Inference+Tests

=__What is an inference test?__ = An inference test is a method of using samples from a population to determine conclusions about the population. It also allows us to determine the strength or weakness of our conclusions.

There are two main types of inference tests:
 * 1) Confidence Intervals
 * 2) Tests of Significance

We are going to focus on **Tests of Significance**.

**__z-tests__**  A z-test is used when:
 * the problem asks for a test of significance
 * the standard deviation of the population is known/given in the problem
 * the data is approximately normal

The most basic form of the z-test is a 1 sample z-test. This is used when one sample is taken from the population - we will draw our conclusions from this one sample statistic, x.

media type="custom" key="5928107" Stat --> Tests --> z-test (#1) If Data is given -  If no data is given, but statistics are given -  **Example of 1 sample z-test** Suppose that in a particular geographic region, the mean and standard deviation of scores on a reading test are 100 points, and 12 points, respectively. Our interest is in the scores of 55 students in a particular school who received a mean score of 96. We can ask whether this mean score is significantly lower than the regional mean — that is, are the students in this school comparable to a simple random sample of 55 students from the region as a whole, or are their scores surprisingly low? __Step One__: Ho: μ = 100 The mean score of students in a particular school is 100 points, the same as the average for the entire geographic region Ha: μ < 100 The mean score of students in a particular school is less than 100 points, significantly lower than the average for the entire geographic region __Step Two:__ SRS? Yes. Normal? Yes, CLT applies. 55 ≥ 20 1 sample z-test __Step Three: z__ = (96-100)/(12/sqrt(55)) = -2.47 p = .0068 __Step Four__: .0068 < .05 p < α Reject the Ho. There is enough evidence to say that the mean score of students in a particular school is less than 100 points, significantly lower than the average for the entire geographic region.
 * To perform a 1 sample z-test in your calculator: **
 * µ0 = Ho
 * σ = σpopulation
 * List L1
 * Frequency = 1
 * ≠ or (Ha symbol)
 * the calculator will automatically double the p-value for you if you are performing a 2-sided test
 * µ0 = Ho
 * <span style="font-family: 'Lucida Sans Unicode','Lucida Grande',sans-serif;">σ = σpopulation
 * <span style="font-family: 'Lucida Sans Unicode','Lucida Grande',sans-serif;">x = sample mean
 * <span style="font-family: 'Lucida Sans Unicode','Lucida Grande',sans-serif;">n = # in sample
 * <span style="font-family: 'Lucida Sans Unicode','Lucida Grande',sans-serif;">≠ or (Ha symbol)

Another form of the z-test compares two sample statistics in order to determine if there is a difference between the two. This is called a 2 sample z-test. It is used when two samples are taken from two different populations - we will draw conclusions from two sample statistics, x1 and x2.

media type="custom" key="5928119"

