How do I use a Z-Test or a Chi-square Test in the following study
question. A survey of 1,233 visitors to a country last year revealed
that 110 visited a small cafe. It is claimed that 10% of visitors to
this country will include a visit to a cafe. Using a 0.05 significance
level, test the claim.

You can do it using a Z-test.

H0 (null hypothesis): pi=.10 (The percentage of visitors the visit
cafes is actually 10%)
H1: pi != .10

As you stated in your question, a=.05

The test proportion here is p=110/1223=.08994

Your test-statistic should be of the formula z=(p-pi)/sqrt( (pi*(1-pi))/N)

Filling in the data gives:
.08994-.10
-----------
root( .1*.9/1223)

Therefore, the test statistic is 1.0484

Since 1.0484 is less than 1.96 (from a=.05) the null hypothesis cannot
be rejected, and the claim appears to be valid.

Hope this helps!

Oops, I mistyped that on my calculator.  The actual test statistic
should be -1.1727, but this is still between -1.96 and 1.96, so the
null hypothesis is not rejected.

Thank you very much. This helped tremendously. Now that I know how to
apply the Z-Test, I can further gain impact from the study questions.