To find the area under the standard normal curve between z = -2.48 and z = -1.02, we follow these steps:
1. Understand the Problem:
We are asked to find the area under the standard normal curve between two z-scores, specifically from z = -2.48 to z = -1.02. This area represents the probability that a value from a standard normal distribution falls within this range.
2. Cumulative Distribution Function (CDF):
The cumulative distribution function for the standard normal distribution, denoted by [tex]\( \Phi(z) \)[/tex], gives the probability that a standard normal variable is less than or equal to a particular value [tex]\( z \)[/tex].
3. Find the CDF values for the given z-scores:
- For [tex]\( z = -2.48 \)[/tex], the cumulative probability [tex]\( \Phi(-2.48) \)[/tex] is approximately 0.006569.
- For [tex]\( z = -1.02 \)[/tex], the cumulative probability [tex]\( \Phi(-1.02) \)[/tex] is approximately 0.153864.
4. Calculate the Area Between the Two z-scores:
To find the area between the two z-scores, subtract the cumulative probability at [tex]\( z = -2.48 \)[/tex] from the cumulative probability at [tex]\( z = -1.02 \)[/tex]:
[tex]\[
\text{Area} = \Phi(-1.02) - \Phi(-2.48)
\][/tex]
Plug in the values we obtained:
[tex]\[
\text{Area} = 0.153864 - 0.006569
\][/tex]
5. Compute the Difference:
- First, subtract the two cumulative probabilities:
[tex]\[
0.153864 - 0.006569 = 0.147295
\][/tex]
6. Result:
Therefore, the area under the standard normal curve between z = -2.48 and z = -1.02 is approximately 0.147295.
In summary, the area under the standard normal curve between z = -2.48 and z = -1.02 is approximately 0.147295. This represents the probability that a value from the standard normal distribution falls within this range.