<span style="font-family: 'Lucida Sans Unicode','Lucida Grande',sans-serif;">Stat --> Tests --> 2-SampZTest (#3) If Data is given - <span style="font-family: 'Lucida Sans Unicode','Lucida Grande',sans-serif;"> If no data is given, but statistics are given -
 * <span style="color: #808080; font-family: 'Lucida Sans Unicode','Lucida Grande',sans-serif; font-size: 150%;">To perform a 2 sample z-test in your calculator: **
 * <span style="font-family: 'Lucida Sans Unicode','Lucida Grande',sans-serif;">σ1 = σ of first population
 * <span style="font-family: 'Lucida Sans Unicode','Lucida Grande',sans-serif;">σ2 = σ of second population
 * <span style="font-family: 'Lucida Sans Unicode','Lucida Grande',sans-serif;">List L1 - first dataset
 * <span style="font-family: 'Lucida Sans Unicode','Lucida Grande',sans-serif;">List L2 - second dataset
 * <span style="font-family: 'Lucida Sans Unicode','Lucida Grande',sans-serif;">Frequency1 = 1
 * <span style="font-family: 'Lucida Sans Unicode','Lucida Grande',sans-serif;">Frequency2 = 1
 * <span style="font-family: 'Lucida Sans Unicode','Lucida Grande',sans-serif;">≠ or (Ha symbol)
 * <span style="font-family: 'Lucida Sans Unicode','Lucida Grande',sans-serif;">the calculator will automatically double the p-value for you if you are performing a 2-sided test
 * <span style="font-family: 'Lucida Sans Unicode','Lucida Grande',sans-serif;">σ1 = σ of first population
 * <span style="font-family: 'Lucida Sans Unicode','Lucida Grande',sans-serif;">σ2 = σ of second population
 * <span style="font-family: 'Lucida Sans Unicode','Lucida Grande',sans-serif;">x1 = first sample mean
 * <span style="font-family: 'Lucida Sans Unicode','Lucida Grande',sans-serif;">x2 = second sample mean
 * <span style="font-family: 'Lucida Sans Unicode','Lucida Grande',sans-serif;">Frequency1 = 1
 * <span style="font-family: 'Lucida Sans Unicode','Lucida Grande',sans-serif;">Frequency2 = 1
 * <span style="font-family: 'Lucida Sans Unicode','Lucida Grande',sans-serif;">n1 = # in first sample
 * <span style="font-family: 'Lucida Sans Unicode','Lucida Grande',sans-serif;">n2 = # in second sample
 * <span style="font-family: 'Lucida Sans Unicode','Lucida Grande',sans-serif;">≠ or (Ha symbol)
 * <span style="font-family: 'Lucida Sans Unicode','Lucida Grande',sans-serif;">the calculator will automatically double the p-value for you if you are performing a 2-sided test

z-tests can also be used to analyze proportions, and whether or not a sample proportion coincides with the stated or expected proportion. In a 1 proportion z-test, an expected p proportion is compared to a p^ sample proportion. media type="custom" key="5937731" Stat --> Tests --> 1-PropZTest (#5)
 * <span style="color: #808080; font-family: 'Lucida Sans Unicode','Lucida Grande',sans-serif; font-size: 20px;">To perform a 1 proportion z-test in your calculator: **
 * p0 = p
 * x = number of successes in the sample
 * n = size of the sample
 * prop ≠ or (same as in alternate hypothesis)

If you are trying to compare two proportions from two populations, then a 2 proportion z-test is appropriate. It uses two samples from two populations to calculate two test statistics, p1 and p2, and then compares the two. media type="custom" key="5937639" Stat --> Tests --> 2-PropZTest (#6)
 * <span style="color: #808080; font-family: 'Lucida Sans Unicode','Lucida Grande',sans-serif; font-size: 20px;">To perform a 2 proportion z-test in your calculator: **
 * x1 = number of successes in the first sample
 * n1 = size of the first sample
 * x2 = number of successes in the second sample
 * n2 = size of the second sample
 * prop ≠ or (same as in alternate hypothesis)

<span style="background-color: #bea9ef; color: #ffffff; font-family: 'Lucida Sans Unicode','Lucida Grande',sans-serif; font-size: 150%;">__t-tests__ T- tests are very similar to z-tests, but they differ in one main point: t-tests can be used when the standard deviation of the **population** of interest is unknown. Thus, when performing t-tests, the standard deviation of the **sample** is used.

