Basic Statistics

SAMPLE SIZE NEEDED FOR A DESIRED CONFIDENCE INTERVAL 83

If a higher confidence level is chosen, we have greater confidence that the interval
contains p; but on the other hand, we pay for this higher level of confidence by having
a longer interval. The confidence levels generally used are 90,95, and 99%.

7.2 SAMPLE SIZE NEEDED FOR A DESIRED CONFIDENCE

INTERVAL

To obtain a short interval and at the same time to have one in which we have a high
level of confidence, we must increase the sample size. In the simplified example
under consideration, we can in the planning stages of the experiment calculate the
length of a 95% confidence interval. Since the interval is xi 1.96a/&, the length
of the entire interval is 2(1.96)a/&;. If we wish this length to be only 60 g, we can
solve the equation

2( 1.96)120

GO = -

dn

2(1.96)(120)

60

fi= = 7.84

n = 61.47 or 62

The required sample size is 62 since in calculating a sample size for a certain
confidence level, one rounds up to a whole number rather than down. In general, if
we call the desired length of the interval L, the formula for n can be written as

where z[Xj is the tabled value from Table A.2 (1.645 for 90%, 1.96 for 95%, and

2.575 for a 99% confidence interval).

Calculations of this sort help us to determine in advance the size of sample needed.

Estimating the necessary sample size is important; there is no “best” sample size

applicable to all problems. Sample size depends on what is being estimated, on the

population standard deviation 0, on the length of the confidence interval, and on the

confidence level.

7.3 THE t DISTRIBUTION

In most research work, the value of the population variance or standard deviation is

unknown and must be estimated from the data. In the example in Section 7.1.1, we

assumed we knew that a equaled 120g and made use of the fact that the quantity

z = (x - p)/(a/fi) has a standard normal distribution whose areas have been

calculated and are available in Table A.2 or from a statistical program. In a more

usual research situation, the population standard deviation a is unknown and must