2.7 Hypothesis Testing: Binomial Random Variables
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study are simply due to the variation inherent in sampling? [Hint: Compare 90 confidence intervals.]
10. n values of X, the number of successes in a binomial process, are used to compute
n 95 confidence intervals for the unknown parameter p: Find the probability that p lies in exactly k of the n confidence intervals.
2.7
c
Hy pothesis Testing: Binomial Random Variables
In the previous section we considered confidence intervals for binomial random vari- ables. The problem of estimating a parameter, in this case the value of p by means of
an interval, is part of statistics or statistical inference. Statistical inference, in simplest terms, is concerned with drawing inferences from data that have been gathered by a
sampling process. Statistical inference is comprised of the theory of estimation and that of hypothesis testing. In the preceding section, we considered the construction of a
confidence interval, which is part of the theory of estimation. The remaining portion of the theory of drawing inferences from samples is called hypothesis testing. We begin
with a somewhat artificial example in order to fix ideas and define some vocabulary before proceeding to other applications.
Ex ample 2.7.1
The manufacturing process of a sensitive component has been producing items of which 20 must be reworked before they can be used. A recent
sample of 20 items shows 6 items that must be reworked. Has the manu- facturing process changed so that 30 of the items must be reworked?
Assume that the production process is binomial, with p; which is of course unknown to us, denoting the probability an item must be reworked.
We begin with a hypothesis or conjecture about the binomial process, that the process has not in fact changed and that the proportion of items
that must be reworked is 20. We denote this by H
o
and call it the null hypothesis. As a result of a test – in this case the result of a sample of the
items – this hypothesis will be accepted that is we will believe that H
o
is true or it will be rejected that is, we will believe that H
o
is not true. In the latter case, when the null hypothesis is rejected, we agree to accept an
alternative hypothesis, H
a
: Here the hypotheses are chosen as follows:
H
o
: p = 0:20 H
a
: p = 0:30: How are sample results in this case, 6 items that must be reworked to
be interpreted? Does this information lead to the acceptance or the rejection of H
o
? We must decide what sample results lead to the acceptance of H
o
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Chapter 2 Discrete Random Variables and Probability Distributions
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