<span style="font-family: Corbel; font-size: 14pt; mso-bidi-font-family: Corbel; mso-fareast-font-family: Corbel; mso-text-raise: -4.0pt; msobidifontfamily: Corbel; msofareastfontfamily: Corbel; msolist: Ignore; msotextraise: -4.0pt; position: relative; text-shadow: auto; textshadow: auto; top: 4pt;">° //__One-Sample t-test Facts**:**__// <span style="color: #ff9900; font-size: 26pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 14.0pt; mso-fareast-font-family: Arial; mso-text-raise: -4.0pt; msobidifontfamily: Arial; msobidifontsize: 14.0pt; msofareastfontfamily: Arial; msolist: Ignore; msotextraise: -4.0pt; position: relative; text-shadow: auto; textshadow: auto; top: 4pt;">▫ //Used to evaluate **differences in means** of **two** groups.// <span style="color: #ff9900; font-size: 26pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 14.0pt; mso-fareast-font-family: Arial; mso-text-raise: -4.0pt; msobidifontfamily: Arial; msobidifontsize: 14.0pt; msofareastfontfamily: Arial; msolist: Ignore; msotextraise: -4.0pt; position: relative; text-shadow: auto; textshadow: auto; top: 4pt;">▫ //as d.f.// //<span style="font-family: Symbol; mso-ascii-font-family: Arial; mso-char-type: symbol; mso-hansi-font-family: Arial; mso-symbol-font-family: Symbol; msoasciifontfamily: Arial; msochartype: symbol; msohansifontfamily: Arial; msosymbolfontfamily: Symbol;">­, the curve approaches normality.// <span style="color: #ff9900; font-size: 26pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 14.0pt; mso-fareast-font-family: Arial; mso-text-raise: -4.0pt; msobidifontfamily: Arial; msobidifontsize: 14.0pt; msofareastfontfamily: Arial; msolist: Ignore; msotextraise: -4.0pt; position: relative; text-shadow: auto; textshadow: auto; top: 4pt;">▫ //the **observed** mean (from a single sample) is compared to an **expected** (or reference) mean of the population.// <span style="color: #ff9900; font-size: 26pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 14.0pt; mso-fareast-font-family: Arial; mso-text-raise: -4.0pt; msobidifontfamily: Arial; msobidifontsize: 14.0pt; msofareastfontfamily: Arial; msolist: Ignore; msotextraise: -4.0pt; position: relative; text-shadow: auto; textshadow: auto; top: 4pt;">▫ //The **variation** in the population is estimated based on the variation in the observed sample.// <span style="color: #ff9900; font-size: 26pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 14.0pt; mso-fareast-font-family: Arial; mso-text-raise: -4.0pt; msobidifontfamily: Arial; msobidifontsize: 14.0pt; msofareastfontfamily: Arial; msolist: Ignore; msotextraise: -4.0pt; position: relative; text-shadow: auto; textshadow: auto; top: 4pt;">▫ //Can be used with __small__ __sample__ __sizes__ **(such as n=10)**// <span style="color: #ff9900; font-size: 26pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 14.0pt; mso-fareast-font-family: Arial; mso-text-raise: -4.0pt; msobidifontfamily: Arial; msobidifontsize: 14.0pt; msofareastfontfamily: Arial; msolist: Ignore; msotextraise: -4.0pt; position: relative; text-shadow: auto; textshadow: auto; top: 4pt;">▫ //Normality can be determined by observing the data as a **histogram** or by performing a **normality test**// <span style="color: #ff9900; font-size: 26pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 14.0pt; mso-fareast-font-family: Arial; mso-text-raise: -4.0pt; msobidifontfamily: Arial; msobidifontsize: 14.0pt; msofareastfontfamily: Arial; msolist: Ignore; msotextraise: -4.0pt; position: relative; text-shadow: auto; textshadow: auto; top: 4pt;">▫ //One **independent variable** (group) required ~// // such as gender: Males vs. Females // <span style="color: #ff9900; font-size: 26pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 14.0pt; mso-fareast-font-family: Arial; mso-text-raise: -4.0pt; msobidifontfamily: Arial; msobidifontsize: 14.0pt; msofareastfontfamily: Arial; msolist: Ignore; msotextraise: -4.0pt; position: relative; text-shadow: auto; textshadow: auto; top: 4pt;">▫ //One **dependent variable** (group) required ~// // such as test scores // <span style="color: #ff9900; font-size: 26pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 14.0pt; mso-fareast-font-family: Arial; mso-text-raise: -4.0pt; msobidifontfamily: Arial; msobidifontsize: 14.0pt; msofareastfontfamily: Arial; msolist: Ignore; msotextraise: -4.0pt; position: relative; text-shadow: auto; textshadow: auto; top: 4pt;">▫ //Analyze/Compare data (means and measures in variations) of the two groups using a **box-and-whiskers plot**//**// ~ Performing inference(significance)test: //** // ( state the null and alternative hypothesis ~ __with context__ ) // Ha: μ ≠ μo : ** // (check conditions for **__EACH__** population) //
 * __ 1-sample t-test __**** :  ** * **table shows probabilities of right tail:**
 * //__ Step #1: __//**
 * H0: μ = μo : **
 * //__ Step #2: __//**
 * // - 1-Sample t-test //**
 * // - SRS: //****// (if…) //**
 * // * ___Yes___ //****//<span style="color: #993366; font-family: Symbol; font-size: 11pt; mso-ascii-font-family: Arial; mso-bidi-font-family: Arial; mso-char-type: symbol; mso-hansi-font-family: Arial; mso-symbol-font-family: Symbol; msoasciifontfamily: Arial; msobidifontfamily: Arial; msochartype: symbol; msohansifontfamily: Arial; msosymbolfontfamily: Symbol;">® //****// __state:__ //**// **“** the sample is Simple Random Sample / is random **”** //
 * // * ___No___ //****//<span style="color: #3366ff; font-family: Symbol; font-size: 11pt; mso-ascii-font-family: Arial; mso-bidi-font-family: Arial; mso-char-type: symbol; mso-hansi-font-family: Arial; mso-symbol-font-family: Symbol; msoasciifontfamily: Arial; msobidifontfamily: Arial; msochartype: symbol; msohansifontfamily: Arial; msosymbolfontfamily: Symbol;">® //****// __state:__ “ //**// proceed with caution **”** //
 * // - Draw Box plot //**
 * // - Independent Sample: //**// ( __yes__ / __no__ ) //
 * // - Normal Sample: //**// ( if CLT applies, than normal sample ~ n ////<span style="color: #666699; font-family: Symbol; font-size: 10pt; mso-ascii-font-family: Arial; mso-bidi-font-family: Arial; mso-char-type: symbol; mso-hansi-font-family: Arial; mso-symbol-font-family: Symbol; msoasciifontfamily: Arial; msobidifontfamily: Arial; msochartype: symbol; msohansifontfamily: Arial; msosymbolfontfamily: Symbol;">³ //// 30 ) //
 * // - degrees of freedom = //**** ( n-1 ) **** = **


