Estimate the Value of a Parameter Using Confidence Intervals
In this chapter we discuss and estimate 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 – ∝)100% of all simple random samples of size n from the population whose parameter is unknown will contain the parameter
constructing a confidence interval for µ, σ unknown
Lower bound: x̄ – (z * σ/√n)
Upper bound: x̄ + (z * σ/√n)
robust procedures
minor departures from normality will not seriously affect the results
constructing a Z-interval
construction of the confidence interval with σ known, using z-scores
margin of error
E = z * (σ/√n)
determining the sample size n
the sample size required to estimate the population mean, µ, with a level of confidence (1 – ∝) * 100% with a specified margin of error, E, is given by
n = [(z*σ)/E]²
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 = {[(x̄ – µ)] / s} / √n
follows Student’s t-distribution with n – 1 degrees of freedom, where x̄ is the sample mean and s is the sample standard deviation
t-statistic
represents the number of sample standard errors x̄ is from the population mean, µ. 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 0, which equal .5.
4. As t increases without bound, the graph approaches, but never equals zero. As t decreases without bound, the graph approaches, but never equals, zero.
5. The area in the tails of the t-distribution is a little greater than the area in the tails of the standard normal distribution, because we are using s as an estimate of σ, thereby introducing further variability into the t-statistic
6. As the sample size n increases, the density cure of t gets closer to the standard normal density curve. This result occurs because, as the sample size increases, the values of s get closer to the value of σ, by the Law of Large Numbers.
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 µ, σ unknown, for a population mean
Lower bound: x̄ – (t * s/√n)
Upper bound: x̄ + (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 p̂
mean: µ-sub-p̂ = p
standard deviation: σ-sub-p̂ = √[(p * (1-p))/n]
provided that np*(1 – p) ≥10
constructing a confidence interval for a population proportion
lower bound: p̂ – z √(p̂ (1-p))/n)
upper bound: p̂ + z √(p̂ (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 = p̂(1 – p̂)(z/E)², rounded up to the next integer, where p̂ is a prior estimate of p.
If a prior estimate of p is unavailable, the sample size required is:
n = 0.25 (z/E)², roundd up to the next integer.