Certainly! Let's answer this question step-by-step while providing a clear explanation.
1. Identify the Problem:
We need to find the area under the standard normal distribution curve between [tex]\( z = -1.30 \)[/tex] and [tex]\( z = 2.02 \)[/tex].
2. Standard Normal Distribution:
The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. The area under this curve within a specific interval represents the probability that a value falls within that interval.
3. Determine the Cumulative Distribution Function (CDF) Values:
The CDF for a given [tex]\( z \)[/tex]-value represents the area under the curve to the left of that [tex]\( z \)[/tex]-value. We need to find the CDF values for [tex]\( z = -1.30 \)[/tex] and [tex]\( z = 2.02 \)[/tex].
Let's denote:
- [tex]\( cdf(z = -1.30) \)[/tex] as [tex]\( cdf_{\text{lower}} \)[/tex]
- [tex]\( cdf(z = 2.02) \)[/tex] as [tex]\( cdf_{\text{upper}} \)[/tex]
Using the standard normal distribution table or computation tools, we get:
- [tex]\( cdf_{\text{lower}} = 0.0968 \)[/tex]
- [tex]\( cdf_{\text{upper}} = 0.9783 \)[/tex]
4. Calculate the Area Between [tex]\( z = -1.30 \)[/tex] and [tex]\( z = 2.02 \)[/tex]:
The area between these two z-values is the difference between their CDF values. Thus,
[tex]\[
\text{Area} = cdf_{\text{upper}} - cdf_{\text{lower}}
\][/tex]
Substituting in our values:
[tex]\[
\text{Area} = 0.9783 - 0.0968 = 0.8815
\][/tex]
5. Round the Result:
The area should be rounded to four decimal places. In this case, the result is already rounded appropriately.
6. Sketch (SALT method):
While I cannot draw here, you would sketch the standard normal distribution curve which is bell-shaped and symmetric around the mean (0). You'd shade the area under the curve between [tex]\( z = -1.30 \)[/tex] and [tex]\( z = 2.02 \)[/tex]. Label these z-values on the horizontal axis, and show the shaded region representing the area (0.8815).
Conclusion:
The area under the standard normal curve between [tex]\( z = -1.30 \)[/tex] and [tex]\( z = 2.02 \)[/tex] is approximately [tex]\( 0.8815 \)[/tex], rounded to four decimal places.