 * //Step #3://**
 * // (find t) //**
 * // t = (x__bar__ – mean of the population)/se //**


 * SEM: //(standard error of mean)// **
 * σ // x =//** //s **/**// ** √ **n


 * // - s//** **//<span style="font-family: Symbol; font-size: 10pt; mso-ascii-font-family: Arial; mso-char-type: symbol; mso-hansi-font-family: Arial; mso-symbol-font-family: Symbol; msoasciifontfamily: Arial; msochartype: symbol; msohansifontfamily: Arial; msosymbolfontfamily: Symbol;">® //****// sample standard deviation//**
 * // - n//** **//<span style="font-family: Symbol; font-size: 10pt; mso-ascii-font-family: Arial; mso-char-type: symbol; mso-hansi-font-family: Arial; mso-symbol-font-family: Symbol; msoasciifontfamily: Arial; msochartype: symbol; msohansifontfamily: Arial; msosymbolfontfamily: Symbol;">® //****// sample size//**
 * // p = //**
 * // d.f. = //**

- p<alpha, Reject Ho: There __ is __ enough evidence to say (__the Ho__) is __ true __ - p > alpha, Fail to Reject Ho: There is __ NOT __ enough evidence to say (__the Ho__) is true __ 2-sample t-test: __ <span style="font-family: Corbel; font-size: 14pt; mso-bidi-font-family: Corbel; mso-fareast-font-family: Corbel; mso-text-raise: -4.0pt; msobidifontfamily: Corbel; msofareastfontfamily: Corbel; msolist: Ignore; msotextraise: -4.0pt; position: relative; text-shadow: auto; textshadow: auto; top: 4pt;">° __ Facts: __ <span style="color: #ff9900; font-size: 26pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 14.0pt; mso-fareast-font-family: Arial; mso-text-raise: -4.0pt; msobidifontfamily: Arial; msobidifontsize: 14.0pt; msofareastfontfamily: Arial; msolist: Ignore; msotextraise: -4.0pt; position: relative; text-shadow: auto; textshadow: auto; top: 4pt;">▫ Used to evaluate differences in means of two groups <span style="color: #ff9900; font-size: 26pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 14.0pt; mso-fareast-font-family: Arial; mso-text-raise: -4.0pt; msobidifontfamily: Arial; msobidifontsize: 14.0pt; msofareastfontfamily: Arial; msolist: Ignore; msotextraise: -4.0pt; position: relative; text-shadow: auto; textshadow: auto; top: 4pt;">▫ as d.f. <span style="font-family: Symbol; mso-ascii-font-family: Arial; mso-char-type: symbol; mso-hansi-font-family: Arial; mso-symbol-font-family: Symbol; msoasciifontfamily: Arial; msochartype: symbol; msohansifontfamily: Arial; msosymbolfontfamily: Symbol;">­, the curve approaches normality <span style="color: #ff9900; font-size: 26pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 14.0pt; mso-fareast-font-family: Arial; mso-text-raise: -4.0pt; msobidifontfamily: Arial; msobidifontsize: 14.0pt; msofareastfontfamily: Arial; msolist: Ignore; msotextraise: -4.0pt; position: relative; text-shadow: auto; textshadow: auto; top: 4pt;">▫ // comparing 2 means (2 populations //2 treatments) <span style="color: #ff9900; font-size: 26pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 14.0pt; mso-fareast-font-family: