Confidence+Intervals

**What is a Confidence Interval?**
A confidence interval is estimated range of values which includes an unknown parameter from the population. The parameter of the population is the mean which is estimated with the sample mean. The Confidence level of a confidence interval gives the probability that the true value of the parameter is included in the interval produced. = __3 pieces you need to construct a CI:__ =
 * ====== The estimate of margin of error or how accurate we believe our estimate to be ======
 * === AP Sentence ===
 * ====== **I am 95% (or whatever your confidence interval is) confident that the true mean** will **be captured in the interval above (or state the interval) in repeated samples** ======
 * === margin of error ===
 * === confidence level ===
 * **statistic**

= __How CI behave__ = = __Important Vocabulary!__ =
 * ====== You want a high CI that will give you a small ME ======
 * ====== ME= **Z* (STD of population)/Square Root (number in sample)** ======
 * ====== As the ME gets smaller ======
 * ====== z* gets smaller ======
 * ====== CI gets smaller ======
 * ====== STDev gets smaller ======
 * ====== n gets larger ======
 * ====== Choosing sample size ======
 * ====== **n= (z*STDev)/(ME)2** ======
 * ====== ME is CI covers only random sampling errors but does not cover bias ======

=
** Central limit theorem ** : States that a mean with a large sample size will have approx normal distribution ======

=
** Simple Random Sample ** : All things with in the population have an equally likely chance to occur ======

=
** Confidence Interval **:  interval where the measurement falls with the corresponding probability ======

=
** T-test **: test to see whether the mean or means is normally distributed about the population with a value given in the null hypothesis ======

=
** Z-test **** : ** test to see whether the mean or means is normally distributed about the sample with a value given in the null hypothesis ======

= __Common Z-values and Confidence Levels__ =


 * Confidence Level z
 * 99% 1.645
 * 95% 1.960
 * 90% 2.576


 * Example:

Confidence Intervals for large samples

 * ====== The central limit theorem shows that the sample distribution of the sample means for any poulation when the same is > 30 is distrbiuted with a mean equal to mean of the population ======

= __Types of CI Test__ =

** //4 Step Procedure for Confidence Intervals for Means media type="youtube" key="Q6Lj_8yt4Qk" height="252" width="280" align="left"// **

 * Identify the population (parameter)


 * Check for Conditions and State the test
 * SRS-
 * If it does not state say "proceed with caution
 * normality
 * stated or state that CLT applied
 * if does not say "don't generalize population"
 * State test: CI For means
 * Z interval= [[image:http://www.stat.yale.edu/Courses/1997-98/101/xbar.gif height="21" align="top"]]+/- Z*√S/N


 * Interpret Sentence
 * I am 95% (or whatever your confidence interval is) confident that the true mean will be captured in the interval above (or state the interval) in repeated sample
 * **Example**
 * Suppose we want to estimate the average weight of an adult male in Dekalb County, Georgia. We draw a random sample of 1,000 men from a population of 1,000,000 men and weigh them. We find that the average man in our sample weighs 180 pounds, and the standard deviation of the sample is 30 pounds. What is the 95% confidence interval.
 * A) 180 + 1.86
 * (B) 180 + 3.0
 * (C) 180 + 5.88
 * (D) 180 + 30
 * (E) None of the above
 * **Answer (A)**


 * 1) ====== **Tell what you are estimating** ======
 * 2) Conditions
 * normal prove by drawing a box plot if skwewed or outlier state this
 * SRS
 * sample T-CI

3. T interval= +/- t*√S/N 4. AP Sentence:
 * I am 95% (or whatever your confidence interval is) confident that the true mean** will **be captured in the interval above (or state the interval) in repeated samples**

media type="custom" key="5936439"


 * 1) ======** Tell what you are estimating **======
 * 2) ** Conditions **
 * ** normal prove by drawing a box plot if skwewed or outlier state this **
 * ** SRS **
 * ** sample 2 sample T-CI **


 * 3. ** [[image:http://upload.wikimedia.org/math/4/7/3/473a40b099857f4a3e808f9b3fac3fcf.png]] where [[image:http://upload.wikimedia.org/math/e/b/8/eb863a7fb14b6994a77d034ab9bfe501.png]]
 * 4. ** ** AP Sentence **

// **CI for proportions** //
> Normality? (use formulas: population ≥ 10 * the sample size [AND] sample size * population ≥ 10 > 1 sample proportion z interval test > p ± z* to find the interval
 * 1) Tell what you are estimating, don't forget to use context!
 * 2) Conditions: Simple Random Sample?
 * 1) Find the estimated value of p
 * 1) **We are % confident that the proportion of (context) will be in the interval above in repeated samples.**

>
 * Example
 * // Find The proportion of left-handed professional baseball players. //  We have a sample of size 59 from this population.
 * There are 15 left-handed baseball players so the sample proportion is [[image:http://www.stat.wmich.edu/s160/book/img194.gif width="143" height="34" align="middle" caption="$hat{p} = 15/59 = .2542$"]]. Thus .2542 is our estimate of the proportion of left-handed professional baseball players. How much did it miss by? [[image:http://www.stat.wmich.edu/s160/book/img196.gif width="373" height="59" caption="begin{displaymath}1.96 sqrt{frac{hat{p}(1-hat{p})}{n}} = 1.96 sqrt{frac{.2542(1-.2542)}{59}} = .1111>> end{displaymath}"]]
 * Hence, the confidence interval is (.2542 - .1111, .2542 + .1111) = (.143, .365)

media type="custom" key="5941081"​​

// **CI for 2 proportions** //
> ex. We are estimating the difference in proportions between (context) and (context) > INDEPENDENT (may have to state why) > Normality? (use formulas: multiply two smallest n-values≥ 5) > 2 sample proportion z interval test > to find the interval
 * 1) Tell what you are estimating, don't forget to use context!
 * 1) Conditions: Simple Random Sample? (may have to tell what makes it am SRS)
 * 1) Find the estimated value of p
 * 1) **We are % confident that the difference proportions of (context and context) will be in the interval above in repeated samples.**

> //Form a 90% confidence interval for the proportion difference.//
 * Example: //In a recent Gallup survey, 50% of 1,242 Democratic voters favored Obama and 42% of 1,242 Democratic voters favored Clinton.//

<span style="color: #c0c0c0; display: block; font-family: Arial,Helvetica,sans-serif; text-align: left;">Sample X N Sample p 1 621 1242 0.500000 2 522 1242 0.420290 Difference = p (1) - p (2) Estimate for difference: 0.079710190% CI for difference: (0.0469177, 0.112503) Test for difference = 0 (vs not = 0): Z = 4.00 P-Value = 0.000 The 90% confidence interval is given by 0.047 < p o - pc < 0.113. Since the interval does not contain 0, we can conclude that there is a difference in the two proportions.

// **CI for regression** //
> ex. We are estimating the (context) > Normality? Assumed > Linear? Assumed > linear regression t-interval > Degrees of Freedom: Number in sample - 2 > b +- t* (SE b ) > To find SE b : s/ (the square root of the sum of ( x i − ¯ x ) 2
 * 1) Tell what you are estimating, don't forget to use context!
 * 1) Conditions:
 * 1) Find interval using formula:
 * 1) **We are % confident that the (context ) will (increase/decrease) in the interval above in repeated samples.**