Revised: April 20, 2020
Brian Earp is associate director of ethics and health at Yale University and an anti-circumcision activist. In 2018 he led a study that produced startling results about American men's knowledge of circumcision and penile anatomy.
Earp reported that the results showed an inverse correlation between knowledge and satisfaction. In other words, the more satisfied a man was about circumcision, the greater the likelihood that he held false beliefs about circumcision. Conversely, the more dissatisfied a man was about circumcision, the greater the likelihood that his beliefs about the procedure were true. The study was appropriate titled, "False beliefs predict increased circumcision satisfaction in a sample of US American men". 
The researchers declared, "These findings provide tentative support for the hypothesis that the lack of harm reported by many circumcised men... may be related to holding inaccurate beliefs concerning [uncircumcised penises] and the consequences of [infant circumcision]."
Penile Anatomy and Circumcision Quiz
A total of 902 men were surveyed. Of the participants, 732 identified themselves as circumcised, and 170 identified themselves as uncircumcised. Readers should be wary of studies based on self-identification, which can be unreliable. A 2002 study of 1,508 adolescent boys found that 23% of circumcised males and 31% of uncircumcised males didn't know their status. For those subjects who didn't know - even after viewing images of a circumcised penis and an uncircumcised penis, just 82% of the circumcised males and 43% of the uncircumcised males were able to correctly identify their status. 
For reasons unstated the researchers excluded men who were circumcised at age 1 or older. This restriction may have formed asymmetry in the group. Men who suffered a foreskin complication that circumcision corrects (e.g. phimosis, UTIs), might be more likely to have a positive view of circumcision; such men were automatically excluded. In contrast, men who suffered a complication commonly linked to circumcision (e.g. meatal stenosis, skin bridges) might be more likely to have a negative view of the procedure; apparently such men were not automatically excluded. 
The measure of beliefs was based on a set of ten statements, each of which was determined to be either true or false. Earp recruited two outside experts - one pro-circumcision, the other anti-circumcision - to review the statements, verify that there was only one correct answer to each question, and suggest any changes to avoid ambiguity. He further stated that "we needed to make sure that there was a reasonable mix of ‘positive’ and ‘negative’ claims about circumcision (and the foreskin), so that participants would not feel that there was a bias to the questions one way or the other, which might raise suspicions or possible experimenter demand issues."
Let's take a look at the set of statements.
According to the American Academy of Pediatrics, there is good evidence that being circumcised is associated with a lower incidence of urinary tract infections (UTIs) in boys under the age of 2. [TRUE]
If a baby boy is not circumcised shortly after birth, he will most likely require a circumcision anyway to correct medical problems before he turns 18. [FALSE]
The percentage of men who are circumcised in the United States is significantly greater than the percentage of men who are circumcised in most other English-speaking countries. [TRUE]
The foreskin is typically the least sensitive part of the penis to light touch. [FALSE]
After birth, a boy who has not been circumcised should have his foreskin “retracted” or pulled back as soon as possible to facilitate cleaning. [FALSE]
Some forms of non-therapeutic (ritual) female genital cutting that are described by the World Health Organization (WHO) as "mutilation" are less physically invasive than male circumcision as it is typically performed in the United States. [TRUE]
Most medical associations around the world that have issued statements on routine newborn male circumcision have concluded that the foreskin does not have any functions. [FALSE]
In the United States, during the late 19th and early 20th centuries, circumcision was advocated by mainstream doctors as a preventative measure against—or a “cure” for - masturbation. [TRUE]
According to most authoritative sources, approximately 100 circumcisions would be needed to prevent 1 urinary tract infection (UTI) among boys with normally developing anatomy. [TRUE]
Most medical associations around the world that have issued statements on routine newborn male circumcision recommend the procedure. [FALSE]
Irrelevant trivia and capricious limitations
One concern that stands out is that some of the questions may not be relevant to circumcision knowledge. How is knowledge of female anatomy germaine to a measurement of knowledge of circumcision and penile anatomy? How is a 150 year-old "cure" for masturbation germane to contemporary circumcision knowledge? I also question the relevance of light touch (a.k.a. fine touch) sensitivity, which is not the type of sensitivity typically involved in sexual performance and pleasure. 
Other questions contain capricious limitations. Take statement #3, which compares the percentage of circumcised American men to the percentage of circumcised men in other English-speaking countries. Why limit the comparison to other English speakers? Does circumcision affect men differently based on which language they speak? In fact, why even assume that all American men speak English? More than 20% of people who live in the United States speak a language other than English in their homes.  Note that if this arbitrary limitation were removed, statement #3 could not be considered true.
British researcher Stephen Moreton found half of the statements in the test to be "irrelevant, ambiguous, or incorrect." He explained that statement #7 is "misleading," #8 is "largely false," #9 is "arguably false," and #10 is "an oversimplification."  Combining Moreton's analysis with this one, 7 of the 10 statements that Earp used to measure circumcision knowledge are problematic.
A measurement of confirmation bias rather than knowledge?
My biggest concern is the truth value of the list, which is 90% negative toward circumcision. For 9 of the 10 statements, correct knowledge would support a belief in the foreskin as beneficial or circumcision as harmful. Only correct knowledge of statement #1 would support a belief in circumcision as beneficial or the foreskin as harmful. And even that statement - about a reduction in UTI risk - is neutered by a later statement (#9) that seems to minimize the value of this benefit. Moreton also noticed this slant, concluding, "It is difficult to escape the perception that the statements have been chosen to suit the first author’s longstanding anti-circumcision agenda." 
