The monthly income of each of the 500 service microenterprises in a city is a random variable with unknown mean μ. In order to simplify tax collection, the SRI has arranged that these companies be taxed monthly with 10% of their income From a random sample of 50 micro-companies, an average monthly income of $ 1000 was obtained with a standard deviation of $ 80.

a. Estimate the mean amount of income of the city's microenterprises with a 95% confidence interval

 b. Estimate the average amount of collection for these microenterprises with a 95% confidence interval

c. If the purpose of the SRI is to achieve a total monthly collection of at least $ 52,000 to these microenterprises, is it feasible that its goals are met?

Answer:

a) The confidence interval is (977.26, 1022.74)

b) The confidence interval is (97.73, 102.27)

c) It is not feasible to meet the collection goal of at least $ 52,000

Step-by-step explanation

To construct a confidence interval, we use the structure: ?x±t∗S E? . Where:

?x? is the midpoint of the interval
t is the critical value
SE is the standard error

The midpoint is represented by the sample mean: ?x=1 0 0 0?

The standard error is: ?S E=s/n=80/(50)^1/2=1 1 . 3 1 4?

To find the critical value, we will use the t-student distribution (degrees of freedom: n - 1 = 50 - 1 = 49):

In EXCEL, we select a cell and place the command: "= INV.T ((1 + 0.95) / 2.49)" and the software shows t = 2.01

The confidence interval is:

?1 0 0 0±2 . 0 1∗1 1 . 3 1 4=( 1 0 0 0-2 . 0 1∗1 1 . 3 1 4 ,1 0 0 0+2 . 0 1∗1 1 . 3 1 4 )=( 9 7 7 . 2 6 ,1 0 2 2 . 7 4 )?

To find the confidence interval of the collection, we calculate the 10% of the extremes of the previous interval:

(977.26 * 0.1, 1022.74 * 0.1) = (97.73, 102.27)

The average collection per company is between $ 97.73 and $ 102.27. If we multiply these numbers by 500 (the number of companies) we obtain an average total collection between $ 48,865 and $ 51,137. So it is not feasible that the fundraising goal of at least $ 52,000 is met