To find the approximate value of [tex]\( P(-0.78 \leq Z \leq 2.00) \)[/tex] for a standard normal distribution, let's follow the steps below:
1. Understand the problem:
We need to find the probability that the standard normal random variable [tex]\( Z \)[/tex] lies between -0.78 and 2.00.
2. Use the standard normal distribution table:
The table provides cumulative probabilities [tex]\( P(Z \leq z) \)[/tex] for given [tex]\( z \)[/tex] values.
3. Find [tex]\( P(Z \leq 2.00) \)[/tex]:
From the table, we see that [tex]\( P(Z \leq 2.00) = 0.9772 \)[/tex].
4. Find [tex]\( P(Z \leq -0.78) \)[/tex]:
Since the table typically provides probabilities for positive values of [tex]\( Z \)[/tex], we need to convert -0.78 to a positive equivalent using the symmetry property of the normal distribution. According to the symmetry property, [tex]\( P(Z \leq -0.78) = 1 - P(Z \leq 0.78) \)[/tex]. From the table, [tex]\( P(Z \leq 0.78) = 0.7823 \)[/tex].
Therefore, [tex]\( P(Z \leq -0.78) = 1 - 0.7823 = 0.2177 \)[/tex].
5. Calculate [tex]\( P(-0.78 \leq Z \leq 2.00) \)[/tex]:
This probability is the difference between the two cumulative probabilities:
[tex]\[
P(-0.78 \leq Z \leq 2.00) = P(Z \leq 2.00) - P(Z \leq -0.78)
\][/tex]
Substituting the values we found:
[tex]\[
P(-0.78 \leq Z \leq 2.00) = 0.9772 - 0.2177 = 0.7595
\][/tex]
Therefore, the approximate value of [tex]\( P(-0.78 \leq Z \leq 2.00) \)[/tex] is [tex]\( 0.7595 \)[/tex], which corresponds to approximately [tex]\( 75.95\% \)[/tex].
This calculated probability does not precisely match any of the given multiple-choice answers directly, but in terms of percentages, it is closest to [tex]\( 78\% \)[/tex].
