Week 5 Lecture 13 This week we look at two different approaches to analyzing data and making inferences about the populations they come from The
Week Lecture This week we look at two different approaches to analyzing data and making inferences about the
Week Lecture This week we look at two different approaches to analyzing data
we look at two different approaches to analyzing data and making inferences about the populations they come from The
Week Lecture This week we look at two different approaches to
analyzing data and making inferences about the populations they come from The
Week Lecture This week we look at two different
Week Lecture This week
Week 5 Lecture 13 This week we look at two different approaches to analyzing data and making inferences about the populations they come from. The...

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TWO QUESTIONS, TOP PART IS THE SAME AS PREVIOUS! 1) READ AND ANSWER LECTURE 13 AND 14 LECTURES ATTACHED! 2ND QUESTION, WRITE QUESTION, AND ANSWER, SAME AS LAST WEEK! 2) Post a question that you had related to the material this week. Conduct research to provide the answer to the question and provide the source. 1) Part One - Confidence Intervals Read Lecture Thirteen. Lecture Thirteen introduces you to confidence intervals. What is a confidence interval, and why do some prefer them to single point estimates? Ask your manager what is preferred and why? What are the strengths and weaknesses of using confidence intervals in making decisions? Part Two - Chi Square Read Lecture Fourteen. As Lecture Fourteen notes, the chi-square test is—in some ways—fundamentally different than the previous tests we have looked at. In what ways and why is this approach important? Examples were shown of gender-degree distributions and employees per grade. How do these tests help with understanding our equal pay for equal work question? Do they change or reinforce our decision from last week? What situations in your personal or professional lives could use a chi-square approach? Part Three - Overall Reactions Has your opinion about statistics changed? How can statistical analysis help your professional career? 2) Post a question that you had related to the material this week. Conduct research to provide the answer to the question and provide the source. Attachment 1 Attachment 2 ATTACHMENT PREVIEW Download attachment .t { position: absolute; -webkit-transform-origin: top left; -moz-transform-origin: top left; -o-transform-origin: top left; -ms-transform-origin: top left; -webkit-transform: scale(0.25); -moz-transform: scale(0.25); -o-transform: scale(0.25); -ms-transform: scale(0.25); z-index: 2; position:absolute; white-space:nowrap; overflow:visible; } // Ensure that we're not replacing any onload events function addLoadEvent(func) { var oldonload = window.onload; if (typeof window.onload != 'function') { window.onload = func; } else { window.onload = function() { if (oldonload) { oldonload(); } func(); } } } addLoadEvent(function(){load1();}); function adjustWordSpacing(widths) { var i, allLinesDone = false; var isDone = []; var currentSpacing = []; var elements = []; // Initialise arrays for (i = 0; i < widths.length; i++) { elements[i] = document.getElementById(widths[i][0]); if (isIE) widths[i][1] = widths[i][1] * 4; if (elements[i].offsetWidth < widths[i][1]) { currentSpacing[i] = Math.floor((widths[i][1] - elements[i].offsetWidth) / elements[i].innerHTML.match(/\s.