Question
Find the critical value zva/2 needed to construct a confidence interval with level 99.5%
Answers
GPT-4o mini
To find the critical value \( z_{\alpha/2} \) for a confidence level of 99.5%, we need to determine the value that corresponds to the tails of the standard normal distribution.
1. **Identify the significance level**:
- A confidence level of 99.5% has a significance level \( \alpha = 1 - 0.995 = 0.005 \).
- Since \( z_{\alpha/2} \) represents the critical value for the upper tail of the standard normal distribution, we need to find \( \alpha/2 \):
\[
\alpha/2 = 0.005 / 2 = 0.0025
\]
2. **Find the critical z-value**:
- We are looking for the z-value such that the area to the right of it is 0.0025. This is equivalent to finding the z-value that leaves 0.9975 to the left:
\[
P(Z < z_{\alpha/2}) = 0.9975
\]
3. **Using the standard normal distribution table or calculator**:
- Using a standard normal distribution table or an inverse normal calculator, we look for the z-value that corresponds to 0.9975.
- This value is typically found to be approximately \( z_{0.0025} \approx 2.576 \).
Therefore, the critical value \( z_{\alpha/2} \) needed to construct a confidence interval with a confidence level of 99.5% is:
\[
z_{\alpha/2} \approx 2.576
\]
1. **Identify the significance level**:
- A confidence level of 99.5% has a significance level \( \alpha = 1 - 0.995 = 0.005 \).
- Since \( z_{\alpha/2} \) represents the critical value for the upper tail of the standard normal distribution, we need to find \( \alpha/2 \):
\[
\alpha/2 = 0.005 / 2 = 0.0025
\]
2. **Find the critical z-value**:
- We are looking for the z-value such that the area to the right of it is 0.0025. This is equivalent to finding the z-value that leaves 0.9975 to the left:
\[
P(Z < z_{\alpha/2}) = 0.9975
\]
3. **Using the standard normal distribution table or calculator**:
- Using a standard normal distribution table or an inverse normal calculator, we look for the z-value that corresponds to 0.9975.
- This value is typically found to be approximately \( z_{0.0025} \approx 2.576 \).
Therefore, the critical value \( z_{\alpha/2} \) needed to construct a confidence interval with a confidence level of 99.5% is:
\[
z_{\alpha/2} \approx 2.576
\]
how do you use a TI-84 calculator to find this
GPT-4o mini
To find the critical value \( z_{\alpha/2} \) for a confidence level of 99.5% using a TI-84 calculator, you can follow these steps:
1. **Turn on your TI-84 calculator.**
2. **Access the distribution menu:**
- Press the `2nd` button, then `VARS` (which is the `DISTR` option). This will open the distribution menu.
3. **Select the inverse normal function:**
- Scroll down to `invNorm(` (often option 2). Select it by pressing `ENTER`.
4. **Input the area:**
- Since you need to find \( z_{\alpha/2} \) for a right tail area of \( 0.0025 \), you need to input the cumulative area to the left of the critical value. As explained earlier, this is \( 1 - 0.0025 = 0.9975 \).
- Type in `0.9975` as the argument:
\[
invNorm(0.9975)
\]
5. **Press `ENTER`:**
- After typing in `invNorm(0.9975)`, press `ENTER` to get the result.
6. **Interpret the result:**
- The calculator will display the \( z \)-value corresponding to the area of 0.9975, which will be roughly \( 2.576 \).
### Example Input on the TI-84:
1. `2nd` -> `VARS`
2. Scroll to `2:invNorm(` and press `ENTER`
3. Type `0.9975` and then press `)`, so it reads `invNorm(0.9975)`
4. Press `ENTER` to see the output.
This should yield the critical z-value you need for a 99.5% confidence level.
1. **Turn on your TI-84 calculator.**
2. **Access the distribution menu:**
- Press the `2nd` button, then `VARS` (which is the `DISTR` option). This will open the distribution menu.
3. **Select the inverse normal function:**
- Scroll down to `invNorm(` (often option 2). Select it by pressing `ENTER`.
4. **Input the area:**
- Since you need to find \( z_{\alpha/2} \) for a right tail area of \( 0.0025 \), you need to input the cumulative area to the left of the critical value. As explained earlier, this is \( 1 - 0.0025 = 0.9975 \).
- Type in `0.9975` as the argument:
\[
invNorm(0.9975)
\]
5. **Press `ENTER`:**
- After typing in `invNorm(0.9975)`, press `ENTER` to get the result.
6. **Interpret the result:**
- The calculator will display the \( z \)-value corresponding to the area of 0.9975, which will be roughly \( 2.576 \).
### Example Input on the TI-84:
1. `2nd` -> `VARS`
2. Scroll to `2:invNorm(` and press `ENTER`
3. Type `0.9975` and then press `)`, so it reads `invNorm(0.9975)`
4. Press `ENTER` to see the output.
This should yield the critical z-value you need for a 99.5% confidence level.