3. A tax department official is interested in determining the mean income of all

Question

3. A tax department official is interested in determining the mean income of all individuals in a particular profession. Suppose there are 5250 such individuals in total of whom 4750 are females and 500 are males. Because there are relatively few male individuals, the tax official believes that a simple random sample might understate the earnings of the males. She thus decides to undertake a stratified random sample of 750 individuals: 500 females and 250 males. The following income data (in S1000s) was obtained: l(Females) 500 31,500 2,056,356 2(Mals 250 68,000 20,089,600 (a) (2 marks). Was the sample of 750 individuals allocated proportionately between the two strata? Verify. (b) (5 marks). Estimate the mean income of individuals in this particular profession and place a bound on the error of estimation

Explanation / Answer

a)

Answer: NO  

Ratio of number male to number of female: Population – 500 : 4750 = 2 : 19

                                                                       Sample       - 250 : 500 = 1 : 2

b)

Mean income = 1000 x [{2(68000/250)} + {19(31500/500)}]/21

= 1000 x {(544 + 1197)/21}

= 1000 x 82.90

= $82900 ANSWER

c)

To test if the mean income is 80000.

Let X = income of individuals in the particular profession

Then, we assume X ~ N(µ, 2)

Claim: Mean income individuals in the particular profession is $80000

Hypotheses:

Null H0: µ = µ0 = 80000 Vs    Alternative HA: µ 80000

Test statistic:

t = (n)(Xbar - µ0)/s, where n = sample size; Xbar = sample average; s = sample standard deviation.

Summary of Excel Calculations is given below:

s2 = {(2056356 + 20089600)/750} – (82.90)2

= 29527.9413 – 6872.41

= 22655.5313

s = 150.5175

t = (750)(82.90 - 80)/150.5175

= 0.5276

Distribution, Critical Value and p-value

Under H0, t ~ tn – 1 [i.e., t-distribution with (n -1) degrees of freedom)

Critical value = upper (/2)% point of tn – 1.

Using Excel Function, tcrit = 1.96 [taking = 5%]

Decision Criterion (Rejection Region)

Reject H0, if | tcal | > tcrit

Decision:

Since | tcal | < tcrit, H0 is accepted.

Conclusion:

There is sufficient evidence to support the claim that the mean income individuals in the particular profession is $80000

DONE

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