Question
(a) How large a sample must be drawn so that a 95% confidence interval for will have a margin of error equal to 4.6? Population standard deviation o=17.2
Round the critical value to no less than three decimal places.
Round the sample size up to the nearest integer.
Round the critical value to no less than three decimal places.
Round the sample size up to the nearest integer.
Answers
GPT-4o mini
To determine the required sample size for a specific margin of error in a confidence interval, we can use the following formula:
\[
n = \left( \frac{Z \cdot \sigma}{E} \right)^2
\]
Where:
- \( n \) = required sample size
- \( Z \) = Z-score corresponding to the desired confidence level
- \( \sigma \) = population standard deviation
- \( E \) = margin of error
Given:
- Confidence level = 95%
- Population standard deviation (\( \sigma \)) = 17.2
- Margin of error (\( E \)) = 4.6
1. **Determine the Z-score for a 95% confidence level**:
For a 95% confidence level, the Z-score is approximately 1.96.
2. **Substitute the known values into the formula**:
\[
n = \left( \frac{1.96 \cdot 17.2}{4.6} \right)^2
\]
3. **Calculate**:
Calculating the numerator:
\[
1.96 \cdot 17.2 \approx 33.712
\]
Now divide by the margin of error:
\[
\frac{33.712}{4.6} \approx 7.32
\]
Now square the result:
\[
n \approx (7.32)^2 \approx 53.5856
\]
4. **Round up to the nearest whole number**:
Since we always round up for sample size to ensure we have enough data, we find:
\[
n = 54
\]
Therefore, the required sample size is **54**.
\[
n = \left( \frac{Z \cdot \sigma}{E} \right)^2
\]
Where:
- \( n \) = required sample size
- \( Z \) = Z-score corresponding to the desired confidence level
- \( \sigma \) = population standard deviation
- \( E \) = margin of error
Given:
- Confidence level = 95%
- Population standard deviation (\( \sigma \)) = 17.2
- Margin of error (\( E \)) = 4.6
1. **Determine the Z-score for a 95% confidence level**:
For a 95% confidence level, the Z-score is approximately 1.96.
2. **Substitute the known values into the formula**:
\[
n = \left( \frac{1.96 \cdot 17.2}{4.6} \right)^2
\]
3. **Calculate**:
Calculating the numerator:
\[
1.96 \cdot 17.2 \approx 33.712
\]
Now divide by the margin of error:
\[
\frac{33.712}{4.6} \approx 7.32
\]
Now square the result:
\[
n \approx (7.32)^2 \approx 53.5856
\]
4. **Round up to the nearest whole number**:
Since we always round up for sample size to ensure we have enough data, we find:
\[
n = 54
\]
Therefore, the required sample size is **54**.
how do you do this on a TI-84 calculator
GPT-4o mini
To calculate the required sample size for a confidence interval on the TI-84 calculator, you can use a combination of the calculator’s functions. Here’s a step-by-step guide:
1. **Find the Z-score**:
For a 95% confidence level, as mentioned before, the Z-score is approximately 1.96. You can verify this using the calculator:
- Press the `2nd` button.
- Then press `VARS` to access the `DISTR` menu.
- Choose `invNorm`.
- For a 95% confidence level, input `0.975` (since you want the area to the left of the Z-score, you take half of the remaining area, which is 0.5), then press `ENTER`.
- The output should give you approximately `1.96`.
2. **Use the sample size formula**:
You can directly calculate the sample size using the formula on the calculator.
- Press `Y=` to enter the function editor.
- Move to `Y1=` and enter the formula for the sample size:
\[
((1.96 \times 17.2) / 4.6)^2
\]
- This will look like:
\[
((1.96 \times 17.2) / 4.6)^2
\]
- After inputting this, press `GRAPH` to visualize it, or you can simply proceed to calculate it directly.
3. **Calculate** directly:
If you prefer, you can simply compute the value directly:
a. Press the `HOME` button.
b. Type the calculation as follows:
```
((1.96 * 17.2) / 4.6)²
```
c. Press `ENTER`. The calculator will return the computed value.
**Note**: Ensure to use the `^` symbol for squaring on the TI-84. The calculator supports using parentheses to structure your formula properly.
4. **Round up**:
Since you found the calculated result for the sample size, let’s say it gave you approximately 53.5856.
- To get the required integer sample size, round up to the nearest whole number.
- On the TI-84, you can use:
- Press `MATH`, then scroll to `NUM` and select `ceil(` to round up, or simply note that 53.5856 rounds up to 54 manually.
### Summary
After completing these steps, you will find that the required sample size is **54**. You can use your TI-84 calculator for intermediate calculations as needed or for verification of the Z-score.
1. **Find the Z-score**:
For a 95% confidence level, as mentioned before, the Z-score is approximately 1.96. You can verify this using the calculator:
- Press the `2nd` button.
- Then press `VARS` to access the `DISTR` menu.
- Choose `invNorm`.
- For a 95% confidence level, input `0.975` (since you want the area to the left of the Z-score, you take half of the remaining area, which is 0.5), then press `ENTER`.
- The output should give you approximately `1.96`.
2. **Use the sample size formula**:
You can directly calculate the sample size using the formula on the calculator.
- Press `Y=` to enter the function editor.
- Move to `Y1=` and enter the formula for the sample size:
\[
((1.96 \times 17.2) / 4.6)^2
\]
- This will look like:
\[
((1.96 \times 17.2) / 4.6)^2
\]
- After inputting this, press `GRAPH` to visualize it, or you can simply proceed to calculate it directly.
3. **Calculate** directly:
If you prefer, you can simply compute the value directly:
a. Press the `HOME` button.
b. Type the calculation as follows:
```
((1.96 * 17.2) / 4.6)²
```
c. Press `ENTER`. The calculator will return the computed value.
**Note**: Ensure to use the `^` symbol for squaring on the TI-84. The calculator supports using parentheses to structure your formula properly.
4. **Round up**:
Since you found the calculated result for the sample size, let’s say it gave you approximately 53.5856.
- To get the required integer sample size, round up to the nearest whole number.
- On the TI-84, you can use:
- Press `MATH`, then scroll to `NUM` and select `ceil(` to round up, or simply note that 53.5856 rounds up to 54 manually.
### Summary
After completing these steps, you will find that the required sample size is **54**. You can use your TI-84 calculator for intermediate calculations as needed or for verification of the Z-score.