Virtually all secondary sources (including Floyd, 1911; see below) accept Anna Maria as a daughter of Andreas & Anna Catharina, though I have yet to find a primary source linking Anna to these parents.  A son of Anna Maria (STRAUB) MARKLE married a HOUGENDOBLER, and a daughter married a FOLLMER.  A son of Andrew STRAUB II also married a FOLLMER.

In the 1790 census, Anna Maria (STRAUB) MARKLE is living in Richmond Twp., Berks Co., PA.  Also living in Richmond Twp. in 1790 is Johannes STRAUB, who is listed next to Widow MERKEL, who is two lines from Casper MERKEL.  Is Johannes a brother of Anna Maria?  It certainly appears that he may be.

I realize some of these sources are highly repetitive.  I include them so you won't waste your time tracking them down, only to find they contain nothing new.  Only males age 16 years or over were included on the ship's passage lists.  There is no proof that the Andreas/Andrew who arrived in 1742 is the one who settled in Lancaster Co., but it is very likely.  This Andreas/Andrew STRAUB is the only one — for whom a record has been found — who arrived early enough to be our subject.

1a.  Israel Daniel Rupp.  1876.  A Collection of upwards of Thirty Thousand Names of German, Swiss, Dutch, French, and Other Immigrants in Pennsylvania from 1727 to 1776... Leary, Stuart, & Co., Philadelphia, PA (reprinted 1965/75 by Genealogical Publ. Co., Baltimore, MD; Broderbund CD-267).

p. 153

Aug. 25, 1742.  Foreigners imported in the brigantine Mary, John Mason, Master, from Rotterdam, last from Cowes... Andreas Straub...

1b.  Israel Daniel Rupp.  1931.  A Collection of Upwards of Thirty Thousand Names of German, Swiss, Dutch, French and Other Immigrants in Pennsylvania from 1727 to 1776, with a Statement of the Names of Ships, Whence They Sailed, and the Date of Their Arrival at Philadelphia, Chronologically Arranged, Together with the Necessary Historical and Other Notes.  Degener & Co., Leipzig, Germany (a revised, enlarged edition of the original 1876 work; 1985, Genealogical Publ. Co., Baltimore; 1998, Broderbund CD-354).

p. 153

Andreas Straub, Philadelphia, 1742.

2a.  William Henry Egle, ed.  1890.  Pennsylvania Archives:  Names of Foreigners Who Took the Oath of Allegiance to the Province and State of Pennsylvania, 1727-1775, with the Foreign Arrivals, 1786-1808.  Ser. 2, Vol. 17.  E.K. Meyers, Harrisburg, PA (1967, Genealogical Publ. Co., Baltimore; 1998, Broderbund CD-354).

p. 227

Andreas Straub, Pennsylvania, 1742.

2b.  William Henry Egle.  1890.  Names of Foreigners Who Took the Oath of Allegiance to the Province and State of Pennsylvania, 1727-1775, with the Foreign Arrivals, 1786-1808.  Pennsylvania Archives, Vol. XVII, 2nd Ser., Harrisburg, PA (reprinted 1967/94, Genealogical Publ. Co., Baltimore, MD; Broderbund CD-267):

p. 227

List of Foreigners Imported in the Brigatine (sic) Mary, John Mason, Master, from Roterdam.  Qualified Aug. 25, 1742... Andreas Straub...

3a.  Ralph Beaver Strassburger.  1934.  Pennsylvania German Pioneers.  Pennsylvania German Society, Norristown, PA (online at Ancestry.com):

List 92 C; at the Courthouse Philada, Augt 25th 1742; the Foreigners whose Names are underwritten, imported in the Brigt Mary, John Mason, Master, from Rotterdam, but last from Cowes, did this day take the usual Oaths or Qualifications to the Government, viz... Andreas STRAUB.

3b.  Ralph Beaver Strassburger.  1934.  Pennsylvania German Pioneers: a Publication of the Original Lists of Arrivals in the Port of Philadelphia, from 1727 to 1808.  Pennsylvania German Society, Norristown, PA (reprinted 1966/92 Genealogical Publ Co., Baltimore, MD; Broderbund CD-267).

p. 322

[List 92 C] At the Courthouse Philada, Augt 25th 1742.   Present: Clement Plumsted, Esqr. Mayor.            Robert Strettell,|            William Till,    | Esqrs.            Abraham Taylor   |   The Foreigners whose Names are underwritten, imported in the Brigt Mary, John Mason, Master, from Rotterdam, but last from Cowes, did this day take the usual Oaths of Qualifications to the Government... Andereas (XX) Straub...

4.   Karl Ehmann.  1977.  Die Auswanderung in die Neuengland-Staaten aus Orten des Enzkreises im 18. Jahrhundert.  Suedwestdeutsche Blaetter fuer Familien- und Wappenkunde, special supplement (1998,  Broderbund CD-354).  On p. 36:

Andreas Straub, America, 1742.

5a.   Pennsylvania German Society.  1983.  Pennsylvania German Church Records: Birth and Baptismal Register of Trinity Lutheran Church, Lancaster, Pa.  Genealogical Publ. Co., Baltimore, MD (1996, Broderbund CD-130).  On p. 12:

p. 4	1748	Andreas Straub, s. Andreas and Catharina; b. February; bap. April 24.
p. 12	1750	Valentin STRAUBE, s. Andreas, Cath., and Anna Catharina, Luth.; b. Oct. 18, 1749; bap. April 16.
p. 70	1766	Maria Eva STRAUB, d. Andreas and Catharina; b. Jan. 23; bap. May 18.

5b.  John T. Humphrey.  1991-1998.  Pennsylvania Births.  14 vols.  Self-published, Washington, DC (Broderbund Family Archives: Birth Index: Southeastern PA, 1680-1800, CD-196):

Child	Andreas Straub	Valentine Straube	Maria Eva Straub
Event	Birth/Baptism	Birth/Baptism	Birth/Baptism
Event Date	Feb 1748	18 Oct 1749	23 Jan 1766
Church	Trinity Lutheran Church	Trinity Lutheran Church	Trinity Lutheran Church
Locaiton	Lancaster	Lancaster	Lancaster
County	Lancaster	Lancaster	Lancaster
State	Pennsylvania	Pennsylvania	Pennsylvania
Father	Andreas Straub	Andreas Straube	Andreas Straub
Mother	Mrs. Catharina Straub	Mrs. Anna Catharina Straube	Mrs. Catharina Straub

6.  From a post by Gwen Boyer Borkman in the Archives of the STRAUB-L mailing list (online at RootsWeb.com):

Hawbaker and Groff: A New Index: Lancaster County, Pennsylvania Before the Federal Census, Volume 3, Index to the 1750 Tax Records, pp. 99-102. STROUT, ANDW. Hempfield twp, 1756  ["Laborrows"]

7.  From a statewide index of 1780 Pennsylvannia tax lists in the Denver Public Library, p. 176 (extracted by Jim Burnell):

1780 Tax List

Strout, Andreas

Lancaster Co.

Hempfield

8.  Anon.  1911.  Genealogical & Biographical Annals of Northumberland County.  J.L. Floyd & Co., Chicago. Biographical sketch of subject's grandson, Jacob STRAUB and his kin, which states that "the emigrant" married in America, settled in Columbia, and lived out his days there, with the implication that his children were born there.  Jacob names his grandfather's children as Andrew, Valentine, Mrs. HOUGENDOBLER,  and Mrs. MERKLE [the former being Maria Eva, the latter Anna Maria] — no son, Johannes, is mentioned, but as Johannes was deceased before Jacob was born, it's possible Jacob wouldn't have remembered this uncle.