To solve the problem of finding the area under the standard normal curve between [tex]\( z = -1.40 \)[/tex] and [tex]\( z = 1.03 \)[/tex], we will use the properties of the standard normal distribution and its cumulative distribution function (CDF).
### Step-by-Step Solution:
1. Understand the Problem:
- We are given two z-scores, [tex]\( z = -1.40 \)[/tex] and [tex]\( z = 1.03 \)[/tex].
- We need to find the probability that a standard normal random variable [tex]\( Z \)[/tex] falls between these two values, i.e., [tex]\( P(-1.40 \leq Z \leq 1.03) \)[/tex].
2. Cumulative Distribution Function (CDF):
- The CDF of a standard normal distribution, denoted as [tex]\( \Phi(z) \)[/tex], gives the probability that [tex]\( Z \)[/tex] is less than or equal to [tex]\( z \)[/tex]. Mathematically, [tex]\( \Phi(z) = P(Z \leq z) \)[/tex].
3. Calculate the CDF Values:
- First, we calculate [tex]\( \Phi(z) \)[/tex] for the upper bound [tex]\( z = 1.03 \)[/tex]. This gives us the probability that [tex]\( Z \)[/tex] is less than or equal to 1.03.
- Next, we calculate [tex]\( \Phi(z) \)[/tex] for the lower bound [tex]\( z = -1.40 \)[/tex]. This gives us the probability that [tex]\( Z \)[/tex] is less than or equal to -1.40.
4. Find the Desired Probability:
- The probability that [tex]\( Z \)[/tex] falls between [tex]\( -1.40 \)[/tex] and [tex]\( 1.03 \)[/tex] is found by taking the difference between the CDF values at these two points:
[tex]\[
P(-1.40 \leq Z \leq 1.03) = \Phi(1.03) - \Phi(-1.40)
\][/tex]
5. Result:
- For [tex]\( z = 1.03 \)[/tex], the CDF value [tex]\( \Phi(1.03) \)[/tex] is approximately [tex]\( 0.84849 \)[/tex].
- For [tex]\( z = -1.40 \)[/tex], the CDF value [tex]\( \Phi(-1.40) \)[/tex] is approximately [tex]\( 0.08075 \)[/tex].
6. Final Calculation:
- Subtract the CDF value at [tex]\( z = -1.40 \)[/tex] from the CDF value at [tex]\( z = 1.03 \)[/tex]:
[tex]\[
P(-1.40 \leq Z \leq 1.03) = 0.84849 - 0.08075
\][/tex]
[tex]\[
P(-1.40 \leq Z \leq 1.03) \approx 0.76774
\][/tex]
Therefore, the area under the standard normal curve between [tex]\( z = -1.40 \)[/tex] and [tex]\( z = 1.03 \)[/tex] is approximately [tex]\( 0.76774 \)[/tex].
