{"id":594,"date":"2020-05-13T18:47:52","date_gmt":"2020-05-13T22:47:52","guid":{"rendered":"http:\/\/pressbooks.library.upei.ca\/montelpare\/?post_type=chapter&p=594"},"modified":"2020-08-24T14:16:49","modified_gmt":"2020-08-24T18:16:49","slug":"computing-the-z-statistic-for-the-one-sample-runs-test","status":"publish","type":"chapter","link":"https:\/\/pressbooks.library.upei.ca\/montelpare\/chapter\/computing-the-z-statistic-for-the-one-sample-runs-test\/","title":{"raw":"Computing the Z Statistic for the One Sample Runs Test","rendered":"Computing the Z Statistic for the One Sample Runs Test"},"content":{"raw":"Often in quantitative methods, we expect that any score we observe occurs at random and is not a result of selection bias.\u00a0 This expectation is of particular importance when we are dealing with strings of binary events, such as viewing the change in a particular measure over time or counting the sequence of similar outcomes without a break.\r\n\r\nWald and Wolfowitz referred to such strings or sequences of similar events as runs<\/em><\/strong>. A run is defined as a sequence of similar data values. A run of an event occurs when a particular outcome of interest is observed within a sampling space. A run can have a sequence of 1 or a run can have a sequence of > 1.\r\n\r\nFor example, consider the toss of a fair coin. If we toss the coin 20 times we could expect to observe the following extreme outcomes:\r\n
    \r\n \t
  1. H,T,H,T,H,T,H,T,H,T,H,T,H,T,H,T,H,T,H,T<\/li>\r\n \t
  2. H,H,H,H,H,H,H,H,H,H,T,T,T,T,T,T,T,T,T,T<\/li>\r\n<\/ol>\r\nIn the first sample space above (1), the outcome was a complete interspersing of each toss of H followed by a T (or T followed by H). In the second sample space above (2) we observe the complete clumping of ten heads followed by ten tails, both of these events can be considered random, but they represent the extremes of what we might observe.\r\n\r\nConsider that the purpose of the runs test is to determine, within a string of events, the randomness of fluctuations.\u00a0 That is, do the observed fluctuations (if any) occur at random or do the fluctuations of observations exhibit some form of clumping together? Does the sequence observed within a sampling space represent a pattern over a given sampling space or period of time.\r\n\r\nThe formula for the runs z-statistic is shown here:\r\n\r\n\"\"\r\n\r\nThere are three parts to the z formula for the One Sample Runs Test.\r\n\r\n1) The first part is to count the number of runs of a given type of events. For example, in the coin toss example, there were 20 tosses of a fair coin, which resulted in 10 runs<\/strong> as shown here:\r\n\r\nH, T, T, T, H, T, H, H, T, H, H, H, T, T, H, H, H, H, T, T\r\n\r\n2) The second part is to compute the mean number of expected runs using the formula:\r\n\r\n\"\"\r\n

    In this scenario, there were 21 reported outcomes whereby we consider the number of heads were counted as n1<\/sub> and the number of tails were counted as n2<\/sub>, so that n1<\/sub>=11 and n2<\/sub>=9.<\/p>\r\n\"\"\r\n\r\n3) The third part of the calculation is to compute the standard deviation of the estimate of runs using the formula:\r\n\r\n\"\"\r\n

    4) the calculation of z for the runs test is then simplified to:\"\"<\/p>\r\nThe evaluation of runs of events is a z test, which means that the evaluation of the null hypothesis associated with this test is based on a normal (z) distribution.\r\n\r\nThe value of z = 0.42 is within the region of acceptance of the null hypothesis, as shown with this graph. The null hypothesis: \u00a0is accepted if the z observed > -1.96 and <1.96.\r\n\r\n\"\"\r\n\r\nTherefore, we accept the null hypothesis that there is no pattern or sequence to THIS<\/strong> toss of a fair coin.\r\n\r\n

