Page 171 - Statistics for Dummies

Chapter 9: The Normal Distribution
The 50th percentile for the normal distribution is the mean (because of

symmetry) and its z-score is zero. Smaller percentiles, like the 10th, lie below
the mean and have negative z-scores. Larger percentiles, like the 75th, lie
above the mean and have positive z-scores.
Here’s another style of wording that has a bit of a twist: Suppose times to
complete a statistics exam have a normal distribution with a mean of 40 min-
utes and standard deviation of 6 minutes. Deshawn’s time comes in at the
90th percentile. What percentage of the students are still working on their
exams when Deshawn leaves? Because Deshawn is at the 90th percentile, 90
percent of the students have exam times lower than hers. That means 90% of
the students left before Deshawn, so 100 – 90 = 10 percent of the students are
still working when Deshawn leaves.
To be able to decipher the language used to imply a percentile problem, look
for clues like the bottom 10% (also known as the 10th percentile) and the top
10% (also known as the 90th percentile). For the best 10%, you must determine
whether low or high numbers qualify as “best.” 155
Normal Approximation to the Binomial
Suppose you flip a fair coin 100 times and you let X equal the number of
heads. What’s the probability that X is greater than 60? In Chapter 8, you
solve problems like this (involving fewer flips) using the binomial distribu-
tion. For binomial problems where n (the number of trials) is small, you
can either use the direct formula (found in Chapter 8), the binomial table
(found in the appendix), or you can use technology if available (such
as a graphing calculator or Microsoft Excel).
However, if n is large the calculations get unwieldy and the binomial table
runs out of numbers. If there’s no technology available (like when taking an
exam), what can you do to find a binomial probability? Turns out, if n is large
enough, you can use the normal distribution to find a very close approximate
answer with a lot less work.
But what do I mean by n being “large enough”? To determine whether n is
large enough to use what statisticians call the normal approximation to the
binomial, both of the following conditions must hold:
✓ n ∗ p ≥ 10 (at least 10), where p is the probability of success
✓ n ∗ (1 – p) ≥ 10 (at least 10), where 1 – p is the probability of failure

3/25/11 8:16 PM
15_9780470911082-ch09.indd 155 3/25/11 8:16 PM
15_9780470911082-ch09.indd 155