Inference Homework Supplement: Apartment Rental Rates

Inference Homeworksupplement 1 Apartment Rental Rates You Want To

Inference Homework Supplement – 1: Apartment Rental Rates: You want to rent an unfurnished two-bedroom apartment in Dallas next year. The mean monthly rent for a random sample of 10 apartments advertised in the local newspaper is $1050. Assume that the monthly rate in Dallas follows a Normal distribution with a standard deviation of $220. Find a 95% confidence interval for the mean monthly rent for unfurnished two-bedroom available in Dallas.

Paper For Above instruction

The task is to determine a 95% confidence interval for the average monthly rent of unfurnished two-bedroom apartments in Dallas, based on a small sample. The data indicate that the sample mean rent is $1050, with a standard deviation (population standard deviation) of $220, and a sample size of 10 apartments.

Understanding the context, the objective is to estimate the true mean rent in the population within a range that has a 95% confidence level. When working with normally distributed data and known population standard deviation, the confidence interval calculation employs the z-distribution.

The formula for the confidence interval for the mean when population standard deviation is known is:

\[ \bar{x} \pm z^* \times \frac{\sigma}{\sqrt{n}} \]

where:

- \( \bar{x} \) is the sample mean

- \( z^* \) is the z-value corresponding to the confidence level

- \( \sigma \) is the population standard deviation

- \( n \) is the sample size

Given data:

- \( \bar{x} = 1050 \)

- \( \sigma = 220 \)

- \( n = 10 \)

Next, determine the z-value for a 95% confidence interval. The z-value for 95% confidence (two-tailed) is approximately 1.96.

Calculating the standard error (SE):

\[ SE = \frac{\sigma}{\sqrt{n}} = \frac{220}{\sqrt{10}} \approx \frac{220}{3.1623} \approx 69.53 \]

Constructing the confidence interval:

\[ 1050 \pm 1.96 \times 69.53 \]

\[ 1050 \pm 136.29 \]

Therefore:

- Lower bound:

\[ 1050 - 136.29 \approx 913.71 \]

- Upper bound:

\[ 1050 + 136.29 \approx 1186.29 \]

Concluding, the 95% confidence interval for the average monthly rent of unfurnished two-bedroom apartments in Dallas is approximately \$913.71 to \$1186.29. This indicates that, with 95% confidence, the true mean rent in Dallas for such apartments falls within this range, providing valuable information for prospective renters or real estate analysts.

This interval accounts for the variability inherent in sampling and assumes the population's rent distribution remains normal, which is reasonable given the central limit theorem and the problem's assumptions. Such insights facilitate better decision-making concerning rent expectations and financial planning in the Dallas housing market.

References

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