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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.

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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!