Confidence Intervals Unit 5

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Confidence Interval Rule

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25 Terms

1

Confidence Interval Rule

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

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Statistical inference

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

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CL

Confidence Level

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If the confidence procedure was repeated many times,

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

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A level C confidence interval for a parameter has two parts:

  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

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Constructing a confidence interval (the construction of a confidence interval for a population mean mu is appropriate when)

  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)

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Critical Values

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

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90% confidence level

.05 tail area & z*=1.6449

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95% confidence level

.025 tail area & z*=1.96

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99% confidence level

.005 tail area & z*=2.5758

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Confidence Interval for a Population Mean

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

<p>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</p><ul><li><p>this interval is exact when the population distribution is normal and approximately correct for large values of n</p></li></ul>
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Conditions for a z interval

  1. Must be an SRS

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

  3. Independence (10n≤N)

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Steps to construct any confidence interval (PANIC)

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 ____.

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Behavior of confidence intervals

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.

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Margin of Error

<p></p>
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Margin of error gets smaller when

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

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Since we use the square root of n, we must

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

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Since we use the square root of n, we must

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

<p>take 4 times the same size to cut the margin of error in half</p>
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Sample size manufacturing

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

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NEVER use probability to

describe a confidence interval

<p>describe a confidence interval</p>
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T-distribution

  • 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

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As the df increases, the t-distribution

closer to normal since S is getting closer to signs

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One sample t procedures are exactly correct only when

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

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One sample t procedures

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=__.

<p>To calculate one sample t-interval, we must still assume or know the following:</p><ol><li><p>The data is from an SRS from the population of interest</p></li><li><p>The sample mean x-bar has a normal distribution (normal population, graphically supported (normal probability plot) or n≥30 for CLT)</p></li><li><p>Independent (10n≤N)</p></li><li><p>NAMING THE TEST: one sample t-interval, CL=__%, df=__.</p></li></ol>
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Z tests have

SIGMA, population SD

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