Arial; mso-text-raise: -4.0pt; msobidifontfamily: Arial; msobidifontsize: 14.0pt; msofareastfontfamily: Arial; msolist: Ignore; msotextraise: -4.0pt; position: relative; text-shadow: auto; textshadow: auto; top: 4pt;">▫ population standard deviation ( σ ) is unknown __ Step #1: __ (state the null and alternative hypothesis ~ __with context__ ) H0: μ1 = μ2 Ha: μ1 ( ≠ //< //> ) μ2 *__Note__: ( ≠ ) ~ 2-sided test ( <span style="color: #333333; font-family: Symbol; font-size: 10pt; mso-ansi-language: ES-MX; mso-ascii-font-family: 'Trebuchet MS'; mso-char-type: symbol; mso-hansi-font-family: 'Trebuchet MS'; mso-symbol-font-family: Symbol; msoansilanguage: ES-MX; msoasciifontfamily: 'Trebuchet MS'; msochartype: symbol; msohansifontfamily: 'Trebuchet MS'; msosymbolfontfamily: Symbol;">³ ; <span style="color: #333333; font-family: Symbol; font-size: 10pt; mso-ansi-language: ES-MX; mso-ascii-font-family: 'Trebuchet MS'; mso-char-type: symbol; mso-hansi-font-family: 'Trebuchet MS'; mso-symbol-font-family: Symbol; msoansilanguage: ES-MX; msoasciifontfamily: 'Trebuchet MS'; msochartype: symbol; msohansifontfamily: 'Trebuchet MS'; msosymbolfontfamily: Symbol;">£ ) ~1 sided test __ Step #2: __ (check conditions for __EACH__ population) (a) 2-Sample t-test (b) Draw Box plots (2) (c) SRS (2): (if…) * ___Yes___ __state:__ “ the sample is Simple Random Sample / is random ” * ___No___ __state:__ “ proceed with caution ” (d) Normal Sample (2) : (if CLT applies, than normal sample ~ n >= 30) (e) Independent Samples (2) (s12/n1 + s22/n2) 2 - Degrees of freedom (d.f.) =  <span style="color: gray; font-size: 10pt; mso-color-alt: #666699; mso-text-raise: 6.0pt; position: relative; text-effect: emboss; top: -6pt;"> __ [ (s12 / n1) 2 / (n1 - 1) ] + [ (s22 / n2) 2 / (n2 - 1) ] Don’t forget to label the populations/variables: __ || μ || σ || n || Xbar || S || || μ1 || σ 1 || n 1 || x 1 || s 1 || || μ2 || σ 2 || n 2 || x 2 || s 2 ||
 * //Step #4://**
 * // (Interpret) //**
 * // pvalue ( //****//__<span style="color: #339966; font-family: Symbol; mso-ascii-font-family: Arial; mso-char-type: symbol; mso-hansi-font-family: Arial; mso-symbol-font-family: Symbol; msoasciifontfamily: Arial; msochartype: symbol; msohansifontfamily: Arial; msosymbolfontfamily: Symbol;">< __ // ** __<span style="color: #339966; font-family: Symbol; mso-ascii-font-family: Arial; mso-char-type: symbol; mso-hansi-font-family: Arial; mso-symbol-font-family: Symbol; msoasciifontfamily: Arial; msochartype: symbol; msohansifontfamily: Arial; msosymbolfontfamily: Symbol;">> __ )  <span style="color: #3366ff; font-family: Symbol; font-size: 20pt; mso-ascii-font-family: 'Comic Sans MS'; mso-bidi-font-family: Arial; mso-char-type: symbol; mso-hansi-font-family: 'Comic Sans MS'; mso-symbol-font-family: Symbol; msoasciifontfamily: 'Comic Sans MS'; msobidifontfamily: Arial; msochartype: symbol; msohansifontfamily: 'Comic Sans MS'; msosymbolfontfamily: Symbol;">µ
 * population
 * 1
 * 2