The researchers believe that they measured knowledge of circumcision. In fact they may have measured confirmation bias, which is a tendency to embrace information that supports one’s views while rejecting information that conflicts with one’s views. Psychologist Raymond S. Nickerson explained that, "Philosophers and psychologists alike have observed that people find it easier to believe propositions they would like to be true than propositions they would prefer to be false." 
In other words, a plausible interpretation of the study would reverse Earp's interpretation. True or false beliefs don't predict satisfaction; but satisfaction predicts which beliefs - true or false - one might accept. The men with high circumcision satisfaction found it easier to believe statements supporting circumcision than statements supporting foreskin. Conversely the men with low circumcision satisfaction found it easier to believe statements supporting foreskin than statements supporting circumcision.
Based on this interpretation, the set of statements itself is inherently flawed. Each response is based on whether a statement confirms or contradicts a man's view of circumcision as positive or negative. I suspect that men with a high level of circumcision satisfaction would perform better on a quiz that contained a set of statements whose overall truth value provides positive reinforcement of circumcision - rather than negative reinforcement.
Given these concerns about the problematic statements and unbalanced truth values, it is unreasonable to say that this study tests relevant knowledge of circumcision and penile anatomy.
An alternate quiz to rule out confirmation bias
There is a simple way to test which interpretation - Earp's or mine - is correct. One could conduct a similar test in which the truth value for a set of statements is weighted heavily in favor of circumcision.
Consider the following list of statements about circumcision and penile anatomy. To address Earp's concern, the list contains a reasonable mix of positive and negative statements. But in contrast to his list, correct knowledge for 9 out of these 10 statements would support a belief that circumcision is beneficial (or that the foreskin is potentially harmful.)
In order to perform a circumcision procedure, it is typically necessary for the penis to be in a state of erection. [FALSE]
Medicaid covers the cost of elective newborn circumcision in approximately 35 out of 50 states in the U.S. [TRUE]
Most private insurance plans in the United States do not cover adult elective circumcision, which typically costs from $2,000-$4,000. [TRUE]
A study in Canada found that the foreskin is typically the most sensitive area of the penis to heat and pain. [FALSE]
According to the World Health Organization, about 70% of males worldwide are not circumcised. [TRUE]
A study in Denmark reported that 1 out of every 20 boys under 18 suffered a foreskin complication that required treatment at a medical center. [TRUE]
According to most authoritative sources, circumcision status does not affect the rate of urinary tract infections in the first year of life. [FALSE]
Most medical organizations around the world that have issued statements on routine newborn male circumcision have concluded that the procedure doesn't provide any medical benefits [FALSE]
Multiple studies show that during the first year of a boy's life, circumcision reduces by up to 90% his risk of a urinary tract infection, one of the causes of sepsis, which kills almost 7,000 children annually in the United States. [TRUE]
Several experts who say that circumcision increases the force necessary for vaginal penetration relied on a study in which the researcher had sexual intercourse with a Styrofoam cup. [TRUE]
Skeptics may argue that some of the statements are based on trivial information. I might concede that point - if those skeptics will concede that sex with a Styrofoam cup (a study cited in two dozen academic papers) has no less relevance to contemporary circumcision knowledge than a 19th century cure for masturbation.
I offer this challenge to Brian Earp. Conduct the same study, but measure circumcision knowledge using these ten statements. If the men with negative feelings about circumcision outperform the men with positive feelings, then the results will support his conclusion that false beliefs predict increased circumcision satisfaction. On the other hand, if the men with positive feelings about circumcision outperform the men with negative feelings, then the results will support my interpretation that circumcision satisfaction predicts true or false beliefs.
I suspect that a group of men who hold a positive view of circumcision would perform significantly better on this quiz. Either way, the results would contribute to the body of knowledge about circumcision.
Stephen Moreton's analysis of Earp's study was published just one week after this post. He provides valuable additional criticism and a second set of statements that Earp could use to test circumcision knowledge.
 Brian D. Earp et al; "False beliefs predict increased circumcision satisfaction in a sample of US American men"; Culture, Health & Sexuality; July 2018
 Risser, et al; “Self-Assessment of Circumcision Status by Adolescents”; American Journal of Epidemiology; 2004
 The decision to exclude men circumcised at ages above 12 months may also affect the religious demographics of the group. There is no set age for Muslim circumcisions, and Muslim boys are typically circumcised up to the teenage years. One study in Pakistan found that the median age for circumcision is one year. If the median age in the U.S. is similar, then Earp may have eliminated from the study half of the prospective circumcised Muslims. Abdul Wahid Anwer; "Reported Male Circumcision Practices in a Muslim-Majority Setting"; BioMed Research International; January 17, 2017  "Of all the nerves"; Circumcision Choice; March 3, 2018
 Christopher Ingraham; "Millions of U.S. citizens don’t speak English to one another. That’s not a problem."; Washington Post; May 21, 2018. "As of 2016, 35 million U.S. citizens over the age of 18, or more than 15 percent of the adult population, speak a language other than English at home, according to the U.S. census... If you include noncitizen residents, the odds are even higher: 66 million U.S. residents ages 5 and older - 21.6 percent of that population - speak a language other than English at home.
 Stephen Moreton; "Do False Beliefs Predict Increased Circumcision Satisfaction in Men?"; Advances in Sexual Medicine; April 17, 2020
[7 Raymond S. Nickerson; "Confirmation Bias: A Ubiquitous Phenomenon in Many Guises"; Review of General Psychology, p 197; 1998