Question
The mean monthly rent for a one-bedroom apartment without a doorman in Manhattan is 2557. Assume the standard deviation is 486. A real estate firm samples 86 apartments. Use the TI-84 Plus calculator.
Answers
GPT-4o mini
To analyze the data regarding the monthly rent for a one-bedroom apartment in Manhattan, we can perform a few statistical calculations using the given information about mean, standard deviation, and sample size. Here’s how you can do this step-by-step on a TI-84 Plus calculator:
### Given Data:
- Mean rent (\( \mu \)) = 2557
- Standard deviation (\( \sigma \)) = 486
- Sample size (\( n \)) = 86
### Calculating the Standard Error of the Mean (SEM)
The standard error of the mean (SEM) is calculated using the formula:
\[
SEM = \frac{\sigma}{\sqrt{n}} = \frac{486}{\sqrt{86}}
\]
1. **Turn on the calculator.**
2. **Calculate \( \sqrt{86} \):**
- Press `√` (the square root key), then `86`, and hit `ENTER`.
- This will give you approximately **9.273**.
3. **Divide the standard deviation by the square root of the sample size:**
- Now perform the division: `486 ÷ 9.273` and hit `ENTER`.
- This will give you the Standard Error of the Mean (SEM).
### Finding the Confidence Interval
If you’d like to find a confidence interval or perform hypothesis testing, you can follow these additional steps:
### Example: 95% Confidence Interval
1. **Find the critical value (z-value) for a 95% confidence level:**
- For a 95% confidence level, you can use a z-table or a calculator. The z-value is approximately **1.96**.
2. **Calculate the margin of error (ME):**
\[
ME = z \times SEM
\]
- Enter `1.96 * (result from SEM calculation)` and hit `ENTER`.
3. **Calculate the confidence interval:**
- Lower limit: \( \mu - ME \)
- Enter `2557 - (result from ME)` and hit `ENTER` for the lower limit.
- Upper limit: \( \mu + ME \)
- Enter `2557 + (result from ME)` and hit `ENTER` for the upper limit.
### Summary of Results
After completing these calculations, you will have:
- The **Standard Error of the Mean (SEM)**
- The **Margin of Error (ME)**
- The **95% Confidence Interval** for the mean rent.
Feel free to perform these calculations and let me know if you need help with specific steps, or if you'd like to calculate a different confidence level or perform other analyses!
### Given Data:
- Mean rent (\( \mu \)) = 2557
- Standard deviation (\( \sigma \)) = 486
- Sample size (\( n \)) = 86
### Calculating the Standard Error of the Mean (SEM)
The standard error of the mean (SEM) is calculated using the formula:
\[
SEM = \frac{\sigma}{\sqrt{n}} = \frac{486}{\sqrt{86}}
\]
1. **Turn on the calculator.**
2. **Calculate \( \sqrt{86} \):**
- Press `√` (the square root key), then `86`, and hit `ENTER`.
- This will give you approximately **9.273**.
3. **Divide the standard deviation by the square root of the sample size:**
- Now perform the division: `486 ÷ 9.273` and hit `ENTER`.
- This will give you the Standard Error of the Mean (SEM).
### Finding the Confidence Interval
If you’d like to find a confidence interval or perform hypothesis testing, you can follow these additional steps:
### Example: 95% Confidence Interval
1. **Find the critical value (z-value) for a 95% confidence level:**
- For a 95% confidence level, you can use a z-table or a calculator. The z-value is approximately **1.96**.
2. **Calculate the margin of error (ME):**
\[
ME = z \times SEM
\]
- Enter `1.96 * (result from SEM calculation)` and hit `ENTER`.
3. **Calculate the confidence interval:**
- Lower limit: \( \mu - ME \)
- Enter `2557 - (result from ME)` and hit `ENTER` for the lower limit.
- Upper limit: \( \mu + ME \)
- Enter `2557 + (result from ME)` and hit `ENTER` for the upper limit.
### Summary of Results
After completing these calculations, you will have:
- The **Standard Error of the Mean (SEM)**
- The **Margin of Error (ME)**
- The **95% Confidence Interval** for the mean rent.
Feel free to perform these calculations and let me know if you need help with specific steps, or if you'd like to calculate a different confidence level or perform other analyses!