| ./g).length);//min if (isIE) currentSpacing[i] = Math.floor(currentSpacing[i] / 4); isDone[i] = false; } else { currentSpacing[i] = 1;//too long isDone[i] = true; } } while (!allLinesDone) { // Add each adjustment to the render queue without forcing a render for (i = 0; i < widths.length; i++) { if (!isDone[i]) { elements[i].style.wordSpacing = currentSpacing[i] + 'px'; } } allLinesDone = true; // If elements still need to be wider, add 1 to the word spacing for (i = 0; i < widths.length; i++) { if (!isDone[i] && currentSpacing[i] < 160) { if (elements[i].offsetWidth >= widths[i][1]) { isDone[i] = true; } else { currentSpacing[i]++; allLinesDone = false; } } } } for (i = 0; i < widths.length; i++) { elements[i].style.wordSpacing = (currentSpacing[i] - 1) + 'px'; } } #t1_1{left:270px;top:74px;} #t2_1{left:113px;top:98px;} #t3_1{left:75px;top:114px;} #t4_1{left:333px;top:114px;} #t5_1{left:435px;top:114px;} #t6_1{left:75px;top:129px;} #t7_1{left:75px;top:145px;} #t8_1{left:432px;top:145px;} #t9_1{left:514px;top:145px;} #ta_1{left:75px;top:161px;} #tb_1{left:75px;top:176px;} #tc_1{left:264px;top:200px;} #td_1{left:113px;top:224px;} #te_1{left:75px;top:240px;} #tf_1{left:248px;top:240px;} #tg_1{left:75px;top:255px;} #th_1{left:75px;top:271px;} #ti_1{left:279px;top:271px;} #tj_1{left:75px;top:286px;} #tk_1{left:113px;top:311px;} #tl_1{left:75px;top:326px;} #tm_1{left:391px;top:326px;} #tn_1{left:75px;top:342px;} #to_1{left:75px;top:357px;} #tp_1{left:113px;top:381px;} #tq_1{left:75px;top:397px;} #tr_1{left:75px;top:412px;} #ts_1{left:75px;top:428px;} #tt_1{left:75px;top:444px;} #tu_1{left:75px;top:459px;} #tv_1{left:75px;top:475px;} #tw_1{left:75px;top:491px;} #tx_1{left:513px;top:491px;} #ty_1{left:75px;top:506px;} #tz_1{left:193px;top:506px;} #t10_1{left:214px;top:506px;} #t11_1{left:532px;top:506px;} #t12_1{left:75px;top:522px;} #t13_1{left:75px;top:546px;} #t14_1{left:113px;top:576px;} #t15_1{left:75px;top:590px;} #t16_1{left:75px;top:605px;} #t17_1{left:75px;top:619px;} #t18_1{left:75px;top:634px;} #t19_1{left:113px;top:663px;} #t1a_1{left:440px;top:663px;} #t1b_1{left:75px;top:677px;} #t1c_1{left:75px;top:692px;} #t1d_1{left:113px;top:721px;} #t1e_1{left:167px;top:721px;} #t1f_1{left:75px;top:735px;} #t1g_1{left:181px;top:735px;} .s1_1{ FONT-SIZE: 50px; FONT-FAMILY: TimesNewRomanPS-BoldMT; color: rgb(0,0,0); FONT-WEIGHT: bold; } .s3_1{ FONT-SIZE: 50px; FONT-FAMILY: TimesNewRomanPS-ItalicMT; color: rgb(0,0,0); FONT-STYLE: italic; } .s2_1{ FONT-SIZE: 50px; FONT-FAMILY: TimesNewRomanPSMT; color: rgb(0,0,0); } Week 5 Lecture 13 This week we look at two different approaches to analyzing data and making inferences about the populations they come from. The first is confidence intervals , a range of values that we expect to contain the actual population mean based on the sample results we obtained. The other is a way to use nominal and ordinal data in a statistical analysis. The Chi Square family of tests looks at patterns within samples and sees whether the underlying populations could contain the same pattern of measure distributions (Lind, Marchel, & Wathen, 2008). Confidence Intervals When we perform a t-test or ANOVA, we are using a single point estimate for the means of the populations we are testing. Some professionals and managers are a bit uncomfortable with this; they understand that the sample has a sampling error – and the actual population mean could be – and most likely is – a bit different. They are interested in getting an estimate of what the sampling error is and how much the population mean could differ from the sample mean. We deal with this through the use of confidence intervals, a range of values that have a specific probability of containing the actual population mean. We have seen one example of a confidence mean already, the intervals used to determine which population means varied when we rejected the null hypothesis for the ANOVA test were confidence intervals. Confidence intervals