Sample Size and Statistical theory - Interval Estimation
Interval Estimation
The sample mean, X, is used to estimate the unknown population mean (p.). Because X varies from sample to sample, it is not, of course, equal to the population mean (u-). There is a sampling error. It is useful to provide an interval estimate around X that reflects our judgment of the extent of this sampling error:
X ± sampling error = the interval estimate of u.
The size of the interval will depend on how confident we want to be tha: the interval contains the true unknown population mean. If it were necessary to be 95 percent confident that the interval estimate contained the true population mean, the interval estimate would be
2o-„
X ± 28^ = X ± —f= = 95 percent interval estimate of p.
(recall that ox = a^/Vri). The interval size is based on 28* because, as Figure 12-3 shows, the probability that X will be within 28* of the population mean is 0.95. In our example, the interval would be
X ± 28x- = 0.5 ± 2 x .47 = 0.5 ± .94
a„
sinceS - = —7= = .47. Note that this interval includes the true population V n
mean (recall from Figure 12-1 that p = 0.3). About 95 percent of samples will generate an interval estimate that will include the true population mean.
If the desire were to be 90 percent confident that the interval estimate
included the true population mean, then the interval estimate would be
X ± f (CTx) = X ± ^j— = 90 percent interval estimate of p.
Again, the interval is based on % (cr^) because, as shown in Figure 12-3, there is a 0.90 probability that X is within % (o^) of the true population mean, p. In our example, the 90 percent interval estimate is
X ± § (oi) = 0.5 ± 5 (.47) = 0.5 ± .78
Note that the interval is smaller, but that we are less confident that it would include the true population mean.
If the population standard deviation (o^ = a) is not known, it is necessary to estimate it with the sample standard deviation, s. Thus, the 95 percent interval estimate would be
2s
X ± ~~7= = 95 percent interval estimate with cr unknown Vn
In our example, it would be
/ 1.27 \ .5 ± 2 (—7=) = .5 ± .80 V VTO /
since from Figure 12-2, s was determined to be 1.27.
X ± sampling error, or X ±
where
z = 2 for a 95 percent confidence level z = | for a 90 percent confidence level
To summarize, the interval estimate of the population mean, p, can be written as
ux = population standard deviation (s is used if ux is unknown) n = the sample size
Thus, the size of the interval estimate will depend on three factors. The first is the confidence level. If we are willing to be less confident that the interval estimate will include the true unknown population mean, then the interval will be smaller. The second factor is the population standard deviation. If there is little variation in the population, then the interval estimate of the population mean will be smaller. The third is the sample size. As the sample size gets larger, the sampling error is reduced and the interval will get smaller.
The sample mean, X, is used to estimate the unknown population mean (p.). Because X varies from sample to sample, it is not, of course, equal to the population mean (u-). There is a sampling error. It is useful to provide an interval estimate around X that reflects our judgment of the extent of this sampling error:
X ± sampling error = the interval estimate of u.
The size of the interval will depend on how confident we want to be tha: the interval contains the true unknown population mean. If it were necessary to be 95 percent confident that the interval estimate contained the true population mean, the interval estimate would be
2o-„
X ± 28^ = X ± —f= = 95 percent interval estimate of p.
(recall that ox = a^/Vri). The interval size is based on 28* because, as Figure 12-3 shows, the probability that X will be within 28* of the population mean is 0.95. In our example, the interval would be
X ± 28x- = 0.5 ± 2 x .47 = 0.5 ± .94
a„
sinceS - = —7= = .47. Note that this interval includes the true population V n
mean (recall from Figure 12-1 that p = 0.3). About 95 percent of samples will generate an interval estimate that will include the true population mean.
If the desire were to be 90 percent confident that the interval estimate
included the true population mean, then the interval estimate would be
X ± f (CTx) = X ± ^j— = 90 percent interval estimate of p.
Again, the interval is based on % (cr^) because, as shown in Figure 12-3, there is a 0.90 probability that X is within % (o^) of the true population mean, p. In our example, the 90 percent interval estimate is
X ± § (oi) = 0.5 ± 5 (.47) = 0.5 ± .78
Note that the interval is smaller, but that we are less confident that it would include the true population mean.
If the population standard deviation (o^ = a) is not known, it is necessary to estimate it with the sample standard deviation, s. Thus, the 95 percent interval estimate would be
2s
X ± ~~7= = 95 percent interval estimate with cr unknown Vn
In our example, it would be
/ 1.27 \ .5 ± 2 (—7=) = .5 ± .80 V VTO /
since from Figure 12-2, s was determined to be 1.27.
X ± sampling error, or X ±
where
z = 2 for a 95 percent confidence level z = | for a 90 percent confidence level
To summarize, the interval estimate of the population mean, p, can be written as
ux = population standard deviation (s is used if ux is unknown) n = the sample size
Thus, the size of the interval estimate will depend on three factors. The first is the confidence level. If we are willing to be less confident that the interval estimate will include the true unknown population mean, then the interval will be smaller. The second factor is the population standard deviation. If there is little variation in the population, then the interval estimate of the population mean will be smaller. The third is the sample size. As the sample size gets larger, the sampling error is reduced and the interval will get smaller.
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