Question
sketching the area under the standard normal curve to the left of z = -1.5
Answers
GPT-4o mini
To sketch the area under the standard normal curve to the left of \( z = -1.5 \), follow these steps:
1. **Draw the Standard Normal Curve**: Start with a bell-shaped curve that represents the standard normal distribution, which has a mean of 0 and a standard deviation of 1.
2. **Label the Axis**: Label the horizontal axis with the z-scores. Make sure to mark the mean (0) at the center.
3. **Locate \( z = -1.5 \)**: Move to the left of the mean (0) and mark the point \( z = -1.5 \) on the horizontal axis.
4. **Shade the Area**: Shade the area under the curve to the left of \( z = -1.5 \). This shaded region represents the cumulative probability of z-scores that are less than -1.5.
5. **Add Labels**: You can label the shaded area with \( P(Z < -1.5) \) to indicate that it represents the probability of a z-score being less than -1.5.
To determine the exact probability of this area, you can look up the standard normal table or use a calculator that provides cumulative probabilities. The probability \( P(Z < -1.5) \) is approximately 0.0668, meaning about 6.68% of the data falls to the left of \( z = -1.5 \).
1. **Draw the Standard Normal Curve**: Start with a bell-shaped curve that represents the standard normal distribution, which has a mean of 0 and a standard deviation of 1.
2. **Label the Axis**: Label the horizontal axis with the z-scores. Make sure to mark the mean (0) at the center.
3. **Locate \( z = -1.5 \)**: Move to the left of the mean (0) and mark the point \( z = -1.5 \) on the horizontal axis.
4. **Shade the Area**: Shade the area under the curve to the left of \( z = -1.5 \). This shaded region represents the cumulative probability of z-scores that are less than -1.5.
5. **Add Labels**: You can label the shaded area with \( P(Z < -1.5) \) to indicate that it represents the probability of a z-score being less than -1.5.
To determine the exact probability of this area, you can look up the standard normal table or use a calculator that provides cumulative probabilities. The probability \( P(Z < -1.5) \) is approximately 0.0668, meaning about 6.68% of the data falls to the left of \( z = -1.5 \).