often provide the added information and comfort about estimates of population parameter values that the single point estimates lack. Since the one thing we do know about a statistic generated from a sample is that it will not exactly equal the population parameter, we can use a confidence interval to get a better feel for the range of values that might be the actual population parameter. They also give us an indication of how much variation exists in the data set. The larger the range (at the same confidence level), the more variation within the sample data set and the less representative the mean would be (Lind, Marchel, & Wathen, 2008). We are going to look at two different kinds of confidence intervals this week – intervals for a one sample mean and intervals for the differences between the means of two samples (Lind, Marchel, & Wathen, 2008). One Sample Confidence Interval for the mean A confidence interval is simply a range of values that could contain the actual population parameter of interest. It is centered on the sample mean, and uses the variation in the sample to estimate a range of possible values (Lind, Marchel, & Wathen, 2008). To construct a confidence interval, we use several pieces of information from the sample and the confidence level we want. From the sample we use the mean, standard deviation, and size. To get the confidence level – a desired probability (usually set at 95%), that the interval does, in fact, contain the population mean. Example. The confidence interval for the female mean salary in the population would be calculated this way. The sample mean value is 38, the standard deviation is 18., and the sample var isIE = false; var f1 = [['t1_1',403],['t2_1',1765],['t3_1',1019],['t4_1',405],['t5_1',455],['t6_1',1904],['t7_1',1413],['t8_1',316],['t9_1',185],['ta_1',1939],['tb_1',1532],['tc_1',448],['td_1',1790],['te_1',667],['tf_1',1264],['tg_1',1960],['th_1',789],['ti_1',1068],['tj_1',1794],['tk_1',1746],['tl_1',1236],['tm_1',636],['tn_1',1907],['to_1',1586],['tp_1',1791],['tq_1',1958],['tr_1',1743],['ts_1',1938],['tt_1',1956],['tu_1',1933],['tv_1',1804],['tw_1',1737],['ty_1',457],['t10_1',1258],['t12_1',568],['t13_1',988],['t14_1',1792],['t15_1',1920],['t16_1',1950],['t17_1',1826],['t18_1',135],['t19_1',1280],['t1a_1',436],['t1b_1',1818],['t1c_1',348],['t1e_1',1578],['t1f_1',397],['t1g_1',1502]]; function load1(){ var timeout = 100; if (navigator.userAgent.match(/iPhone|iPad|iPod|Android/i)) timeout = 500; setTimeout(function() {adjustWordSpacing(f1);},timeout); } View the Answer #t1_2{left:75px;top:74px;} #t2_2{left:263px;top:74px;} #t3_2{left:75px;top:89px;} #t4_2{left:286px;top:89px;} #t5_2{left:435px;top:89px;} #t6_2{left:447px;top:89px;} #t7_2{left:75px;top:103px;} #t8_2{left:99px;top:103px;} #t9_2{left:113px;top:132px;} #ta_2{left:75px;top:147px;} #tb_2{left:75px;top:176px;} #tc_2{left:75px;top:205px;} #td_2{left:281px;top:205px;} #te_2{left:75px;top:219px;} #tf_2{left:214px;top:219px;} #tg_2{left:368px;top:219px;} #th_2{left:75px;top:234px;} #ti_2{left:113px;top:263px;} #tj_2{left:364px;top:263px;} #tk_2{left:75px;top:277px;} #tl_2{left:377px;top:277px;} #tm_2{left:75px;top:292px;} #tn_2{left:333px;top:292px;} #to_2{left:75px;top:306px;} #tp_2{left:75px;top:321px;} #tq_2{left:75px;top:350px;} #tr_2{left:113px;top:379px;} #ts_2{left:75px;top:393px;} #tt_2{left:211px;top:393px;} #tu_2{left:426px;top:393px;} #tv_2{left:75px;top:408px;} #tw_2{left:75px;top:422px;} #tx_2{left:170px;top:422px;} #ty_2{left:75px;top:437px;} #tz_2{left:113px;top:466px;} #t10_2{left:75px;top:480px;} #t11_2{left:109px;top:480px;} #t12_2{left:75px;top:495px;} #t13_2{left:75px;top:509px;} #t14_2{left:255px;top:509px;} #t15_2{left:113px;top:538px;} #t16_2{left:226px;top:538px;} #t17_2{left:339px;top:538px;} #t18_2{left:113px;top:567px;} #t19_2{left:339px;top:567px;} #t1a_2{left:113px;top:596px;} #t1b_2{left:151px;top:626px;} #t1c_2{left:75px;top:655px;} #t1d_2{left:113px;top:684px;} .s3_2{ FONT-SIZE: 50px; FONT-FAMILY: TimesNewRomanPS-BoldMT; color: rgb(0,0,0); FONT-WEIGHT: bold; } .s2_2{ FONT-SIZE: 50px; FONT-FAMILY: TimesNewRomanPS-ItalicMT; color: rgb(0,0,0); FONT-STYLE: italic; } .s1_2{ FONT-SIZE: 50px; FONT-FAMILY: TimesNewRomanPSMT; color: rgb(0,0,0); } size is 25 3 (from Week 1 material). Once we determine the confidence level we want, we use the associated 2-tail t value to achieve it. The t-value is found with the fx function t.inv.2t (Prob, df). For a 95% confidence interval, we would use t.inv.2t(0.05, 24), this equals 2.064 (rounded). We now have all the information we need to construct a 95% confidence interval for the female salary mean: CI = mean +/- t * stdev/sqrt(sample size) = 38 +/- 2.064*18.3/sqrt(25) = 38 +/- 7.6. This is typically written as 30.4 to 45.6. Note: the standard deviation divided by the square root of the sample size is called the standard error of the mean , and is the variation measure of the sample used in several statistical tests, including the t-test and confidence intervals. The associated 95% CI for males is 44.6 to 59.3. Note that the endpoints overlap – male smallest vale is 44.6 while the female largest value is 45.6. This suggests that both population average salaries could be the same and around 45. However, just as the two one-sample t-tests gave us misleading information on possible equality, using two confidence intervals to compare two populations also is not the best approach. The Confidence Interval for mean differences. When comparing multiple samples, it is always best to use all the possible information in a single test or procedure. The same is true for confidence intervals. If we are interested in seeing if sample means could be equal, we look to see if the difference between the averages could be 0 or not. If so, then the means could be the same; if not, then the means must be significantly different. The formula for the mean difference confidence interval is mean difference +/- t*standard error. The standard error for the difference of two populations is found by adding the variance/sample size (which is the standard error squared) for each and taking the square root (Lind, Marchel, & Wathen, 2008). For our salary data set we have the following values: Female mean = 38 Male mean = 52 t = t.inv.2t(0.05, 48) = 2.106 Female Stdev = 18.3 Maler Stdev = 17.8 Sample size = 50, df = 48 Standard error = sqrt(Variance (female)/25 + Variance (male)/25) = Sqrt(334.7/25 + 316/25) = 5.10. This gives us a 95% confidence interval for the difference equaling: (52-38) +/- 2.106 * 5.10 = 14 +/- 10.7 = 3.3 to 24.7. var isIE = false; var f2 = [['t1_2',725],['t2_2',1153],['t3_2',817],['t4_2',582],['t6_2',460],['t8_2',1850],['t9_2',1773],['ta_2',404],['tb_2',1674],['tc_2',798],['td_2',1105],['te_2',539],['tf_2',618],['tg_2',717],['th_2',1676],['ti_2',977],['tj_2',778],['tk_2',1182],['tl_2',689],['tm_2',1006],['tn_2',875],['to_2',1927],['tp_2',916],['tq_2',991],['tr_2',1794],['ts_2',517],['tt_2',833],['tu_2',437],['tv_2',1863],['tw_2',352],['tx_2',1422],['ty_2',449],['tz_2',1808],['t11_2',1584],['t12_2',1879],['t13_2',692],['t14_2',1062],['t15_2',373],['t16_2',328],['t17_2',573],['t18_2',841],['t19_2',516],['t1a_2',1363],['t1b_2',641],['t1c_2',1364],['t1d_2',1043]]; function load2(){ var timeout = 100; if (navigator.userAgent.match(/iPhone|iPad|iPod|Android/i)) timeout = 500; setTimeout(function() {adjustWordSpacing(f2);},timeout); } Show entire document