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34% of the area falls between 0 and +1 standard deviations from the mean. So, 68% of the observations fall within +/- one standard deviation of the mean.
45% of the area falls between 0 and +1.65 standard deviations from the mean. So, 90% of the observations fall within +/- 1.65 standard deviations of the mean.
47.5% of the area falls between 0 and 1.96 standard deviations from the mean. So, 95% of the observations fall within +/- 1.96 standard deviations of the mean.
m: Construct confidence intervals for a normally distributed random variable.
Often we must use an approximation of µ and σ with the sample mean and sample standard deviation denoted as "X bar" and "s." These are point estimates. We often frame probability statements for a random variable using confidence intervals that are built around these point estimates. Some important examples of confidence intervals are below. We should note the similarity between these and the statements above using µ and σ.
P(X will be within X bar +/- 1.65 * s) = 90%. We say: The 90 percent confidence interval for X is X bar - 1.65 * s to x bar + 1.65 * s.
P(X will be within X bar +/- 1.96 * s) = 95%. We say: The 95 percent confidence interval for X is X bar - 1.96 * s to X bar + 1.96 * s.
P(X will be within X bar +/- 2.58 * s) = 99%. We say: The 99 percent confidence interval for X is X bar - 2.58 * s to X bar + 2.58 * s.
