Question

Find each of the shaded areas under the standard normal curve.
-1.92
Answer
to the right of 1.03
Answer
to the left of -1.92
Answer
between -1.76 and -0.65
Answer
the area of the shaded regions to the left of -0.26 AND to the right of 1.98
Answered by GPT-4o mini
To find the shaded area under the standard normal curve at a z-score of -1.92, we can look up this z-score in the standard normal distribution table or use a calculator that provides the cumulative distribution function (CDF) for the standard normal distribution.

1. **Using the Z-table**: The Z-table provides the area to the left of a given z-score in the standard normal distribution.

When you look up -1.92 in the Z-table, you will find the cumulative probability:

\[
P(Z < -1.92) \approx 0.0274
\]

This means that approximately 2.74% of the data falls to the left of z = -1.92 on the standard normal curve.

2. **Using a calculator**: You could also use a calculator with a normal distribution function. Most scientific calculators and statistics software allow you to input the z-score, and they will return the cumulative area.

In either case, the shaded area to the left of z = -1.92 is about **0.0274** or **2.74%**.

If you are interested in the area to the right of -1.92, you can calculate it as:

\[
P(Z > -1.92) = 1 - P(Z < -1.92) \approx 1 - 0.0274 = 0.9726
\]

So, the area to the right would be approximately **0.9726** or **97.26%**.
Answered by GPT-4o mini
To find the area under the standard normal curve to the right of a z-score of 1.03, we can use the cumulative distribution function (CDF). This tells us the area to the left of a given z-score.

1. **Using the Z-table**: Look up the z-score of 1.03 in the Z-table. The value typically found is:

\[
P(Z < 1.03) \approx 0.8485
\]

This indicates that approximately 84.85% of the data falls to the left of z = 1.03.

2. **Finding the area to the right**: To find the area to the right of z = 1.03, you subtract the cumulative probability from 1:

\[
P(Z > 1.03) = 1 - P(Z < 1.03) \approx 1 - 0.8485 = 0.1515
\]

Therefore, the area to the right of z = 1.03 is approximately **0.1515** or **15.15%**.
Answered by GPT-4o mini
To find the area under the standard normal curve to the left of a z-score of -1.92, we can again refer to the cumulative distribution function (CDF).

1. **Using the Z-table**: Look up -1.92 in the Z-table. The cumulative probability for this z-score is:

\[
P(Z < -1.92) \approx 0.0274
\]

This means that approximately 2.74% of the data falls to the left of z = -1.92.

So, the area to the left of z = -1.92 is approximately **0.0274** or **2.74%**.
Answered by GPT-4o mini
To find the area between two z-scores, -1.76 and -0.65, we first need to find the cumulative probabilities (areas) for each z-score and then subtract them.

1. **Find the cumulative probability for z = -1.76**:
- Using a Z-table or calculator, we find:
\[
P(Z < -1.76) \approx 0.0392
\]
This means approximately 3.92% of the data falls to the left of z = -1.76.

2. **Find the cumulative probability for z = -0.65**:
- Again, using a Z-table or calculator, we find:
\[
P(Z < -0.65) \approx 0.2578
\]
This means approximately 25.78% of the data falls to the left of z = -0.65.

3. **Calculate the area between the two z-scores**:
To find the area between -1.76 and -0.65, we subtract the cumulative probability of -1.76 from that of -0.65:
\[
P(-1.76 < Z < -0.65) = P(Z < -0.65) - P(Z < -1.76)
\]
\[
= 0.2578 - 0.0392 \approx 0.2186
\]

Thus, the area between z = -1.76 and z = -0.65 is approximately **0.2186** or **21.86%**.
Answered by GPT-4o mini
To find the area of the shaded regions to the left of \( z = -0.26 \) and to the right of \( z = 1.98 \), we need to calculate the cumulative probabilities for each z-score and then sum those areas.

1. **Area to the left of \( z = -0.26 \)**:
- Look up the cumulative probability for \( z = -0.26 \):
\[
P(Z < -0.26) \approx 0.3925
\]
This indicates that approximately 39.25% of the data falls to the left of \( z = -0.26 \).

2. **Area to the right of \( z = 1.98 \)**:
- Look up the cumulative probability for \( z = 1.98 \):
\[
P(Z < 1.98) \approx 0.9761
\]
This indicates that approximately 97.61% of the data falls to the left of \( z = 1.98 \).
- To find the area to the right of \( z = 1.98 \), we subtract this probability from 1:
\[
P(Z > 1.98) = 1 - P(Z < 1.98) \approx 1 - 0.9761 = 0.0239
\]
This means approximately 2.39% of the data falls to the right of \( z = 1.98 \).

3. **Calculate the total area of the shaded regions**:
Now, we add the two areas together:
\[
\text{Total Area} = P(Z < -0.26) + P(Z > 1.98)
\]
\[
= 0.3925 + 0.0239 \approx 0.4164
\]

Thus, the total area of the shaded regions to the left of \( z = -0.26 \) and to the right of \( z = 1.98 \) is approximately **0.4164** or **41.64%**.