1. Find the cumulative probability for [tex]\(z = 0.6\)[/tex].
2. Find the cumulative probability for [tex]\(z = 2.12\)[/tex].
3. Subtract the cumulative probability for [tex]\(z = 0.6\)[/tex] from that for [tex]\(z = 2.12\)[/tex] to get the probability that [tex]\(z\)[/tex] lies between these two values.
### Step-by-Step Solution:
1. Finding [tex]\(P(z \leq 0.6)\)[/tex]:
- The cumulative probability [tex]\(P(z \leq 0.6)\)[/tex] is approximately [tex]\(0.7257\)[/tex].
2. Finding [tex]\(P(z \leq 2.12)\)[/tex]:
- The cumulative probability [tex]\(P(z \leq 2.12)\)[/tex] is approximately [tex]\(0.9830\)[/tex].
3. Calculate [tex]\(P(0.6 \leq z \leq 2.12)\)[/tex]:
- We subtract the cumulative probability at [tex]\(z = 0.6\)[/tex] from the cumulative probability at [tex]\(z = 2.12\)[/tex]:
[tex]\[ P(0.6 \leq z \leq 2.12) = P(z \leq 2.12) - P(z \leq 0.6) \][/tex]
[tex]\[ P(0.6 \leq z \leq 2.12) = 0.9830 - 0.7257 = 0.2573 \][/tex]
4. Convert the probability to a percentage:
- [tex]\(0.2573\)[/tex] in percentage terms is [tex]\(25.73\%\)[/tex].
Therefore,
[tex]\[ P(0.6 \leq z \leq 2.12) \approx 25.73\% \][/tex]
Given the choices:
- [tex]\(16 \%\)[/tex]
- [tex]\(26 \%\)[/tex]
- [tex]\(73 \%\)[/tex]
- [tex]\(98 \%\)[/tex]
The closest answer is:
[tex]\[ \boxed{26 \%} \][/tex]
