Question

A family is relocating from St. Louis, Missouri, to California. Due to an increasing inventory of houses in St. Louis, it is taking longer than before to sell a house. The wife is concerned and wants to know when it is optimal to put their house on the market. Her realtor friend informs them that the last 18 houses that sold in their neighborhood took an average time of 100 days to sell. The realtor also tells them that based on her prior experience, the population standard deviation is 20 days.a)What assumption regarding the population is necessary for making an interval estimate for the population mean?b) Construct the 99% confidence interval for the mean sale time for all homes in the neighborhood. (Round intermediate calculations to at least 4 decimal places. Round "z" value to 3 decimal places and final answers to 2 decimal places.)

Answer 1

The necessary assumption for estimating the population mean is normal distribution, especially for small samples. For a 99% confidence interval with a mean of 100 days, a standard deviation of 20 days, and a sample size of 18, we calculate a range of (87.85, 112.15) days.

Step-by-step explanation:

99% Confidence Interval for Mean Sale Time

To answer the student's query, we'll first discuss the assumption necessary for making the interval estimate, and then we'll construct the confidence interval for the average sale time.

Population Assumption

The assumption necessary for making an interval estimate for the population mean is that the population from which the sample is drawn is normally distributed. This assumption is crucial, especially when the sample size is small (n < 30). However, due to the Central Limit Theorem, if the sample size is large (n ≥ 30), the distribution of the sample mean will tend to be normally distributed regardless of the population distribution.

Confidence Interval Calculation

To calculate a 99% confidence interval for the mean sale time for all homes, we will use the formula for a confidence interval when the population standard deviation (σ) is known:

Confidence Interval = μ ± (Z* σ / √n)

Where: μ is the sample mean, Z* is the z-value corresponding to the confidence level, σ is the population standard deviation, and n is the sample size.

For our case with a mean (μ) of 100 days, a population standard deviation (σ) of 20 days, and a sample size (n) of 18, we need to find the Z value for 99% confidence, which is approximately 2.576 (rounded to three decimal places).

Using these values, the confidence interval is calculated as follows:

100 ± (2.576 * 20 / √18) = 100 ± (2.576 * 20 / 4.2426)

After calculations:

100 ± 12.15 = (87.85, 112.15)

Therefore, the 99% confidence interval for the average sale time is 87.85 to 112.15 days. This means we can be 99% confident that the true mean sale time for all homes in the neighborhood lies within this range.

Answer 2

To construct the 99% confidence interval for the mean sale time of houses in St. Louis, the assumption of a normally distributed population or a large enough sample is required. Using a sample mean of 100 days, population standard deviation of 20 days, and sample size of 18 houses, the 99% confidence interval is calculated to be (87.86, 112.14) days.

Step-by-step explanation:

The realtor's question about when it is optimal to put their house on the market in St. Louis, Missouri, assuming a normal distribution of selling times for houses, involves constructing a confidence interval for the population mean sale time.

Assumption for Interval Estimate:

The necessary assumption for making an interval estimate for the population mean is that the sale times for houses are normally distributed or that the sample size is large enough for the Central Limit Theorem to apply.

Constructing the 99% Confidence Interval:

To construct the 99% confidence interval for the mean sale time, we use the sample mean (μ) of 100 days, the population standard deviation (σ) of 20 days, and the sample size (n) of 18 houses. First, find the z-value associated with a 99% confidence level, which is commonly about 2.576. Then use the formula for the confidence interval:

• 
  
• Confidence Interval = μ ± (z * (σ / √n))

The calculation will be:

1. 
   
3. Standard Error (SE) = σ / √n = 20 / √18 = 4.7140 approximately.
4. 
   
6. Margin of Error (ME) = z * SE = 2.576 * 4.7140 = 12.1406 approximately.
7. 
   
9. Lower Limit of CI = μ - ME = 100 - 12.14 = 87.86 approximately.
10. 
    
12. Upper Limit of CI = μ + ME = 100 + 12.14 = 112.14 approximately.

Therefore, the 99% confidence interval for the mean sale time for all homes in the neighborhood is (87.86, 112.14) days.