Step #3: (Solve for ‘t’) t = [ ( x 1 - x 2) – Mean of pop. ] / SE pvalue = <span style="color: #ff3399; mso-bidi-font-family: Arial; mso-text-raise: 10.0pt; position: relative; top: -10pt;">SE = √ (s12 / n1 ) + (s22 / n2<span style="color: black; mso-bidi-font-family: Arial; mso-text-raise: 10.0pt; msobidifontfamily: Arial; msospacerun: yes; msotextraise: 10.0pt; position: relative; top: -10pt;"> ) ] <span style="color: black; font-size: 11pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 12.0pt; mso-text-raise: -3.0pt; position: relative; top: 3pt;"> __Step #4:__ (Interpret) *state the alpha ( µ ) significance level being used (( 2 options : )) pvalue ( __<span style="color: fuchsia; font-family: Symbol; mso-ascii-font-family: Arial; mso-char-type: symbol; mso-hansi-font-family: Arial; mso-symbol-font-family: Symbol; msoasciifontfamily: Arial; msochartype: symbol; msohansifontfamily: Arial; msosymbolfontfamily: Symbol;"><  __ ) <span style="color: #3366ff; font-family: Symbol; font-size: 20pt; mso-char-type: symbol; mso-symbol-font-family: Symbol; msoasciifontfamily: 'Comic Sans MS'; msobidifontfamily: Arial; msochartype: symbol; msohansifontfamily: 'Comic Sans MS'; msosymbolfontfamily: Symbol;">µ Reject Ho: - There is __ enough evidence __ to say (the Ho) is true Pvalue ( __<span style="color: fuchsia; font-family: Symbol; mso-ascii-font-family: Arial; mso-char-type: symbol; mso-hansi-font-family: Arial; mso-symbol-font-family: Symbol; msoasciifontfamily: Arial; msochartype: symbol; msohansifontfamily: Arial; msosymbolfontfamily: Symbol;">>  __ ) <span style="color: #3366ff; font-family: Symbol; font-size: 20pt; mso-char-type: symbol; mso-symbol-font-family: Symbol; msoasciifontfamily: 'Comic Sans MS'; msobidifontfamily: Arial; msochartype: symbol; msohansifontfamily: 'Comic Sans MS'; msosymbolfontfamily: Symbol;">µ Fail to Reject Ho: (Ha is true) - There is __ NOT ____ enough evidence __ to say (__the Ho__) is true

media type="custom" key="5935439"

EXAMPLE Matched Pairs A random sample of 15 trainees who took this test where then given a week-long memory training course. They were then retested. The results are shown in the table below. **Test, at the 5% level of significance, that the memory course improved the ability of the trainees to correctly identify license plates.
 * Police trainees were seated in a darkened room facing a projector screen. Ten different license plates were projected on the screen, one at a time, for 5 seconds each, separated by 15-second intervals. After the last 15-second interval, the lights were turned on and the police trainees were asked to write down as many of the 10 license plate numbers as possible, in any order at all.


