To find the approximate value of [tex]\( P(-0.78 \leq z \leq 1.16) \)[/tex] for a standard normal distribution, we will follow these steps:
1. Find the cumulative probability up to [tex]\(z = -0.78\)[/tex]:
- From the standard normal table, the cumulative probability up to [tex]\(z = 0.78\)[/tex] is 0.7823.
- Since the normal distribution is symmetric about the mean (0), the cumulative probability for [tex]\(z = -0.78\)[/tex] is [tex]\(1 - 0.7823 = 0.2177\)[/tex].
2. Find the cumulative probability up to [tex]\(z = 1.16\)[/tex]:
- From the standard normal table, the cumulative probability up to [tex]\(z = 1.16\)[/tex] is 0.8770.
3. Calculate the probability between [tex]\(z = -0.78\)[/tex] and [tex]\(z = 1.16\)[/tex]:
- The probability between these two points is the difference between their cumulative probabilities:
[tex]\[
P(-0.78 \leq z \leq 1.16) = P(z \leq 1.16) - P(z \leq -0.78)
\][/tex]
Substituting the values we found from the table:
[tex]\[
P(-0.78 \leq z \leq 1.16) = 0.8770 - 0.2177 = 0.6593
\][/tex]
Therefore, the approximate value of [tex]\( P(-0.78 \leq z \leq 1.16) \)[/tex] is 0.6593. This corresponds to 65.93%, making the closest match from the listed options [tex]\(66 \%\)[/tex]. Hence, the correct answer is [tex]\( 66\% \)[/tex].
