http://www.ck12.org Chapter 7. Sampling Distributions and Estimations
Example:
The following questions are to be answered with reference to the above diagram.
a) Were all four sample means within 1. 96 √σn, or 1. 96 σx ̄, of the population mean? Explain.
b) Did all four confidence intervals capture the population mean? Explain.
c) In general, what percentage of ̄x′s should be within 1. 96 √σnof the population mean?
d) In general, what percentage of the confidence intervals should contain the population mean?
Solution:
a) The sample mean(x ̄)for Sample 3 is not within 1. 96 √σnof the population mean. It does not fall within the two
vertical lines on the left and right of the sampling distribution of the sample mean.
b) The confidence interval for Sample 3 does not enclose the population mean. This interval is just to the left of
the population mean(μ), which is labeled as the vertical line found in the middle of the sampling distribution of the
sample mean.
c) 95%
d) 95%
When the sample size is large(n≥ 30 ), the confidence interval for the population mean is calculated as shown
below:
x ̄±z√σ
n
wherezis 1.96 for a 95% confidence interval; 1.645 for a 90% confidence interval and 2.56 for a 99%
confidence interval.