find the area under the standard normal curve that lies between the following z-values. Round the answers to at least four decimal places.

To find the area under the standard normal curve that lies between \( z = -2.49 \) and \( z = 0.61 \), we can use the standard normal distribution table (Z-table) or a calculator that provides the cumulative distribution function (CDF) for the standard normal distribution.

1. Find the cumulative area for \( z = -2.49 \):

   • Using a Z-table or CDF calculator, the cumulative area for \( z = -2.49 \) is approximately \( 0.0064 \).

2. Find the cumulative area for \( z = 0.61 \):

   • The cumulative area for \( z = 0.61 \) is approximately \( 0.7291 \).

3. Calculate the area between the two z-values: \[ \text{Area} = P(Z \leq 0.61) - P(Z \leq -2.49) \] \[ \text{Area} = 0.7291 - 0.0064 = 0.7227 \]

Thus, the area under the standard normal curve between \( z = -2.49 \) and \( z = 0.61 \) is approximately \( 0.7227 \) when rounded to four decimal places.

To find the area under the standard normal curve that lies between \( z = -2.32 \) and \( z = -0.82 \), we will use the cumulative distribution function (CDF) of the standard normal distribution.

1. Find the cumulative area for \( z = -2.32 \):

   • Using a Z-table or CDF calculator, the cumulative area for \( z = -2.32 \) is approximately \( 0.0106 \).

2. Find the cumulative area for \( z = -0.82 \):

   • The cumulative area for \( z = -0.82 \) is approximately \( 0.2061 \).

3. Calculate the area between the two z-values: \[ \text{Area} = P(Z \leq -0.82) - P(Z \leq -2.32) \] \[ \text{Area} = 0.2061 - 0.0106 = 0.1955 \]

Thus, the area under the standard normal curve between \( z = -2.32 \) and \( z = -0.82 \) is approximately 0.1955 when rounded to four decimal places.