Question
(a) Find the area under the standard normal curve to the right of z=0.78.
The area to the right of z=0.78 is
The area to the right of z=0.78 is
Answers
Answer
b) Find the area under the standard normal curve to the right of z=-1.81.
Answer
(a) Find the area under the standard normal curve that lies between -1.71 and 1.25.
Answer
(b) Find the area under the standard normal curve that lies between -1.92 and -1.17.
Answer
Use the TI-84 Plus calculator to find the z-score for which the area to its left is 0.59. Round the answer to two decimal places.
The z-score for the given area is ?
The z-score for the given area is ?
Answer
Use the TI-84 Plus calculator to find the z-scores that bound the middle 88% of the area under the standard normal curve. Enter the answers in ascending order and round to two decimal places.
The z-scores for the given area are [?] and [?]
The z-scores for the given area are [?] and [?]
Answered by
GPT-4o mini
To find the area under the standard normal curve to the right of \( z = 0.78 \), we can use the standard normal distribution table (Z-table) or a calculator.
1. **Look up the Z-score**: The Z-table provides the area to the left of a given Z-score. For \( z = 0.78 \), the area to the left is approximately \( 0.7823 \).
2. **Calculate the area to the right**: Since the total area under the normal curve is 1, to find the area to the right, we subtract the area to the left from 1:
\[
\text{Area to the right} = 1 - \text{Area to the left}
\]
\[
\text{Area to the right} = 1 - 0.7823 = 0.2177
\]
Thus, the area to the right of \( z = 0.78 \) is approximately **0.2177**.
1. **Look up the Z-score**: The Z-table provides the area to the left of a given Z-score. For \( z = 0.78 \), the area to the left is approximately \( 0.7823 \).
2. **Calculate the area to the right**: Since the total area under the normal curve is 1, to find the area to the right, we subtract the area to the left from 1:
\[
\text{Area to the right} = 1 - \text{Area to the left}
\]
\[
\text{Area to the right} = 1 - 0.7823 = 0.2177
\]
Thus, the area to the right of \( z = 0.78 \) is approximately **0.2177**.
Answered by
GPT-4o mini
To find the area under the standard normal curve to the right of \( z = -1.81 \), you can follow a similar process:
1. **Look up the Z-score**: Check the Z-table for the area to the left of \( z = -1.81 \). The area to the left (cumulative probability) is approximately \( 0.0351 \).
2. **Calculate the area to the right**: To find the area to the right, subtract the area to the left from 1:
\[
\text{Area to the right} = 1 - \text{Area to the left}
\]
\[
\text{Area to the right} = 1 - 0.0351 = 0.9649
\]
Thus, the area to the right of \( z = -1.81 \) is approximately **0.9649**.
1. **Look up the Z-score**: Check the Z-table for the area to the left of \( z = -1.81 \). The area to the left (cumulative probability) is approximately \( 0.0351 \).
2. **Calculate the area to the right**: To find the area to the right, subtract the area to the left from 1:
\[
\text{Area to the right} = 1 - \text{Area to the left}
\]
\[
\text{Area to the right} = 1 - 0.0351 = 0.9649
\]
Thus, the area to the right of \( z = -1.81 \) is approximately **0.9649**.
Answered by
GPT-4o mini
To find the area under the standard normal curve that lies between \( z = -1.71 \) and \( z = 1.25 \), you can follow these steps:
1. **Find the area to the left of \( z = 1.25 \)**:
- Using a Z-table or calculator, the area to the left of \( z = 1.25 \) is approximately \( 0.8944 \).
2. **Find the area to the left of \( z = -1.71 \)**:
- The area to the left of \( z = -1.71 \) is approximately \( 0.0436 \).
3. **Calculate the area between \( z = -1.71 \) and \( z = 1.25 \)**:
- To find the area between these two Z-scores, subtract the area to the left of \( z = -1.71 \) from the area to the left of \( z = 1.25 \):
\[
\text{Area between} = P(Z < 1.25) - P(Z < -1.71)
\]
\[
\text{Area between} = 0.8944 - 0.0436 = 0.8508
\]
Therefore, the area under the standard normal curve that lies between \( z = -1.71 \) and \( z = 1.25 \) is approximately **0.8508**.
1. **Find the area to the left of \( z = 1.25 \)**:
- Using a Z-table or calculator, the area to the left of \( z = 1.25 \) is approximately \( 0.8944 \).
2. **Find the area to the left of \( z = -1.71 \)**:
- The area to the left of \( z = -1.71 \) is approximately \( 0.0436 \).
3. **Calculate the area between \( z = -1.71 \) and \( z = 1.25 \)**:
- To find the area between these two Z-scores, subtract the area to the left of \( z = -1.71 \) from the area to the left of \( z = 1.25 \):
\[
\text{Area between} = P(Z < 1.25) - P(Z < -1.71)
\]
\[
\text{Area between} = 0.8944 - 0.0436 = 0.8508
\]
Therefore, the area under the standard normal curve that lies between \( z = -1.71 \) and \( z = 1.25 \) is approximately **0.8508**.
Answered by
GPT-4o mini
To find the area under the standard normal curve that lies between \( z = -1.92 \) and \( z = -1.17 \), follow these steps:
1. **Find the area to the left of \( z = -1.17 \)**:
- Using a Z-table or calculator, the area to the left of \( z = -1.17 \) is approximately \( 0.1210 \).
2. **Find the area to the left of \( z = -1.92 \)**:
- The area to the left of \( z = -1.92 \) is approximately \( 0.0274 \).
