chapter 8: confidence intervals Flashcards
what is a confidence interval for a population mean?
interval constructed around the sample mean so that we are reasonably sure, or confident, that this interval contains the population mean
when we actually select the sample, we will observe one particular sample from the extremely large number of possible samples
Therefore, we will obtain one particular confidence interval from the extremely large number of possible confidence intervals
we call the interval [x- ± .22]
supposedly, we do not know for sure the true value of the population mean
95 percent confidence interval for μ
95 percent of all intervals that we might obtain contain μ, and 5 percent of these intervals do not contain σ
the confidence coefficient
The probability that a confidence interval for a population parameter will contain the population parameter
(before sampling)
we start by choosing the probability (for example, .95 or .99) that the confidence interval will contain the population mean
the margin of error
expresses the farthest that the sample mean x- might be from the population mean μ
for a given level of confidence
an be expressed in the form [x- ± margin of error].
what are the advantages of increasing a confidence level
the advantage of making us more confident that μ is contained in the confidence interval
what are the disadvantages of increasing a confidence level
has the disadvantage of increasing the margin of error and thus providing a less precise estimate of the true value of μ
what are 95 percent confidence intervals used for
to make conclusions
why would we use a 99 confidence interval
to make conclusions of stronger evidence
how do w denote a confidence interval that will not contain the population mean?
α
This implies that 1 – α is the probability that the confidence interval will contain the population mean
what happens to the confidence interval when we increase the level of confidence?
the confidence interval becomes longer (wider)
what do we do if we don’t know the population standard deviation?
we us the sample standard deviation
what is a t distribution
If the sampled population is normally distributed, then for any sample size n this sampling distribution is what is called a t distribution
curve of the t distribution has a shape similar to that of the standard normal curve
what is the mean of any t distribution?
why?
0 bruh
A t curve is symmetrical about zero
which is more spread out, the t distribution or the standard normal distribution?
the t distribution
what does the spread (standard deviation) of the t distribution depend on?
depends on a parameter that is called the number of degrees of freedom (denoted df)
varies depending on the problem
what happens to the spread of the distribution when sample size n increases
it decreases
what happens to the t distribution when the number of degrees of freedom approaches infinity?
the t distribution becomes more like a normal curve
what is t point that is denoted tα?
the point on the horizontal axis under the curve of the t distribution that gives a right-hand tail area equal to α
how do we find the degrees of freedom for a t distribution?
n - 1
what might make the t based confidence not valid
if n is small and the distribution is non mount shaped
what happens if n is small and distribution is not mound shaped
might make the t based confidence not valid
we can use a nonparametric method
nonparametric method
a method that makes no assumption about the shape of the sampled population and is valid for any sample size
a tolerance interval
an interval that contains a specified percentage of the individual measurements in a population
what is the form of a tolerance interval if the distribution is normally distributed
[μ ± zα/2*σ]
