Hp 49g+ User Manual download pdf (Page 590)

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101

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to estimate is its mean value, µ. We will use as an estimator the mean value

of the sample, X, defined by (a rule):

∑

⋅=

For the sample under consideration, the estimate of µ is the sample statistic x

= (2.2+2.5+2.1+2.3+2.2)/5 = 2.36. This single value of X, namely x =

2.36, constitutes a point estimation of the population parameter µ.

Estimation of Confidence Intervals

The next level of inference from point estimation is interval estimation, i.e.,

instead of obtaining a single value of an estimator we provide two statistics, a

and b, which define an interval containing the parameter θ with a certain

level of probability. The end points of the interval are known as confidence

limits, and the interval (a,b) is known as the confidence interval.

Definitions

Let (C

) be a confidence interval containing an unknown parameter θ.

• Confidence level or confidence coefficient is the quantity (1-α), where 0 <

α < 1, such that P[C

< θ < C

] = 1 - α, where P[ ] represents a

probability (see Chapter 17). The previous expression defines the so-

called two-sided confidence limits.

• A lower one-sided confidence interval is defined by Pr[C

< θ] = 1 - α.

• An upper one-sided confidence interval is defined by Pr[θ < C

] = 1 - α.

• The parameter α is known as the significance level. Typical values of α

are 0.01, 0.05, 0.1, corresponding to confidence levels of 0.99, 0.95,

and 0.90, respectively.

Confidence intervals for the population mean when the

population variance is known

Let X be the mean of a random sample of size n, drawn from an infinite

population with known standard deviation σ. The 100(1-α) % [i.e., 99%,

95%, 90%, etc.], central, two-sided confidence interval for the population

mean µ is (X−z

⋅σ/√n , X+z

⋅σ/√n ), where z

is a standard normal

variate that is exceeded with a probability of α /2. The standard error of

the sample mean, X, is ⋅σ/√n.

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