Statistics Ch. 9 - Estimating the Value of a Parameter Using Confidence Intervals

point estimate

the value of a statistic that estimates the value of a parameter

confidence interval

an interval of numbers for an unknown parameter

level of confidence

Represents the expected proportion of intervals that will contain the parameter if a large number of different samples is obtained. Denoted (1 - ?) * 100%

margin of error

a measure of how accurate the point estimate is and depends on three factors:
1. level of confidence
2. sample size
3. standard deviation of the population

critical value

the value of the statistic associated with the alpha level. the value that must be exceeded to reach significance.

interpretation of a confidence interval

a (1 - ?)
100% confidence interval indicates that (1 - ?)

constructing a confidence interval for �, ? unknown

Lower bound: x? - (z * ?/?n)
Upper bound: x? + (z * ?/?n)

robust procedures

minor departures from normality will not seriously affect the results

constructing a Z-interval

construction of the confidence interval with ? known, using z-scores

margin of error

E = z * (?/?n)

determining the sample size n

the sample size required to estimate the population mean, �, with a level of confidence (1 - ?) * 100% with a specified margin of error, E, is given by
n = [(z*?)/E]�
where n is rounded
up
to the nearest whole number

Student's t-distribution

If the population from which a simple random sample of size n is drawn from a population follows a normal distribution, the distribution of
t = {[(x? - �)] / s} / ?n
follows Student's t-distribution with n - 1 degrees of freedom, where x? is the sample me

t-statistic

represents the number of sample standard errors x? is from the population mean, �. depends on the sample size, n

properties of the t-distribution

1. The t-distribution is different for different degrees of freedom
2. The t-distribution is centered at 0 and is symmetric about 0
3. The area under the curve is 1. The are under the curve to the right of 0 equals the area under the curve to the left of

z-sub-?

the z-score whose area under the normal curve to the right of z-sub-alpha is ?

t-sub-?

the t-value whose area under the t-distribution to the right of to the right of t-sub-alpha is ?

constructing a confidence interval for �, ? unknown, for a population mean

Lower bound: x? - (t * s/?n)
Upper bound: x? + (t * s/?n)

t-interval

confidence interval using a t-distribution

nonparametric procedures

procedures that do not require normality, and the methods are resistant to outliers

sampling distribution of p?

mean: �-sub-p? = p
standard deviation: ?-sub-p? = ?[(p * (1-p))/n]
provided that np*(1 - p) ?10

constructing a confidence interval for a population proportion

lower bound: p? - z
?(p?
? * (1-p))/n)
upper bound: p? + z
?(p?
? * (1-p))/n)

sample size needed for estimating the population proportion p

The sample size required to obtain a confidence interval for p with a margin of error E is given by
n = p?(1 - p?)(z/E)�, rounded up to the next integer, where p? is a prior estimate of p.
If a prior estimate of p is unavailable, the sample size required