ATTACHMENT PREVIEW 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format("woff"); } Week 5 Lecture 14 The Chi Square Test Quite often, patterns of responses or measures give us a lot of information. Patterns are generally the result of counting how many things fit into a particular category. Whenever we make a histogram, bar, or pie chart we are looking at the pattern of the data. Frequently, changes in these visual patterns will be our first clues that things have changed, and the first clue that we need to initiate a research study (Lind, Marchel, & Wathen, 2008). One of the most useful test in examining patterns and relationships in data involving counts (how many fit into this category, how many into that, etc.) is the chi-square. It is extremely easy to calculate and has many more uses than we will cover. Examining patterns involves two uses of the Chi-square - the goodness of fit and the contingency table . Both of these uses have a common trait: they involve counts per group. In fact, the chi-square is the only statistic we will look at that we use when we have counts per multiple groups (Tanner & Youssef-Morgan, 2013). Chi Square Goodness of Fit Test The goodness of fit test checks to see if the data distribution (counts per group) matches some pattern we are interested in. Example: Are the employees in our example company distributed equal across the grades? Or, a more reasonable expectation for a company might be are the employees distributed in a pyramid fashion – most on the bottom and few at the top? The Chi Square test compares the actual versus a proposed distribution of counts by generating a measure for each cell or count: (actual – expected) 2 /actual. Summing these for all of the cells or groups provides us with the Chi Square Statistic. As with our other tests, we determine the p-value of getting a result as large or larger to determine if we reject or not reject our null hypothesis. An example will show the approach using Excel. Regardless of the Chi Square test, the chi square related functions are found in the fx Statistics window rather than the Data Analysis where we found the t and ANOVA test functions. The most important for us are: • CHISQ.TEST (actual range, expected range) – returns the p-value for the test • CHISQ.INV.RT(p-value, df) – returns the actual Chi Square value for the p-value or probability value used. • CHISQ.DIST.RT(X, df) – returns the p-value for a given value. When we have a table of actual and expected results, using the =CHISQ.TEST(actual range, expected range) will provide us with the p-value of the calculated chi square value (but does not give us the actual calculated chi square value for the test). We can compare this value against our alpha criteria (generally 0.05) to make our decision about rejecting or not rejecting the null hypothesis. var isIE = false; var f1 = [['t1_1',403],['t2_1',445],['t3_1',1763],['t4_1',1876],['t5_1',1953],['t6_1',1930],['t7_1',1340],['t8_1',1696],['t9_1',1773],['ta_1',1864],['tb_1',748],['tc_1',368],['te_1',429],['tf_1',190],['tg_1',1951],['th_1',1777],['ti_1',508],['tj_1',697],['tk_1',1768],['tl_1',893],['tm_1',887],['tn_1',714],['to_1',1180],['tp_1',1852],['tq_1',1687],['tr_1',1275],['tu_1',445],['tv_1',1267],['tw_1',538],['tx_1',1914],['ty_1',397],['tz_1',985],['t10_1',1651],['t12_1',1753],['t13_1',838],['t15_1',1556],['t17_1',1644],['t18_1',513],['t1a_1',1279],['t1b_1',1719],['t1c_1',1887],['t1d_1',1913],['t1e_1',1896],['t1f_1',417]]; function load1(){ var timeout = 100; if (navigator.userAgent.match(/iPhone|iPad|iPod|Android/i)) timeout = 500; setTimeout(function() {adjustWordSpacing(f1);},timeout); } View the Answer #t1_2{left:113px;top:74px;} #t2_2{left:75px;top:89px;} #t3_2{left:75px;top:103px;} #t4_2{left:75px;top:118px;} #t5_2{left:75px;top:132px;} #t6_2{left:75px;top:146px;} #t7_2{left:113px;top:176px;} #t8_2{left:202px;top:176px;} #t9_2{left:75px;top:190px;} #ta_2{left:75px;top:204px;} #tb_2{left:264px;top:204px;} #tc_2{left:75px;top:219px;} #td_2{left:88px;top:250px;} #te_2{left:147px;top:250px;} #tf_2{left:198px;top:250px;} #tg_2{left:248px;top:250px;} #th_2{left:298px;top:250px;} #ti_2{left:348px;top:250px;} #tj_2{left:399px;top:250px;} #tk_2{left:441px;top:250px;} #tl_2{left:89px;top:264px;} #tm_2{left:145px;top:264px;} #tn_2{left:195px;top:264px;} #to_2{left:245px;top:264px;} #tp_2{left:299px;top:264px;} #tq_2{left:349px;top:264px;} #tr_2{left:399px;top:264px;} #ts_2{left:447px;top:264px;} #tt_2{left:113px;top:290px;} #tu_2{left:88px;top:322px;} #tv_2{left:147px;top:322px;} #tw_2{left:198px;top:322px;} #tx_2{left:248px;top:322px;} #ty_2{left:298px;top:322px;} #tz_2{left:348px;top:322px;} #t10_2{left:399px;top:322px;} #t11_2{left:441px;top:322px;} #t12_2{left:89px;top:335px;} #t13_2{left:145px;top:335px;} #t14_2{left:198px;top:335px;} #t15_2{left:248px;top:335px;} #t16_2{left:299px;top:335px;} #t17_2{left:346px;top:335px;} #t18_2{left:399px;top:335px;} #t19_2{left:447px;top:335px;} #t1a_2{left:75px;top:362px;} #t1b_2{left:75px;top:391px;} #t1c_2{left:113px;top:420px;} #t1d_2{left:75px;top:449px;} #t1e_2{left:75px;top:478px;} #t1f_2{left:75px;top:508px;} #t1g_2{left:205px;top:508px;} .s2_2{ FONT-SIZE: 50px; FONT-FAMILY: TimesNewRomanPS-BoldMT; color: rgb(0,0,0); FONT-WEIGHT: bold; } .s3_2{ FONT-SIZE: 41px; FONT-FAMILY: Arial, Helvetica, sans-serif; color: rgb(0,0,0); } .s1_2{ FONT-SIZE: 50px; FONT-FAMILY: TimesNewRomanPSMT; color: rgb(0,0,0); } If, after finding the p-value for our chi square test, we want to determine the calculated value of the chi square statistic, we can use the =CHISQ.INV.RT(probability, df) function, the value for probability is our chi square test outcome, and the degrees of freedom (df) equals the number of cells in our actual table minus 1 (6 – 1 =5 for an problem working with our 6 grade levels). Finally, if we are interested in the probability of exceeding a particular chi square value, we can use the CHIDIST or CHISQ.DIST.RT function. Excel Example. To see if our employees are distributed in a traditional pyramid shape, we would use the Chi Square Goodness of Fit test as we are dealing both with count data and with a proposed distribution pattern. For this test, let us assume the following table shows the expected distribution of our 50 employees in a pyramid organizational structure. Grade: A B C D E F Total Count: 15 12 10 6 4 3 50 The actual or observed distribution within our sample is shown below. Grade: A B C D E F Total Count: 15 7 5 5 12 6 50 The research question: Are employees distributed in a pyramidal fashion? Step 1: Ho: No difference exists between observed and expected frequency counts Ha: Observed and Expected frequencies differ. Step 2: Reject the null hypothesis if the p-value < alpha = .05. Step 3: Chi Square Goodness of Fit test. Step 4: Conduct the test. Below is a screen short of an Excel solution. var isIE = false; var f2 = [['t1_2',1745],['t2_2',1898],['t3_2',1899],['t4_2',1890],['t5_2',1944],['t6_2',1117],['t7_2',331],['t8_2',1410],['t9_2',1873],['ta_2',729],['tb_2',1135],['tc_2',1615],['tt_2',1416],['t1a_2',1483],['t1b_2',1654],['t1c_2',943],['t1d_2',1246],['t1e_2',807],['t1f_2',493],['t1g_2',895]]; function load2(){ var timeout = 100; if (navigator.userAgent.match(/iPhone|iPad|iPod|Android/i)) timeout = 500; setTimeout(function() {adjustWordSpacing(f2);},timeout); } Show entire document

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