Math 2200

8 Confidence Intervals

Definition: Point estimate
For a parameter, is the value of a statistic that is used to estimate the parameter.

Definition: Confidence interval
An interval of numbers obtained from a point estimate of the parameter, along with a percentage that specifies how confident we are that the parameter lies in the interval.

Definition: Confidence level
The percentage, as stated in confidence interval.

Theorem: Assume a large sample ( n ³ 30)
1 Find za/2 for a confidence level of 1 - a .
2. The confidence interval for m is:    - za/2(s /Ö n)    to    + za/2(s /Ö n)
If s is unknown, use s.

Theorem: For a fixed n, a greater confidence level (1- a) means a greater confidence interval length.


Definition: Margin of error (E)
For an estimate of m ,  E = za/2(s/Ö n) (i.e., 1/2 the length of the confidence interval)

Theorem: The sample size n required for a 1 - a confidence level for m is: n = ((za/2s)/E)2 rounded to the nearest whole number.

Student's t- DISTRIBUTION

NOTE - The t-curve is very similar to the z-curve. However, you only use it when dealing with confidence intervals.

Definition: Student's t-distribution
t = ( - m)/(s/Ö n) for a population that is normally distributed.

Theorem: Suppose a random sample of size n is taken from a normally distributed population with mean m. Then t = ( - m)/(s/Ö n) has the t-distribution with n - 1 degrees of freedom (df).

Properties of t curves:
i. total area underneath is 1.
ii. as t goes to + and - infinity, the t-curve approaches the horizontal axis asymptotically.
iii. symmetric about zero.
iv. as df increases, the t-curve approaches the normal curve.

Procedure for determining confidence:
Assume a normal population (with sample size = n )
1 Find ta/2 for a confidence level of 1 - a, with df = n - 1
2. The confidence interval for m is: - ta/2(s/Ö n) to + ta/2(s/Ö n)

When to use t-distribution:
• If probability plot shows outliers or population is far from normal, don’t use
• If outliers are present, but their removal is justified.
• The smaller the sample size, the more normal should be the distribution. Larger sample sizes can deviate slightly.
• ***If population is normal, use it.
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