    \r\n\r\n

    Your Turn: Compute the One Sample Runs Test<\/h2>\r\nConsider the runs of increases and decreases in the daily weather pattern for one month in the seaside Village of Cavendish, Prince Edward Island. A run is defined as a sequence of similar data values. The sequence can be a single entry, or a string of entries occupying the entire set of observations. Since you are a golfer, who likes to play when the weather is hot, you hope that there is only one run and that it is a positive increase to warmer weather each day.\r\n\r\nUse the approach explained <\/strong>for the \u201cone sample runs test\u201d to compute the significance of the runs of temperatures in the following example. In your response state the null hypothesis for this question, and in addition to the results of your computations, include a statement about your decision of whether to accept or reject the null hypothesis.\r\n\r\nData Set #3 One sample runs test data for the month of July\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
    Date & Temperature<\/td>\r\nChange<\/td>\r\nDate & Temperature<\/td>\r\nChange<\/td>\r\n<\/tr>\r\n
    June 30th 20\u00ba C<\/td>\r\n\u00b7<\/td>\r\nJuly 16th 20\u00ba C<\/td>\r\n-<\/td>\r\n<\/tr>\r\n
    July 1st 21\u00ba C<\/td>\r\n+<\/td>\r\nJuly 17th\u00a0 21\u00ba C<\/td>\r\n+<\/td>\r\n<\/tr>\r\n
    July 2nd 22\u00ba C<\/td>\r\n+<\/td>\r\nJuly 18th\u00a0 22\u00ba C<\/td>\r\n+<\/td>\r\n<\/tr>\r\n
    July 3rd\u00a0 22.5\u00ba C<\/td>\r\n+<\/td>\r\nJuly 19th\u00a0 23\u00ba C<\/td>\r\n+<\/td>\r\n<\/tr>\r\n
    July 4th 23\u00ba C<\/td>\r\n+<\/td>\r\nJuly 20th\u00a0 25\u00ba C<\/td>\r\n+<\/td>\r\n<\/tr>\r\n
    July 5th\u00a0 24\u00ba C<\/td>\r\n+<\/td>\r\nJuly 21st 23\u00ba C<\/td>\r\n-<\/td>\r\n<\/tr>\r\n
    July 6th\u00a0 25\u00ba C<\/td>\r\n+<\/td>\r\nJuly 22nd\u00a0 23.5\u00ba C<\/td>\r\n+<\/td>\r\n<\/tr>\r\n
    July 7th\u00a0 26\u00ba C<\/td>\r\n+<\/td>\r\nJuly 23rd\u00a0 22\u00ba C<\/td>\r\n-<\/td>\r\n<\/tr>\r\n
    July 8th\u00a0 24\u00ba C<\/td>\r\n-<\/td>\r\nJuly 24th\u00a0 21\u00ba C<\/td>\r\n-<\/td>\r\n<\/tr>\r\n
    July 9th\u00a0 21\u00ba C<\/td>\r\n-<\/td>\r\nJuly 25th\u00a0 20\u00ba C<\/td>\r\n-<\/td>\r\n<\/tr>\r\n
    July 10th\u00a0 19\u00ba C<\/td>\r\n-<\/td>\r\nJuly 26th\u00a0 24\u00ba C<\/td>\r\n+<\/td>\r\n<\/tr>\r\n
    July 11th\u00a0 18\u00ba C<\/td>\r\n-<\/td>\r\nJuly 27th\u00a0 25\u00ba C<\/td>\r\n+<\/td>\r\n<\/tr>\r\n
    July 12th\u00a0 21\u00ba C<\/td>\r\n+<\/td>\r\nJuly 28th\u00a0 26\u00ba C<\/td>\r\n+<\/td>\r\n<\/tr>\r\n
    July 13th\u00a0 22\u00ba C<\/td>\r\n+<\/td>\r\nJuly 29th\u00a0 27\u00ba C<\/td>\r\n+<\/td>\r\n<\/tr>\r\n
    July 14th\u00a0 22.5\u00ba C<\/td>\r\n+<\/td>\r\nJuly 30th\u00a0 25\u00ba C<\/td>\r\n-<\/td>\r\n<\/tr>\r\n
    July 15th\u00a0 21\u00ba C<\/td>\r\n-<\/td>\r\nJuly 31st 24\u00ba C<\/td>\r\n-<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\n\r\n\r\n\r\n
    Null hypothesis<\/td>\r\nAverage run\r\n\r\nmr<\/td>\r\nStandard deviation\r\n\r\n(sr)<\/td>\r\nZ runs test<\/td>\r\nDecision concerning the null hypothesis<\/td>\r\n<\/tr>\r\n
    <\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>","rendered":"

    Often in quantitative methods, we expect that any score we observe occurs at random and is not a result of selection bias.\u00a0 This expectation is of particular importance when we are dealing with strings of binary events, such as viewing the change in a particular measure over time or counting the sequence of similar outcomes without a break.<\/p>\n

