3. Piyush, who had joined the infrastructure department of the bank, was asked to estimate the rental value of commercial property at Ranchi, for the purpose of identifying suitable premises for locating a new branch. Piyush assessed the standard deviation of the value of the property at s = Rs. 35,000 (the value of the property being normally distributed). A random sample of 16 properties gave a sample mean of Rs. 6,97,630.
a. Give a confidence interval at 95% level of significance for the average value of all properties of this kind. Also interpret the results.
b. Assuming that the area of the property considered is 6,000 sq. ft. and the bank is looking forward to rental values in the range of Rs. 100 per sq. ft.
Are the results arrived in option (a) in line with the expectations of the bank?
To calculate a confidence interval for the average value of all properties, we can use the standard formula:
Confidence Interval = X̄ ± Z * (s / √n)
Where:
- X̄ is the sample mean
- Z is the z-score corresponding to the desired confidence level
- s is the standard deviation of the population
- n is the sample size
In this case, the sample mean (X̄) is Rs. 6,97,630 and the standard deviation (s) is Rs. 35,000. The sample size (n) is 16.
a. To calculate the confidence interval at the 95% level of significance, we need to find the z-score for a 95% confidence level. The z-score for a 95% confidence level is 1.96.
Confidence Interval = 6,97,630 ± 1.96 * (35,000 / √16)
Simplifying the equation:
Confidence Interval = 6,97,630 ± 1.96 * (8,750)
Confidence Interval = 6,97,630 ± 17,100
Therefore, the confidence interval is (Rs. 6,80,530, Rs. 7,14,730).
Interpretation: This means that we are 95% confident that the true average value of all properties of this kind is between Rs. 6,80,530 and Rs. 7,14,730.
b. To check if the results in option (a) are in line with the expectations of the bank, we can compare the confidence interval with the expected range of rental values.
Assuming the area of the property is 6,000 sq. ft. and the bank is looking for rental values in the range of Rs. 100 per sq. ft., we can calculate the expected rental value as follows:
Expected Rental Value = Area * Rental Value per sq. ft.
Expected Rental Value = 6,000 * 100
Expected Rental Value = Rs. 6,00,000
Since the expected rental value of Rs. 6,00,000 falls within the confidence interval of the average property value, the results in option (a) are in line with the expectations of the bank.