Certainly! To find the area under the standard normal curve between two z-scores, [tex]\( z_1 = 0.52 \)[/tex] and [tex]\( z_2 = 2.46 \)[/tex], we follow these steps:
1. Understand the Standard Normal Curve: The standard normal curve, also known as the z-distribution, has a mean of 0 and a standard deviation of 1. The area under this curve represents probabilities.
2. Use the Cumulative Distribution Function (CDF): The cumulative distribution function (CDF) of the standard normal distribution gives us the area under the curve to the left of a given z-score.
3. Calculate the CDF for [tex]\( z_1 \)[/tex]: Find the cumulative probability (area under the curve) to the left of [tex]\( z_1 = 0.52 \)[/tex]. Denote this as [tex]\( \Phi(z_1) \)[/tex].
4. Calculate the CDF for [tex]\( z_2 \)[/tex]: Similarly, find the cumulative probability to the left of [tex]\( z_2 = 2.46 \)[/tex]. Denote this as [tex]\( \Phi(z_2) \)[/tex].
5. Find the Area Between [tex]\( z_1 \)[/tex] and [tex]\( z_2 \)[/tex]: Subtract the cumulative probability at [tex]\( z_1 \)[/tex] from the cumulative probability at [tex]\( z_2 \)[/tex]. This gives the area (probability) between these two z-scores.
So, step-by-step:
- Step 1: The cumulative probability to the left of [tex]\( z_1 = 0.52 \)[/tex] is [tex]\( \Phi(0.52) \)[/tex].
- Step 2: The cumulative probability to the left of [tex]\( z_2 = 2.46 \)[/tex] is [tex]\( \Phi(2.46) \)[/tex].
- Step 3: The area between [tex]\( z_1 \)[/tex] and [tex]\( z_2 \)[/tex] is calculated as [tex]\( \Phi(2.46) - \Phi(0.52) \)[/tex].
After performing these calculations (which would typically be done using a standard normal distribution table or statistical software), we find the area to be approximately [tex]\( 0.2946 \)[/tex].
Thus, the area under the standard normal curve between [tex]\( z = 0.52 \)[/tex] and [tex]\( z = 2.46 \)[/tex] is [tex]\( 0.2946 \)[/tex].
