Question

Nikita is a government official looking to find evidence on whether the mean taxable income for an individual taxpayer in the region dropped since the previous year. She surveyed 32 individual taxpayers in the region and found the taxable income of each individual as shown in the data set provided. Instead of using the standard deviation from the survey, Nikita decided to use the census data for the region to assume that the population standard deviation of income is $26,744. The mean taxable income in the region was $62,712 for the previous year.

(a) H0:μ=$62,712; Ha:μ<$62,712, which is a left-tailed test.

(b) Taxable income of each individual is given below.

Use Excel to test whether this year's mean taxable income for an individual taxpayer in the region is less than the mean taxable income from the previous year, and then draw a conclusion in the context of the problem, where α=0.10. Calculate the test statistic, z, rounding to two decimal places, and the p-value, rounding to three decimal places.

Taxable income ($)

Taxable income ($)
12193
64586
60363
59639
103402
24127
53641
38963
78596
49328
33983
64023
49771
89073
59433
21930
53789
30236
41639
87935
73396
69640
69371
104896
112354
96396
83921
72539
52186
91483
32069
17296

Answer 1

The null and alternative hypothesis for the test is given as:

i.e., the true mean taxable income for individual is $62712 and its not dropped since previous year.

i.e., the true mean taxable income for individual is less than $62712 and its dropped since previous year.

Since the population standard deviation is given, i.e., , so we have to use a Z-statistic.

where,

sample mean of 32 taxpayers taxable income

sample size

population standard deviation

population mean

Sample mean

Calculation for Z-statistic:

The test-statistic is calculated as Z=-0.36

p-value:

Decision:

Since,

In words, at the data does not provide enough evidence to support the alternative hypothesis, i.e., . So we FTR(fail to reject) null hypothesis, . Since we FTR(fail to reject) H₀ , so we can conclude that the true mean taxable income for the individual taxpayers is not dropped since the previous year.