Hello.
Okay, I'm going to assume normal distribution. If that's not okay,
please use the "request clarification" feature to let me know.
To calculate the p-values, one must consult a standard normal table,
such as this one from San Jose State University.
http://www.sjsu.edu/faculty/gerstman/EpiInfo/z-table.htm
a. Right-tail test and Z = 1.54 .
When we look up 1.54 in the normal table, we see a value of 0.9382 ,
this value is equal to the entire area under the normal curve to the
left of 1.54. The p-value for the right tail test is the area to the
right of 1.54. Thus, calculating the area to the right of 1.54 is
simply a matter of subtracting 0.9382 from 1.
p-value = 1 - 0.9382 = 0.0618
b. Left-tail test and Z = -1.03 .
When we look up the absolute value of -1.03 (i.e., 1.03) in the normal
table, we see a value of 0.8485 , this value is equal to the entire
area under the normal curve to the left of 1.03. The p-value for the
left tail test is the area to the left of -1.03 (which is also equal
to the area to right of 1.03 because the normal curve is symetrical.
Thus, calculating the area to the left of -1.03 is simply a matter of
subtracting 0.8485 from 1.
p-value = 1 - 0.8485 = 0.1515
c. Two-tail test and Z = -1.83 .
When we look up the absolute value of -1.83 (i.e., 1.83) in the normal
table, we see a value of 0.9664 , this value is equal to the entire
area under the normal curve to the left of 1.83. The p-value for the
two tail test is the area to the left of -1.83, plus the area to the
right of 1.83. Thus, calculating the area to the right of 1.83 and
the left of -1.83 is simply a matter of subtracting 0.9664 from 1, and
then multiplying by 2.
p-value = 2 * (1 - 0.9664) = 2 * (0.0336) = 0.0672
sources for the concepts discussed:
"In a left-tail test, the P-value is the area under the normal curve
to the left of x: had we chosen the significance level p so that zp =
x, we would have rejected the null hypothesis, but we would not have
rejected it for any smaller value of p, because for all smaller values
of p, xp < x. Similarly, for a right-tail z test, the P-value is the
area under the normal curve to the right of x. For a two-tail z test,
the P-value is the sum of the area under the normal curve to the left
of -|x| and the area under the normal curve to the right of |x|."
Source: "Approximate Hypothesis Tests: the z-test and the t-test,"
hosted by Berkeley.edu:
http://stat-www.berkeley.edu/users/stark/SticiGui/Text/ch22.htm
"Lesson 19 - Part II: Testing Hypotheses," hosted by southwestern.edu
http://www.southwestern.edu/~owensp/Statintro/Lectures/19b/19b.htm
search strategy: "left tail", "right tail", "two tail"
I hope this helps. |
