Question

The data below give the dusted gross income and the amount of Bemized deductions taken by taxpayeny. Data wereported You may need to use the thousands of dolar Adjusted Gross Income ($1,000) Itemized Deductions ($1,000s) 22 9.6 27 9.6 32 10.1 48 11.1 65 13.5 BS 17.7 120 23.5 with the estimated regression equation 9 = 5.33 +0.14%, the point estimate of a reasonable level of total itemized deductions for a taxpayer with an adjusted gross income of $47,500 is $11.980. (a) Develop a 95% confidence interval for the mean amount of total itemized deductions (in thousands of dollars) for all taxpayers with an adjusted gross income of $47,500 (Round your answers to two decimal places.) thousand to $ thousand $ (b) Develop a 95% prediction interval estimate for the amount of total itemized deductions (in thousands of dolars) for a particular taxpayer with an adjusted gross income of $47,500. (Round your answers to two decimal places.) thousand to $ thousand (c) of the particular taxpayer referred to in part (b) claimed total itemized deductions of $18,400, would the IRS agent's request for an audit appear to be justified? Since the predicted amount of temized deductions is significantly greater than the actual claimed itemized deductions, the agent's request for an audit appears to be justified. Since the predicted amount of Remzed deductions is roughly the same as the actual claimed itemized deductions, the agent's request for an audit does not appear to be justified. Since the predicted amount of itemized deductions is significantly less than the actual claimed itemized deductions, the agent's request for an audt appears to be justified. Since the predicted amount of iterized deductions is roughly the same as the actual claimed itemized deductions, the agent's request for an audit appears to be justified. Since the predicted amount of Iterized deductions is significantly greater than the actual claimed itemized deductions, the agent's request for an audit does not appear to be justified. (d) Use your answer to part (b) to give the IRS agent a guideline as to the amount of total itemized deductions (in thousands of dollars) a taxpayer with an adjusted gross income of $47,500 could claim before an audit is recommended. (Round your answer to two decimal places.) Any deductions exceeding the $ thousand -Select-limt could suggest an audit.

please help

Answer 1

X	Y	XY	X²	Y²
22	9.6	211.2	484	92.16
27	9.6	259.2	729	92.16
32	10.1	323.2	1024	102.01
48	11.1	532.8	2304	123.21
65	13.5	877.5	4225	182.25
85	17.7	1504.5	7225	313.29
120	23.5	2820	14400	552.25

Ʃx =	399
Ʃy =	95.1
Ʃxy =	6528.4
Ʃx² =	30391
Ʃy² =	1457.33
Sample size, n =	7
x̅ = Ʃx/n = 399/7 =	57
y̅ = Ʃy/n = 95.1/7 =	13.58571429
SSxx = Ʃx² - (Ʃx)²/n = 30391 - (399)²/7 =	7648
SSyy = Ʃy² - (Ʃy)²/n = 1457.33 - (95.1)²/7 =	165.3285714
SSxy = Ʃxy - (Ʃx)(Ʃy)/n = 6528.4 - (399)(95.1)/7 =	1107.7

Slope, b = SSxy/SSxx = 1107.7/7648 = 0.14

y-intercept, a = y̅ -b* x̅ = 13.58571 - (0.14)*57 = 5.33

Regression equation :

ŷ = 5.33 + (0.14) x

Predicted value of y at x = 47.5

ŷ = 5.33 + (0.14) * 47.5 = 11.98

Sum of Square error, SSE = SSyy -SSxy²/SSxx = 165.32857 - (1107.7)²/7648 = 4.89456

Standard error, se = √(SSE/(n-2)) = √(4.89456/(7-2)) = 0.9894

a)

Critical value, t_c = T.INV.2T(0.05, 5) = 2.5706

95% Confidence interval :

Lower limit = ŷ - tc*se*√((1/n) + ((x-x̅)²/(SSxx)))

= 11.98 - 2.5706*0.9894*√((1/7) + ((47.5 - 57)²/(7648))) = 10.98

Upper limit = ŷ + tc*se*√((1/n) + ((x-x̅)²/(SSxx)))

= 11.98 + 2.5706*0.9894*√((1/7) + ((47.5 - 57)²/(7648))) = 12.98

b)

95% Prediction interval :

Lower limit = ŷ - tc*se*√(1 + (1/n) + ((x-x̅)²/(SSxx)))

= 11.98 - 2.5706*0.9894*√(1 + (1/7) + ((47.5 - 57)²/(7648))) = 9.25

Upper limit = ŷ + tc*se*√(1 + (1/n) + ((x-x̅)²/(SSxx)))

= 11.98 + 2.5706*0.9894*√(1 + (1/7) + ((47.5 - 57)²/(7648))) = 14.71

c)

Since the predicted amount of temized deductions is significantly greater than the actual claimed itemized deductions, the agent's request for an audit appears to be justified.

d)

Any deductions exceeding the upper limit of $ 1471 could suggest an audit.