    Wald and Wolfowitz referred to such strings or sequences of similar events as runs<\/em><\/strong>. A run is defined as a sequence of similar data values. A run of an event occurs when a particular outcome of interest is observed within a sampling space. A run can have a sequence of 1 or a run can have a sequence of > 1.<\/p>\n

    For example, consider the toss of a fair coin. If we toss the coin 20 times we could expect to observe the following extreme outcomes:<\/p>\n

      \n
    1. H,T,H,T,H,T,H,T,H,T,H,T,H,T,H,T,H,T,H,T<\/li>\n
    2. H,H,H,H,H,H,H,H,H,H,T,T,T,T,T,T,T,T,T,T<\/li>\n<\/ol>\n

      In the first sample space above (1), the outcome was a complete interspersing of each toss of H followed by a T (or T followed by H). In the second sample space above (2) we observe the complete clumping of ten heads followed by ten tails, both of these events can be considered random, but they represent the extremes of what we might observe.<\/p>\n

      Consider that the purpose of the runs test is to determine, within a string of events, the randomness of fluctuations.\u00a0 That is, do the observed fluctuations (if any) occur at random or do the fluctuations of observations exhibit some form of clumping together? Does the sequence observed within a sampling space represent a pattern over a given sampling space or period of time.<\/p>\n

      The formula for the runs z-statistic is shown here:<\/p>\n

      \"\"<\/p>\n

      There are three parts to the z formula for the One Sample Runs Test.<\/p>\n

      1) The first part is to count the number of runs of a given type of events. For example, in the coin toss example, there were 20 tosses of a fair coin, which resulted in 10 runs<\/strong> as shown here:<\/p>\n

      H, T, T, T, H, T, H, H, T, H, H, H, T, T, H, H, H, H, T, T<\/p>\n

      2) The second part is to compute the mean number of expected runs using the formula:<\/p>\n

      \"\"<\/p>\n

      In this scenario, there were 21 reported outcomes whereby we consider the number of heads were counted as n1<\/sub> and the number of tails were counted as n2<\/sub>, so that n1<\/sub>=11 and n2<\/sub>=9.<\/p>\n

      \"\"<\/p>\n

      3) The third part of the calculation is to compute the standard deviation of the estimate of runs using the formula:<\/p>\n

      \"\"<\/p>\n

      4) the calculation of z for the runs test is then simplified to:\"\"<\/p>\n

      The evaluation of runs of events is a z test, which means that the evaluation of the null hypothesis associated with this test is based on a normal (z) distribution.<\/p>\n

      The value of z = 0.42 is within the region of acceptance of the null hypothesis, as shown with this graph. The null hypothesis: \u00a0is accepted if the z observed > -1.96 and <1.96.<\/p>\n

      \"\"<\/p>\n

      Therefore, we accept the null hypothesis that there is no pattern or sequence to THIS<\/strong> toss of a fair coin.<\/p>\n

      \n

      Your Turn: Compute the One Sample Runs Test<\/h2>\n

      Consider the runs of increases and decreases in the daily weather pattern for one month in the seaside Village of Cavendish, Prince Edward Island. A run is defined as a sequence of similar data values. The sequence can be a single entry, or a string of entries occupying the entire set of observations. Since you are a golfer, who likes to play when the weather is hot, you hope that there is only one run and that it is a positive increase to warmer weather each day.<\/p>\n

      Use the approach explained <\/strong>for the \u201cone sample runs test\u201d to compute the significance of the runs of temperatures in the following example. In your response state the null hypothesis for this question, and in addition to the results of your computations, include a statement about your decision of whether to accept or reject the null hypothesis.<\/p>\n

