Question

a. 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 (therms) used during the month of January is determined for each house. The resulting observations are

                        103 156 118 89 125 147 122 109 138 99

Let 𝜇 denote the average gas usage during January by all houses in this area. Compute a point estimate of 𝜇

b. Suppose there are 10,000 houses in this area that use natural gas for heating. Let t denote the total amount of gas used by all of these houses during January. Estimate t using the data of part (a). What estimator did you use in computing your estimate?

c. Use the data in part (a) to estimate p, the proportion of all houses that used at least 100 therms.

d. Give a point estimate of the population median usage (the middle value in the population of all houses) based on the sample of part (a). What estimator did you use? 

Answer 1

Solution

a. Compute a point estimate of 𝜇

Let 𝜇 denote the average gas usage gas during: January by all houses in this area

By the formula:

                                             

Then, we get:

b. Estimate 𝒕 using the data of part (a)

Let 𝑡 denote that total amount of gas used by all of these houses during January

                       

In this computing, we use the point estimate of the average gas usage during January by 10 houses in particular area.

c. Use the data in part (a) to estimate 𝒑, the proportion of all houses that used at least 100 therms

By formula :                                   p=x/n

Since: x=8 and n=10

Then:                                             p=8/10=0.8

Therefore,  p=0.8

d. Give a point estimate of the population median usage.

By formula:

                                 

Since: n=10

                         

Therefore,    Me=120