To find the approximate value of [tex]\( P(-0.78 \leq z \leq 1.16) \)[/tex] for a standard normal distribution, we'll follow these steps:
1. Locate the given z-scores in the provided z-table:
- [tex]\(z = 0.78\)[/tex] has a probability of 0.7823.
- [tex]\(z = 1.16\)[/tex] has a probability of 0.8770.
2. Use symmetry of the standard normal distribution to find the probability for the negative z-score:
- The table generally provides values for positive z-scores.
- For negative z-scores, we utilize the fact that the normal distribution is symmetric around 0. Therefore,
[tex]\[
P(Z \leq -0.78) = 1 - P(Z \leq 0.78).
\][/tex]
3. Calculate the probability for [tex]\(z = -0.78\)[/tex]:
- We know [tex]\(P(Z \leq 0.78) = 0.7823\)[/tex].
- Therefore,
[tex]\[
P(Z \leq -0.78) = 1 - 0.7823 = 0.2177.
\][/tex]
4. Find the required probability [tex]\(P(-0.78 \leq z \leq 1.16)\)[/tex]:
- This is determined by subtracting the probability of [tex]\(z \leq -0.78\)[/tex] from the probability of [tex]\(z \leq 1.16\)[/tex], because:
[tex]\[
P(-0.78 \leq z \leq 1.16) = P(Z \leq 1.16) - P(Z \leq -0.78).
\][/tex]
- So,
[tex]\[
P(-0.78 \leq z \leq 1.16) = 0.8770 - 0.2177 = 0.6593.
\][/tex]
In conclusion, the approximate value of [tex]\(P(-0.78 \leq z \leq 1.16)\)[/tex] is [tex]\(0.6593\)[/tex].
