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