Questions: Question 4 of 10 (1 point) Question Attempt 1 of Unlimited High-rent districts The mean monthly rent for a one-bedroom apartment without a doorman in Manhattan is 2699. Assume the standard deviation is 3513. 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. 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. Part 3 of 5 (c) Find the 55th percentile of the sample mean. Round the answer to at least two decimal places. 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

Question 4 of 10 (1 point) Question Attempt 1 of Unlimited High-rent districts The mean monthly rent for a one-bedroom apartment without a doorman in Manhattan is 2699. Assume the standard deviation is 3513. 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. 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. Part 3 of 5 (c) Find the 55th percentile of the sample mean. Round the answer to at least two decimal places. 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: Question 4 of 10 (1 point) | Question Attempt 1 of Unlimited High-rent districts The mean monthly rent for a one-bedroom apartment without a doorman in Manhattan is $2699. Assume the standard deviation is 3513. 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. 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. Part 3 of 5 (c) Find the $55^{\text{th}}$ percentile of the sample mean. Round the answer to at least two decimal places. 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
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Solution

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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:

\[ P(X > 2769) = 1 - P(X \leq 2769) = 1 - \Phi(Z_{end}) \]

Where \( Z_{end} \) is the Z-score corresponding to \$2769. The calculated probability is:

\[ P(X > 2769) = 0.5801 \]

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

Next, we determine the probability that the sample mean rent is between \$2568 and \$2668:

\[ P(2568 < X < 2668) = P(X < 2668) - P(X < 2568) = \Phi(Z_{end}) - \Phi(Z_{start}) \]

The calculated probability for this range is:

\[ P(2568 < X < 2668) = 0.1118 \]

Step 3: 55th Percentile of the Sample Mean

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

\[ Z_{55} = 0.1257 \]

Then, we use this Z-score to find the corresponding sample mean:

\[ X_{55} = \mu + Z_{55} \cdot \sigma_{X} = 2699 + 0.1257 \cdot \left(\frac{3513}{\sqrt{103}}\right) \]

The calculated 55th percentile of the sample mean rent is:

\[ X_{55} = 2742.5 \]

Final Answer

The results for each part are as follows:

  • Probability that the sample mean rent is greater than \$2769: \\(\boxed{0.5801}\\)
  • Probability that the sample mean rent is between \$2568 and \$2668: \\(\boxed{0.1118}\\)
  • 55th percentile of the sample mean rent: \\(\boxed{2742.50}\\)
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