Answer to Question #179569 in Statistics and Probability for Mariam

Question

2. The Ghana Revenue Authority (GRA) publishes annual figures on individual income tax returns in
Statistics of Income, Individual Income Tax Returns. For the year 2009, the GRA reported that
the mean tax of individual income tax returns was GH₡ 10,319. In actuality, the GRA reported
the mean tax of a sample of 292,966 individual income tax returns from a total of more than 130
million such returns.
a. Identify the population under consideration.
b. Identify the variable under consideration.
c. Is the mean tax reported by the GRA a sample mean or the population mean?
d. Should we expect the mean tax

Answer 1

a) The population is more than 130 million individual income tax returns.

b) The variable is individual income tax return.

c) The mean tax reported by the GRA is a sample mean (a sample of 292,966 individual income tax returns).

d) If the population standard deviation, sigma is known, then the mean has a normal (Z) distribution:

$E=Z_{\alpha/2}\sigma/\sqrt{n}$

$\overline{x}-E<\mu<\overline{x}+E$ (confidence interval)

where $\overline{x}$ is sample mean, $\mu$ is population mean.

If the population standard deviation, sigma is unknown, then the mean has a student's t (t) distribution:

$E=t_{\alpha/2}s/\sqrt{n}$

where $s$ is sample standard deviation.

$\overline{x}-E<\mu<\overline{x}+E$

If population mean $\mu$ lies in the confidence interval, then we can expect claimed mean.

In our case we don't know any standard deviation, so we cannot say anything about expected mean.

If we consider sample size, then there is a rule that a good maximum sample size is usually around 10% of the population, as long as this does not exceed 1000.

In our case we have sample size $n>1000$ , that is enough for population mean.

So, we can say that we can expect the claimed mean tax.