ch 8 confidence levels and intervals

CI formula

CI= statistic±[(criticalvalue)·(standarddeviationofstatistic)

The confidence level is sometimes called the "capture rate." Explain why this is an appropriate term.

It can be interpreted as the percentage of repeated samples that "capture" the true mean within the interval.

Describe the differences between a standard normal distribution and a t distribution.

- The spread of the t distributions is a bit greater than that of the standard Normal distribution - The t distributions have more probability in the tails and less in the center than does the standard Normal -Shapes are similar

A confidence interval takes the form of:

An interval calculated from the data; estimate +- moe statistic +- (z*)(SD)

If, instead of constructing a 95% confidence interval, the physician constructed a 90% confidence interval, would the 90% interval be wider, narrower, or the same width as the 95% interval? Explain.

If our interval has to capture the true mean only 90% of the time in the repeated sample instead of 95%, *a narrower interval can be constructed.*

What does it mean if an inference procedure is robust?

If the probability calculations involved in the procedure remain fairly accurate when a condition for using the procedures is violated. Fortunately, the t procedures are quite robust against non-Normality of the population except when outliers or strong skewness are present. Larger samples improve the accuracy of critical values from the t distributions when the population is not Normal.

confidence level context (given an interval)

If this method of constructing an interval were repeated many, many times, about _% of intervals constructed would contain the pop mean (context)

How can you arrange to have both high confidence and a small margin of error?

Increase the sample size

What happens to the margin of error as z* gets smaller?

MOE also gets smaller.

What happens to the margin of error, as sigma gets smaller? As

MOE also gets smaller.

What happens to the margin of error, as n gets larger? By how many times must the sample size n increase in order to cut the margin of error in half?

MOE reduces. 4 times.

Does her interval provide evidence that the true proportion of students at her school who would agree that a third party is needed is 57%?

No. Confidence intervals don't give us evidence that a parameter equals a specific value; they give us a range of plausible values. Diedra's interval says that the true proportion of students who agree could be as low as 55.7% or as high as 74.3%, and that values outside of this interval are not likely. So it wouldn't be appropriate to say this interval supports the value of 57%.

NAME OF TEST

One-sample Z interval for a population proportion

RIN = the three conditions for constructing CI p or µ

Random Independence = must pass 10% rule -- 10n ≤ N or sampled *with replacement* Normality = must pass np hat ≥ 10 (successes) and n(1-p hat) (failures) or in the case of x bar, n ≥ 30, CLT

SE of a stat

SE describes that how close the sample proportion pˆ will be , on average , to the population proportion p in repeated SRSs of size n.

4 step process

State: We want to estimate w/ _% confidence level the prop (phat) of (context) We want to estimate w/ _% confidence level the true mean µ (context) Plan: Check conditions. ( RIN) Random: If the sample was taken randomly or not Normal: Check: np and n(1-p) are greater than 10 or not OR check CLT (n≥30) if n< 30, make graphs to check normality = dot plot (no skew), box plot (no skew), NPP (linearity) ^"There's no reason to doubt the normality of the pop. based on the graphs" Independent: Check 10% condition if without replacement or else if it obvious or given , then mention that (10n ≤ N) Do: If the conditions are met, perform calculations. Identify the appropriate inference method. (One Sample z-interval for a population proportion) CI = phat +- z* (SD) (One Sample t-interval for a population mean) CI = x bar +- z* (sigma / √n) = sigma known CI = x bar +- t* (Sx / √n) = sigma unknown df = n-1 Conclude: We are _____% confident that the interval from _____ to _________ captures the true proportion of ____( context)

MOE

Tells us how close the estimate tends to be to the unknown parameter (µ) in repeated random sampling

How does the standard deviation differ to the standard error for the sampling distribution of pˆ ?

The SD measures the amount of variability or dispersion for a subject set of data from the mean, while the SEM measures how far the sample mean of the data is likely to be from the true pop mean or prop. The SEM is always smaller than SD

Explain the two conditions when the margin of error gets smaller.

The confidence level decreases-- but you want high confidence but you pay a price for it (wider interval) The sample size n increases

Does the confidence level tell us the chance that a particular confidence interval captures the population parameter? If not, what does it tell us?

The confidence level tells us how likely it is that the method we are using will produce an interval that captures the population parameter if we use it many times.

Large samples greater than 30:

The t procedures can be used even for clearly skewed distributions when the sample is large, roughly n ≥ 30.

Sample size at least 15:

The t procedures can be used except in the presence of outliers or strong skewness.

What is the formula for the margin of error of the confidence interval for the population mean µ?

z* (sigma / √n) ≤ moe

Inferences for proportions use _____ and inferences for means use _____

z* and t*

Sample size less than 15:

Use t procedures if the data appear close to Normal (roughly symmetric, single peak, no outliers). If the data are clearly skewed or if outliers are present, do not use t.

Define standard error.

When the standard deviation of a statistic is estimated from data

level C confidence interval

gives the overall success rate of the method for calculating the confidence interval. (i.e. In C% of all possible samples, the method would yield an interval that captures the true parameter value.)

estimate a pop. prop

phat, n

It is the size of the _____ that determines the margin of error. The size of the ________ does not influence the sample size we need.

sample; pop

How would the width of confidence interval change if the we took a larger sample? Explain.

the SD of the sampling distribution will be smaller. So the confidence interval will be narrower.

What happens to the t distribution as the degrees of freedom increase?

the t density curve approaches the standard Normal curve ever more closely.

confidence interval context (given an interval)

we are _ % confident that the interval _ to _ (units) captures the true mean of (context)

P hat =

x / n

estimate a pop. mean

µ, sigma, n