Confidence Intervals Unit 5

The larger your confidence, the wider the interval (the more narrow the interval, the lower the confidence)

Provides methods of drawing conclusions about a population from sample data; in formal inference, we use PROBABILITY to express the strength of our conclusions about a population from sample data

Confidence Level

CL% of the resulting confidence intervals would contain the population mean

1. An interval calculated from the data, usually out of the form estimate ± margin of error

2. A confidence level C, which gives the probability that the interval will capture the true parameter value in repeated samples

1. The data comes from an SRS from the population of interest

2. The sampling distribution of x-bar is approximately normal (if the population is normal or CLT for x-bar; rules of thumb for p-hat)

3. The sample is independent from the population (10n≤N)

The number z* (z star) with probability “p” lying to its right under the standard normal curve (called the upper p critical value of the standard normal deviation)

• First (1-C)/2 → Use Inversenorm

• C being the confidence level

• Critical values are ALWAYS positive

.05 tail area & z*=1.6449

.025 tail area & z*=1.96

.005 tail area & z*=2.5758

An SRS of size n from a population having an unknown mean mu and a known standard deviation sigma, has a level C confidence interval for mu

• this interval is exact when the population distribution is normal and approximately correct for large values of n

1. Must be an SRS

2. Must be approximately normal (CLT or population is approx normal)

3. Independence (10n≤N)

P - Parameter of interest (The parameter of interest is ____)

A - Assumptions

N - Name the type of interval (__ sample, (z or t)__ interval, CL=___%, (if a t-dist, df=__)

I - Interval (calculate)

C - Conclusion in Context (I am ___% confident that the true mu of (parameter in context) lies between ___ and ____.

A small margin of error says that we have gotten really close to the parameter. We want high confidence and small margin of error which will make small intervals.

z* get smaller and confidence level C gets smaller and sigma gets smaller and n gets larger

take 4 times the same size to cut the margin of error in half

take 4 times the same size to cut the margin of error in half

To determine the sample size n that will yield a confidence interval for population mean with specified margin of error, ME, set the expression for the margin of error to be less than or equal to mu and solve

• ALWAYS ROUND UP

describe a confidence interval

• t is a standard value like z, t tells us how many standard deviations x-bar is from mu

• It is symmetric about a zero and is bell shaped, but there is more variation, so the spread is greater

• Must identify the degrees of freedom (n-1) because there is a different t statistic for each sample size

• Use t-table (gives are to the RIGHT of t)

• NO SIGMA; DOES NOT HAVE POPULATION STANDARD DEVIATION

closer to normal since S is getting closer to signs

the population is normal (we assume it’s normal to justify it’s use)

To calculate one sample t-interval, we must still assume or know the following:

1. The data is from an SRS from the population of interest

2. The sample mean x-bar has a normal distribution (normal population, graphically supported (normal probability plot) or n≥30 for CLT)

3. Independent (10n≤N)

4. NAMING THE TEST: one sample t-interval, CL=__%, df=__.

SIGMA, population SD