3. **Calculate the area between \( z = -1.92 \) and \( z = -1.17 \)**:
- To find the area between these two Z-scores, subtract the area to the left of \( z = -1.92 \) from the area to the left of \( z = -1.17 \):
\[
\text{Area between} = P(Z < -1.17) - P(Z < -1.92)
\]
\[
\text{Area between} = 0.1210 - 0.0274 = 0.0936
\]
Therefore, the area under the standard normal curve that lies between \( z = -1.92 \) and \( z = -1.17 \) is approximately **0.0936**.
1. **Find the area to the left of \( z = -1.17 \)**:
- Using a Z-table or calculator, the area to the left of \( z = -1.17 \) is approximately \( 0.1210 \).
2. **Find the area to the left of \( z = -1.92 \)**:
- The area to the left of \( z = -1.92 \) is approximately \( 0.0274 \).
3. **Calculate the area between \( z = -1.92 \) and \( z = -1.17 \)**:
- To find the area between these two Z-scores, subtract the area to the left of \( z = -1.92 \) from the area to the left of \( z = -1.17 \):
\[
\text{Area between} = P(Z < -1.17) - P(Z < -1.92)
\]
\[
\text{Area between} = 0.1210 - 0.0274 = 0.0936
\]
Therefore, the area under the standard normal curve that lies between \( z = -1.92 \) and \( z = -1.17 \) is approximately **0.0936**.
Answered by
GPT-4o mini
To find the z-score corresponding to an area of \( 0.59 \) to its left using a TI-84 Plus calculator, you can follow these steps:
1. **Turn on the calculator** and press the `2nd` button.
2. **Access the DISTR menu** by pressing the `VARS` key.
3. **Select the inverse normal function**:
- Scroll to `invNorm(` or just press `2`.
4. **Input the area** and the parameters for the standard normal distribution:
- Type `invNorm(0.59)`.
5. **Close the parentheses** and press `ENTER`.
The TI-84 Plus will return the z-score corresponding to an area of \( 0.59 \).
After performing these steps, you should find that the z-score is approximately **0.28** when rounded to two decimal places.
Thus, the z-score for which the area to its left is \( 0.59 \) is **0.28**.
1. **Turn on the calculator** and press the `2nd` button.
2. **Access the DISTR menu** by pressing the `VARS` key.
3. **Select the inverse normal function**:
- Scroll to `invNorm(` or just press `2`.
4. **Input the area** and the parameters for the standard normal distribution:
- Type `invNorm(0.59)`.
5. **Close the parentheses** and press `ENTER`.
The TI-84 Plus will return the z-score corresponding to an area of \( 0.59 \).
After performing these steps, you should find that the z-score is approximately **0.28** when rounded to two decimal places.
Thus, the z-score for which the area to its left is \( 0.59 \) is **0.28**.
Answered by
GPT-4o mini
To find the z-scores that bound the middle 88% of the area under the standard normal curve using a TI-84 Plus calculator, you can follow these steps:
1. **Calculate the area in each tail**: Since you want to find the z-scores that bound the middle 88% of the area, the remaining area in the tails is \( 100\% - 88\% = 12\% \). Since there are two tails, you divide this by 2:
\[
\text{Area in each tail} = \frac{12\%}{2} = 6\% = 0.06
\]
2. **Find the area to the left of the lower z-score**: This is the area to the left of the lower z-score:
\[
\text{Area to the left of lower z-score} = 0.06
\]
3. **Find the area to the left of the upper z-score**: This is the area to the left of the upper z-score:
\[
\text{Area to the left of upper z-score} = 1 - 0.06 = 0.94
\]
4. **Use invNorm to find the z-scores**:
- For the lower z-score:
- Press `2nd` then `VARS` to access the DISTR menu.
- Select `invNorm(` and enter `0.06`:
\[
\text{Lower z-score} = \text{invNorm}(0.06)
\]
- For the upper z-score:
- Again, go to the DISTR menu.
- Select `invNorm(` and enter `0.94`:
\[
\text{Upper z-score} = \text{invNorm}(0.94)
\]
5. **Calculate the z-scores**:
- After pressing `ENTER` for both, you will find:
- Lower z-score \( \approx -1.55 \)
- Upper z-score \( \approx 1.55 \)
Thus, the z-scores that bound the middle 88% of the area under the standard normal curve are **-1.55** and **1.55** (rounded to two decimal places).
So your final answer is:
- Lower z-score: **-1.55**
- Upper z-score: **1.55**
1. **Calculate the area in each tail**: Since you want to find the z-scores that bound the middle 88% of the area, the remaining area in the tails is \( 100\% - 88\% = 12\% \). Since there are two tails, you divide this by 2:
\[
\text{Area in each tail} = \frac{12\%}{2} = 6\% = 0.06
\]
2. **Find the area to the left of the lower z-score**: This is the area to the left of the lower z-score:
\[
\text{Area to the left of lower z-score} = 0.06
\]
3. **Find the area to the left of the upper z-score**: This is the area to the left of the upper z-score:
\[
\text{Area to the left of upper z-score} = 1 - 0.06 = 0.94
\]
4. **Use invNorm to find the z-scores**:
- For the lower z-score:
- Press `2nd` then `VARS` to access the DISTR menu.
- Select `invNorm(` and enter `0.06`:
\[
\text{Lower z-score} = \text{invNorm}(0.06)
\]
- For the upper z-score:
- Again, go to the DISTR menu.
- Select `invNorm(` and enter `0.94`:
\[
\text{Upper z-score} = \text{invNorm}(0.94)
\]
5. **Calculate the z-scores**:
- After pressing `ENTER` for both, you will find:
- Lower z-score \( \approx -1.55 \)
- Upper z-score \( \approx 1.55 \)
Thus, the z-scores that bound the middle 88% of the area under the standard normal curve are **-1.55** and **1.55** (rounded to two decimal places).
So your final answer is:
- Lower z-score: **-1.55**
- Upper z-score: **1.55**
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