### 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.