1. ## Statistics: Point Estimations

A random sample of 10 houses in a particular area, each of which is heated with natural gas, is selected and the amount of gas (in therms) used during the month of January is determined for each house. The resulting observations are as follows:

103, 156, 118, 89, 125, 147, 122, 109, 138, 99

a. Let μj denote the average gas usage during January by all houses in this area. Compute a point estimate of μj.
b. Suppose that 10,000 houses in this area use natural gas for heating. Let
τ denote the total amount of gas used by all of these houses during January. Estimate τ using the given data. What statistic did you use in computing your estimate?
c. Use the given data to estimate π, the proportion of all houses that used at least 100 therms.
d. Give a point estimate of the population median usage based on the given sample. Which statistic did you use?

Thanks for the help!

2. ## Response

I hope it's not too late.

A random sample of 10 houses in a particular area, each of which is heated with natural gas, is selected and the amount of gas (in therms) used during the month of January is determined for each house. The resulting observations are as follows:

103, 156, 118, 89, 125, 147, 122, 109, 138, 99

a. Let μj denote the average gas usage during January by all houses in this area. Compute a point estimate of μj.
b. Suppose that 10,000 houses in this area use natural gas for heating. Let
τ denote the total amount of gas used by all of these houses during January. Estimate τ using the given data. What statistic did you use in computing your estimate?
c. Use the given data to estimate π, the proportion of all houses that used at least 100 therms.
d. Give a point estimate of the population median usage based on the given sample. Which statistic did you use?

a. So, for part a. you need to find the average gas usage. You can do this by adding up all the above values and dividing by 10. It would look like:

uj = (103+ 156+ 118+ 89+ 125+ 147+ 122+ 109+ 138+ 99)/10
uj = 120.6

b. To estimate Tau, I think you could just multiply the average usage, denoted uj, by the number 10,000 to get an estimate of the amount of gas used
So it would be Tau = 120.6 * 10,000 = 1206000 units

c. To find Pi, the proportion of all houses that used at least 100 therms, you could examine the above data and count the number. Then divide it by the total for the sample, which was 10.
It would be all houses using greater than or equal to 100:
Pi = 8/10 = 4/5

d. Finally, for the median usage, you would just examine the above data and find the middle range value. It would help to put them in ascending order:
89, 99, 103, 109, 118, 122, 125, 138, 147, 156

Since we have an even number of values, we take the average of the two middles, which is:
(118 + 122) /2 = 120