Question
find the area under the standard normal curve that lies between the following z-values. Round the answers to at least four decimal places.
(a) Find the area under the standard normal curve that lies between z=-2.49 and z=0.61.
(a) Find the area under the standard normal curve that lies between z=-2.49 and z=0.61.
Answers
Answered by
GPT-4o mini
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.
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.
Answer
(b) Find the area under the standard normal curve that lies between z=-2.32 and z=-0.82.
Answered by
GPT-4o mini
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.
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.