This looks like the right family:
Name: H J Lutz
Age in 1860: 36
Birth Year: abt 1824
Birthplace: Virginia
Home in 1860: Lamoine, McDonough, Illinois
Gender: Male
Post Office: Colmer
Value of real estate: View image
Household Members:
Name Age
H J Lutz 36
Sarah J Lutz 32
Saml B Lutz 12
Phlanie R Lutz 9
William D Lutz 7
Lucretta Lutz 4
Rachael L Lutz 2

"United States Census, 1870," Henry J Lutz in household of Henry J Lutz, Illinois, United States
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About this collection
name: Henry J Lutz
estimated birth year: 1824
gender: Male
age in 1870: 46y
color (white, black, mulatto, chinese, indian): White
birthplace: Virginia
home in 1870: Illinois, United States
Household Gender Age Birthplace
Henry J Lutz M 46y Virginia
Sarah J Lutz F 41y Virginia
Samuel B Lutz M 21y Virginia
Floris R Lutz M 19y Ohio
Wm S Lutz M 17y Ohio
Susan L Lutz F 14y Ohio (She seems to have gone by Susan, too)
Rachel Lutz F 11y Illinois
John C Lutz M 9y Illinois
Sarah Lutz F 5y Illinois

You can see by the birth places that the family started in Virginia, where Samuel was born, then moved to Ohio where 3 children were born, and then back to Illinois where 3 more children were born. The transcriptions on these are pretty bad. If you go to the source you can see that it isn't very accurate.