You want to rent an unfurnished one-bedroom apartment in Boston next year. The mean monthly rent for a random sample of 14 apartments advertised in the local newspaper is $1200. Assume that the standard deviation is $230. Find a 95% confidence interval for the mean monthly rent for unfurnished one-bedroom apartments available for rent in this community. (Round your answers to two decimal places.)
I did the work and got ($ 1079.48, $ 1320.50 ) but it says that this answer is wrong.
CI95 = mean ± 1.96 (sd/√n)
With your data:
CI95 = 1200 ± 1.96 (230/√14)
Calculating...
1200 + 120.48 = 1320.48
1200 - 120.48 = 1079.52
I found approximately the same answer you did.
Thank you!
To find the 95% confidence interval for the mean monthly rent for unfurnished one-bedroom apartments in Boston, you can use the formula:
CI = X ± (Z * (σ/√n))
Where:
X = sample mean,
Z = Z-score corresponding to the desired confidence level (for 95% confidence, Z = 1.96),
σ = population standard deviation,
n = sample size.
Given the following information:
Sample mean (X) = $1200,
Standard deviation (σ) = $230,
Sample size (n) = 14.
Calculating the confidence interval:
CI = 1200 ± (1.96 * (230/√14))
CI = 1200 ± (1.96 * (230/√14))
CI = 1200 ± (1.96 * 230/√14)
CI = 1200 ± (1.96 * 230/√14)
CI = 1200 ± (1.96 * 230/√14)
CI = 1200 ± (1.96 * 230/√14)
CI = 1200 ± (1.96 * 230/√14)
CI = 1200 ± (1.96 * 230/√14)
CI = 1200 ± 164.56
CI = (1200 - 164.56, 1200 + 164.56)
CI = ($1035.44, $1364.56)
Therefore, the 95% confidence interval for the mean monthly rent for unfurnished one-bedroom apartments available for rent in this community is ($1035.44, $1364.56).
To calculate the 95% confidence interval for the mean monthly rent for unfurnished one-bedroom apartments in Boston, you can use the formula:
Confidence Interval = (Sample Mean) +/- (Critical Value) * (Standard Deviation / sqrt(Sample Size))
Let's calculate it step by step:
Step 1: Given data
Sample Mean (x̄) = $1200
Standard Deviation (σ) = $230
Sample Size (n) = 14
Step 2: Find the critical value
Since the sample size is small (n < 30) and the population standard deviation is unknown, we need to use a t-distribution. The critical value will depend on the desired confidence level and the degrees of freedom, which is n-1. In this case, the degrees of freedom is 14-1 = 13.
To find the critical value, we refer to the t-distribution table or use a calculator. For a 95% confidence level with 13 degrees of freedom, the critical value is approximately 2.160.
Step 3: Calculate the standard error
The standard error represents the standard deviation of the sample mean and is calculated by dividing the standard deviation by the square root of the sample size.
Standard Error (SE) = σ / sqrt(n)
SE = 230 / sqrt(14)
Step 4: Calculate the margin of error
The margin of error is found by multiplying the critical value by the standard error.
Margin of Error (ME) = Critical Value * SE
ME = 2.160 * (230 / sqrt(14))
Step 5: Calculate the confidence interval
Finally, you can calculate the confidence interval by adding and subtracting the margin of error from the sample mean.
Confidence Interval = Sample Mean ± Margin of Error
CI = $1200 ± (2.160 * (230 / sqrt(14)))
Now let's perform the calculations:
CI = $1200 ± (2.160 * (230 / sqrt(14)))
CI ≈ $1072.7 to $1327.3
Rounding to two decimal places, the 95% confidence interval for the mean monthly rent for unfurnished one-bedroom apartments available for rent in Boston is approximately ($1072.7, $1327.3).
Please note that there might be a computational difference, but the steps and concept explained here should help you arrive at the correct answer.