      Data Set #3 One sample runs test data for the month of July<\/p>\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
      Date & Temperature<\/td>\nChange<\/td>\nDate & Temperature<\/td>\nChange<\/td>\n<\/tr>\n
      June 30th 20\u00ba C<\/td>\n\u00b7<\/td>\nJuly 16th 20\u00ba C<\/td>\n–<\/td>\n<\/tr>\n
      July 1st 21\u00ba C<\/td>\n+<\/td>\nJuly 17th\u00a0 21\u00ba C<\/td>\n+<\/td>\n<\/tr>\n
      July 2nd 22\u00ba C<\/td>\n+<\/td>\nJuly 18th\u00a0 22\u00ba C<\/td>\n+<\/td>\n<\/tr>\n
      July 3rd\u00a0 22.5\u00ba C<\/td>\n+<\/td>\nJuly 19th\u00a0 23\u00ba C<\/td>\n+<\/td>\n<\/tr>\n
      July 4th 23\u00ba C<\/td>\n+<\/td>\nJuly 20th\u00a0 25\u00ba C<\/td>\n+<\/td>\n<\/tr>\n
      July 5th\u00a0 24\u00ba C<\/td>\n+<\/td>\nJuly 21st 23\u00ba C<\/td>\n–<\/td>\n<\/tr>\n
      July 6th\u00a0 25\u00ba C<\/td>\n+<\/td>\nJuly 22nd\u00a0 23.5\u00ba C<\/td>\n+<\/td>\n<\/tr>\n
      July 7th\u00a0 26\u00ba C<\/td>\n+<\/td>\nJuly 23rd\u00a0 22\u00ba C<\/td>\n–<\/td>\n<\/tr>\n
      July 8th\u00a0 24\u00ba C<\/td>\n–<\/td>\nJuly 24th\u00a0 21\u00ba C<\/td>\n–<\/td>\n<\/tr>\n
      July 9th\u00a0 21\u00ba C<\/td>\n–<\/td>\nJuly 25th\u00a0 20\u00ba C<\/td>\n–<\/td>\n<\/tr>\n
      July 10th\u00a0 19\u00ba C<\/td>\n–<\/td>\nJuly 26th\u00a0 24\u00ba C<\/td>\n+<\/td>\n<\/tr>\n
      July 11th\u00a0 18\u00ba C<\/td>\n–<\/td>\nJuly 27th\u00a0 25\u00ba C<\/td>\n+<\/td>\n<\/tr>\n
      July 12th\u00a0 21\u00ba C<\/td>\n+<\/td>\nJuly 28th\u00a0 26\u00ba C<\/td>\n+<\/td>\n<\/tr>\n
      July 13th\u00a0 22\u00ba C<\/td>\n+<\/td>\nJuly 29th\u00a0 27\u00ba C<\/td>\n+<\/td>\n<\/tr>\n
      July 14th\u00a0 22.5\u00ba C<\/td>\n+<\/td>\nJuly 30th\u00a0 25\u00ba C<\/td>\n–<\/td>\n<\/tr>\n
      July 15th\u00a0 21\u00ba C<\/td>\n–<\/td>\nJuly 31st 24\u00ba C<\/td>\n–<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n\n\n\n
      Null hypothesis<\/td>\nAverage run<\/p>\n

      mr<\/td>\n

      		Standard deviation<\/p>\n

      (sr)<\/td>\n

      		Z runs test<\/td>\nDecision concerning the null hypothesis<\/td>\n<\/tr>\n
      <\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n","protected":false},"author":56,"menu_order":4,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-594","chapter","type-chapter","status-publish","hentry"],"part":38,"_links":{"self":[{"href":"https:\/\/pressbooks.library.upei.ca\/montelpare\/wp-json\/pressbooks\/v2\/chapters\/594","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.library.upei.ca\/montelpare\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.library.upei.ca\/montelpare\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.library.upei.ca\/montelpare\/wp-json\/wp\/v2\/users\/56"}],"version-history":[{"count":12,"href":"https:\/\/pressbooks.library.upei.ca\/montelpare\/wp-json\/pressbooks\/v2\/chapters\/594\/revisions"}],"predecessor-version":[{"id":613,"href":"https:\/\/pressbooks.library.upei.ca\/montelpare\/wp-json\/pressbooks\/v2\/chapters\/594\/revisions\/613"}],"part":[{"href":"https:\/\/pressbooks.library.upei.ca\/montelpare\/wp-json\/pressbooks\/v2\/parts\/38"}],"metadata":[{"href":"https:\/\/pressbooks.library.upei.ca\/montelpare\/wp-json\/pressbooks\/v2\/chapters\/594\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.library.upei.ca\/montelpare\/wp-json\/wp\/v2\/media?parent=594"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.library.upei.ca\/montelpare\/wp-json\/pressbooks\/v2\/chapter-type?post=594"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.library.upei.ca\/montelpare\/wp-json\/wp\/v2\/contributor?post=594"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.library.upei.ca\/montelpare\/wp-json\/wp\/v2\/license?post=594"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}