Chapter 8: INTERVAL ESTIMATE

Point Estimator

A function of the random sample used to make inferences about the value of an unknown population parameter.

When examining the possible outcome of an election, what type of confidence interval is most suitable for estimating the current support for a candidate?

Confidence interval for the population proportion

interval estimate

Confident interal

Two basic methodologies emerge from the inferential branch of statistic:

Estimation and hypothesis testing

For a given confidence and sample size n, the width of the confidence interval for the population mean is narrower, the greater the population standard deviation σ.

F

The value of the point estimator derived from

a given sample

Margin of errors

accounts for standard error of the estimator and desired confident level of the interval.

Alpha represent for

level of Significant. will not contain M

Point estimator in proportion parameter

p bar

Common construct confident interval

point estimate+ - margin error

t df distribution and z distribution

t df distribution tail is broader than z distribution they are identical it t df get tp infinity

X bar follow the normal distribution when

the underlying population is normally distributed or when the sample size is sufficiently large (n>=30)

X bar follow the normal distribution

underlying population is distributed or the sample size is sufficiently large(n>=30)

Also referred to as an interval estimate

Confident interval

For given confident level and sample size

The larger population standard deviation, the wider the confident interval

For the given confident level and the population standard deviation

The smaller sample size n, the wider the confident interval

The normality condition is evaluated at the sample proportion p bar . So, for constructing a confidence interval for the population proportion p the following is required:

np bar>5 and n(1-p bar)>5

parameter p represent

proportion of success in the population where the success represent the number of outcome

Confident interval

provides a range of values that, with a certain level of confidence, contains the population parameter of interest

Construct Confident interval

Point estimate + or - Margin error

when estimating the population mean, the T distribution is used when

Population variance unknown

For a given confidence level population standard deviation σ, the width of the confidence interval for the population mean is wider, the smaller the sample size n.

T

Relation of degree of freedom and the tail

The fewer of degree of freedom the broader of the tail

For the given sample size n, and the population standard deviation

The greater the confident level, the wider the confident interval

reduces the margin of error for the interval estimates.

larger n

For a given sample size and population standard deviation, which of the following is true in the interval estimation of the population mean

If the confidence level is greater, the interval is wider.

t df distribution

bell-shaped, symmetric around 0, and asymmetric at tails ( tails get closer and closer to horizontal axis but never touch it)

Degree of freedom df

determine the extent of the broadness of the tail of the distribution. n-1

two pieces of information when calculate t(alpha, df)

df (sample size n-1) and alpha

Precision in interval estimates

implied by a low margin of error.

Reduces the margin of errors in the interval estimate

larger n