 * (L1): # plates correctly identified after training.

||  || (L2): # plates correctly identified before training.

||  || DIFFERENCE (L1) - (L2)

||


 * 6

||  || 6

||  || 0

||


 * 8

||  || 5

||  || 3

||


 * 6

||  || 6

||  || 0

||


 * 7

||  || 5

||  || 2

||


 * 9

||  || 7

||  || 2

||


 * 8

||  || 5

||  || 3

||


 * 9

||  || 4

||  || 5

||


 * 6

||  || 6

||  || 0

||


 * 7

||  || 7

||  || 0

||


 * 5

||  || 8

||  || -3

||


 * 9

||  || 4

||  || 5

||


 * 8

||  || 5

||  || 3

||


 * 6

||  || 4

||  || 2

||


 * 8

||  || 6

||  || 2

||


 * 6

||  || 7

||  || -1

||


 * Mean of DIFFERENCE column>

||  ||  1.5333

||


 * s for DIFFERENCE column>

||  ||  2.1996

|| We will run a one-sample t test on the DIFFERENCE column. In this situation, m is the mean improvement that would be achieved if the entire population of police trainees took the memory training course. __Step 1 (USE CONTEXT):__ Ho: m = 0 (There is no mean improvement in ability to identify plates after training). Ha: m > 0 (There is a mean improvement in ability to identify plates after training). Sample mean = 1.5333. Sample s = 2.1996. __Step 2:__ Type of test: Matched Pairs t test. Degrees of freedom = 15 - 1 = 14. __Step 3:__ Calculated t statistic: t = (1.5333 - 0)/0.5679 = 2.70. Critical values of t: t > 1.761 P value: approx .01 __Step 4:__ .01<.05. We reject Ho at the 5% level of significance. There is enough evidence to suggest that the training will result in an improvement in the ability to identify plates.

<span style="color: #ffffff; font-family: 'Lucida Sans Unicode','Lucida Grande',sans-serif; font-size: 22px; line-height: 32px;"> <span style="background-color: #bea9ef; color: #ffffff; font-family: 'Lucida Sans Unicode','Lucida Grande',sans-serif;">__Chi-squared tests__ <span style="color: #ffffff; font-family: 'Lucida Sans Unicode','Lucida Grande',sans-serif; font-size: 22px; line-height: 32px;">

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<span style="background-color: #bea9ef; color: #ffffff; font-family: 'Lucida Sans Unicode','Lucida Grande',sans-serif; font-size: 22px; line-height: 32px;"> <span style="font-family: 'Lucida Sans Unicode','Lucida Grande',sans-serif; font-size: 90%;">Type I and Type II Error

<span style="background-color: #bea9ef; color: #ffffff; font-family: 'Lucida Sans Unicode','Lucida Grande',sans-serif; font-size: 22px; line-height: 32px;"> * Errors: || // ( __Ho True__ ):   // || // ( __Ho False__ ):   // || || // __ TYPE I Error __ // ||     :) || || <span style="background-color: #ffffff; color: #000000; msochartype: symbol; msosymbolfontfamily: Wingdings;">  :) || __ // TYPE I Error //  __ || __Type I error__ __Alpha__ **( α ) = (**significance level**)
 * <span style="background-color: #ffffff; color: #000000; display: block; font-family: Arial; font-size: 12pt; text-align: center;">
 * // ( __Reject Ho__ ):   //
 * // ( __Fail to Reject Ho__ ):   //
 * <span style="background-color: #ffffff; color: #000000; font-family: 'Lucida Sans Unicode','Lucida Grande',sans-serif; font-size: 90%; text-decoration: none;">

α** % of the time **We //REJECT// the Ho **//<span style="color: #666699; font-family: 'MS Reference Sans Serif'; font-size: 12pt; mso-bidi-font-family: Arial;">( say Ha is True //Ho is False ) //~//** when the Ho is True The probability of incorrectly rejecting a true statistical null hypothesis.

