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