Questions: High-rent districts The mean monthly rent for a one-bedroom apartment without a doorman in Manhattan is 2699. Assume the standard deviation is 513. A real estate firm samples 103 apartments. Use Excel. Part 1 of 5 (a) What is the probability that the sample mean rent is greater than 2769? Round the answer to at least four decimal places. The probability that the sample mean rent is greater than 2769 is 0.0831. Part 2 of 5 (b) What is the probability that the sample mean rent is between 2568 and 2668? Round the answer to at least four decimal places. The probability that the sample mean rent is between 2568 and 2668 is 0.2651. Part 3 of 5 (c) Find the 55th percentile of the sample mean. Round the answer to at least two decimal places. The 55th percentile of the sample mean rent is 2705.35. Part 4 of 5 (d) Would it be unusual if the sample mean were greater than 2795? Round the answer to at least four decimal places. (Choose one) because the probability that the sample mean is greater than 2795 is

High-rent districts The mean monthly rent for a one-bedroom apartment without a doorman in Manhattan is 2699. Assume the standard deviation is 513. A real estate firm samples 103 apartments. Use Excel. Part 1 of 5 (a) What is the probability that the sample mean rent is greater than 2769? Round the answer to at least four decimal places. The probability that the sample mean rent is greater than 2769 is 0.0831. Part 2 of 5 (b) What is the probability that the sample mean rent is between 2568 and 2668? Round the answer to at least four decimal places. The probability that the sample mean rent is between 2568 and 2668 is 0.2651. Part 3 of 5 (c) Find the 55th percentile of the sample mean. Round the answer to at least two decimal places. The 55th percentile of the sample mean rent is 2705.35. Part 4 of 5 (d) Would it be unusual if the sample mean were greater than 2795? Round the answer to at least four decimal places. (Choose one) because the probability that the sample mean is greater than 2795 is
Transcript text: High-rent districts The mean monthly rent for a one-bedroom apartment without a doorman in Manhattan is $\$ 2699$. Assume the standard deviation is $\$ 513$. A real estate firm samples 103 apartments. Use Excel. Part 1 of 5 (a) What is the probability that the sample mean rent is greater than $\$ 2769$ Round the answer to at least four decimal places. The probability that the sample mean rent is greater than $\$ 2769$ is 0.0831 . Part 2 of 5 (b) What is the probability that the sample mean rent is between $\$ 2568$ and $\$ 2668$ Round the answer to at least four decimal places. The probability that the sample mean rent is between $\$ 2568$ and $\$ 2668$ is 0.2651 . Part 3 of 5 (c) Find the $55^{\text {th }}$ percentile of the sample mean. Round the answer to at least two decimal places. The $55^{\text {th }}$ percentile of the sample mean rent is $\$ 2705.35$. Part 4 of 5 (d) Would it be unusual if the sample mean were greater than $\$ 2795$ Round the answer to at least four decimal places. (Choose one) because the probability that the sample mean is greater than $\$ 2795$ is
failed

Solution

failed
failed

Solution Steps

Step 1: Probability that the Sample Mean Rent is Greater than $2769

To find the probability that the sample mean rent is greater than \( \$2769 \), we calculate the Z-score for \( 2769 \):

\[ Z_{end} = \frac{2769 - 2699}{\frac{513}{\sqrt{103}}} \approx 1.3848 \]

Using the cumulative distribution function \( \Phi \), we find:

\[ P = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(\infty) - \Phi(1.3848) = 0.0831 \]

Thus, the probability that the sample mean rent is greater than \( \$2769 \) is \( 0.0831 \).

Step 2: Probability that the Sample Mean Rent is Between $2568 and $2668

Next, we calculate the probability that the sample mean rent is between \( \$2568 \) and \( \$2668 \). We find the Z-scores for both bounds:

\[ Z_{start} = \frac{2568 - 2699}{\frac{513}{\sqrt{103}}} \approx -2.5916 \] \[ Z_{end} = \frac{2668 - 2699}{\frac{513}{\sqrt{103}}} \approx -0.6133 \]

Using the cumulative distribution function \( \Phi \):

\[ P = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(-0.6133) - \Phi(-2.5916) = 0.2651 \]

Thus, the probability that the sample mean rent is between \( \$2568 \) and \( \$2668 \) is \( 0.2651 \).

Step 3: Finding the 55th Percentile of the Sample Mean

To find the 55th percentile of the sample mean, we first calculate the Z-score corresponding to the 55th percentile:

\[ Z_{55} \approx 0.12566134685507416 \]

Next, we calculate the standard error of the mean:

\[ SE = \frac{513}{\sqrt{103}} \approx 50.5 \]

Now, we can find the 55th percentile of the sample mean:

\[ P_{55} = 2699 + Z_{55} \cdot SE \approx 2705.35 \]

Thus, the 55th percentile of the sample mean rent is \( 2705.35 \).

Final Answer

The results are summarized as follows:

  • The probability that the sample mean rent is greater than \( \$2769 \) is \( 0.0831 \).
  • The probability that the sample mean rent is between \( \$2568 \) and \( \$2668 \) is \( 0.2651 \).
  • The 55th percentile of the sample mean rent is \( 2705.35 \).

\[ \boxed{P > 2769: 0.0831, \quad P(2568 < X < 2668): 0.2651, \quad 55^{th} \text{ percentile: } 2705.35} \]

Was this solution helpful?
failed
Unhelpful
failed
Helpful

Questions asked by other users

failedAnswered
Question 38 Isovolumentric relaxation occurs during: none of these early diastole late systole mid-to late diastole Question 39 The portion of the intrinsic conduction system located in the superior interventricular septum is the
failedAnswered
What were investors afraid the Federal Reserve Board would do in March 1929? stop sales of RCA stock make it harder to buy and sell stocks on the stock market force the stock market's paper value to match its real value end its policy of deregulation of the stock market
failedAnswered
A rectangular yard measuring 27 ft by 43 ft is bordered (and surrounded) by a fence. Inside, a walk that is 2 ft wide goes all the way along the fence. Find the area of this walk. Be sure to include the correct unit in your answer.
failedAnswered
If a car is climbing a hill at a 15° angle, how many horizontal feet will the car travel every 1,000 feet of road if the car gains 259 feet of vertical feet? Round your answer to the nearest foot.
failedAnswered
Why are ducks important to freshwater biomes? They clean the water surface. They spread plant seeds. They provide food for plants. They are not important.