Type II error <span style="background-color: #ffffff; color: #000000; font-family: 'Lucida Sans Unicode','Lucida Grande',sans-serif; font-size: 90%;"> <span style="background-color: #ffffff; font-family: 'Lucida Sans Unicode','Lucida Grande',sans-serif;"><span style="font-family: Arial; font-size: 12pt; line-height: 115%; mso-ansi-language: EN-US; mso-bidi-language: AR-SA; mso-fareast-font-family: Calibri; mso-fareast-language: EN-US;"> __//-//**__ //<span style="font-family: Arial; font-size: 11pt; line-height: 115%; mso-ansi-language: EN-US; mso-bidi-language: AR-SA; mso-fareast-font-family: Calibri; mso-fareast-language: EN-US;">__say something is true, when really it isn’t__ // <span style="color: #ff9900; font-size: 26pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 12.0pt; mso-fareast-font-family: Arial; msobidifontfamily: Arial; msobidifontsize: 12.0pt; msofareastfontfamily: Arial; msolist: Ignore;">▫ the ability to detect deviations from the Ho <span style="color: #ff9900; font-size: 26pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 12.0pt; mso-fareast-font-family: Arial; msobidifontfamily: Arial; msobidifontsize: 12.0pt; msofareastfontfamily: Arial; msolist: Ignore;">▫  A __good__ test has **//high power//** <span style="color: #ff9900; font-size: 26pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 12.0pt; mso-fareast-font-family: Arial; msobidifontfamily: Arial; msobidifontsize: 12.0pt; msofareastfontfamily: Arial; msolist: Ignore;">▫ Power increases **↑** when you **increase ↑** n <span style="color: #ff9900; font-size: 26pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 12.0pt; mso-fareast-font-family: Arial; msobidifontfamily: Arial; msobidifontsize: 12.0pt; msofareastfontfamily: Arial; msolist: Ignore;">▫  Power decreases ↓ when you decrease ↓ α <span style="font-family: Wingdings; mso-bidi-font-family: Wingdings; mso-fareast-font-family: Wingdings; msobidifontfamily: Wingdings; msofareastfontfamily: Wingdings; msolist: Ignore;">§ Power = 1 – β (or a Type II error)
 * __Power__ **** : **

<span style="color: #ff9900; font-size: 26pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 12.0pt; mso-fareast-font-family: Arial; msobidifontfamily: Arial; msobidifontsize: 12.0pt; msofareastfontfamily: Arial; msolist: Ignore;">▫ If Type I error is worse then α needs to be smaller <span style="color: #ff9900; font-size: 26pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 12.0pt; mso-fareast-font-family: Arial; msobidifontfamily: Arial; msobidifontsize: 12.0pt; msofareastfontfamily: Arial; msolist: Ignore;">▫ If Type II error is worse then α needs to be larger ** To //__Increase__// **( **↑** ) ** Power: ** 1.  ** Increase α ** - //(alpha)// – a __5__% test of significance will have a greater chance of rejecting Ha   2.   Consider a particular alternative that is farther away from Ho (pick mean farther away) 3.  ** Increase ** **n** //(sample size)//
 * 4.  ****  Decrease  ****//<span style="color: red; font-family: Symbol; font-size: 12pt; line-height: 115%; mso-ascii-font-family: 'Times New Roman'; mso-char-type: symbol; mso-fareast-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'; mso-symbol-font-family: Symbol; msoasciifontfamily: 'Times New Roman'; msochartype: symbol; msofareastfontfamily: 'Times New Roman'; msohansifontfamily: 'Times New Roman'; msosymbolfontfamily: Symbol;">s //**<span style="color: red; font-family: 'Times New Roman'; font-size: 12pt; line-height: 115%; mso-fareast-font-family: 'Times New Roman';"> // (standard deviation) //

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Vocab:__ Statistical Inference__- provides a method for drawing conclusions about a population from samples__ Parameter- __<span style="color: #808080; font-family: arial,helvetica,sans-serif;">Quantity which measures an aspect of a population __Statistic-__ Quantity which measures an aspect of a sample

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