I have "solved" a question in mathematical statistics, but unfortunately received an incorrect answer and I would appreciate any help in understanding why my method did not work in this case. The question is as follows:In city A, 500 randomly selected people were tested for antibodies to covid-19, which resulted in 110 giving positive results. In city B, a random sample of 500 people was also drawn, and here 91 had positive results. Can it be said that it is statistically certain that city A has a larger population share with antibodies than city B? (Justify your answer with an appropriate test. Use 5% significance level.)I started by producing P ∗ ( A ) = 110 500 = 0.22 and respectively P ∗ ( B ) = 91 500 = 0.182.Then I chose to continued with a one-sided hypothesis testing where I assumed that the data was binomially distributed according to B i n ( n = 500 , p ), with H 0 : P = 0.182 and H 1 :&gt; 0.182.And finally brought out the one-sided interval according to: I p = λ α P ∗ ( A ) ( 1 − P ∗ ( A ) ) n ⇒ [ 0.189 , ∞ ] ..And concluded that H 0 should be rejected since 0.182 is not included in the interval. But according to the results, H 0 should not be rejected and they have also used a different method than I did. Why is this method not applicable in my case?Note: At first I thought that errors were because I only used α and not α 2 in the calculation, however, even if I were to replace this, 0.182 is not in the interval.

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I have &quot;solved&quot; a question in mathematical statistics, but unfortunately received an incorrect answer and I would appreciate any help in understanding why my method did not work in this case. The question is as follows:In city A, 500 randomly selected people were tested for antibodies to covid-19, which resulted in 110 giving positive results. In city B, a random sample of 500 people was also drawn, and here 91 had positive results. Can it be said that it is statistically certain that city A has a larger population share with antibodies than city B? (Justify your answer with an appropriate test. Use 5% significance level.)I started by producing       P    ∗    (  A  )  =      110    500    =  0.22 and respectively       P    ∗    (  B  )  =      91    500    =  0.182.Then I chose to continued with a one-sided hypothesis testing where I assumed that the data was binomially distributed according to   B  i  n  (  n  =  500  ,  p  ), with      H    0    :  P  =  0.182   and       H    1    :&amp;gt;  0.182.And finally brought out the one-sided interval according to:      I    p    =      λ    α                                P          ∗                (        A        )        (        1        −                  P          ∗                (        A        )        )            n        ⇒  [  0.189  ,  ∞  ]  ..And concluded that       H    0   should be rejected since 0.182 is not included in the interval. But according to the results,       H    0   should not be rejected and they have also used a different method than I did. Why is this method not applicable in my case?Note: At first I thought that errors were because I only used α and not       α    2   in the calculation, however, even if I were to replace this, 0.182 is not in the interval.

redclick53 · Accepted Answer

Your standard deviation formula is off by the so-called finite sample correction factor. In essence, if you sample a large piece of the total parent of size N, there is an associated reduction in expected variability (going to even zero for a sample equal size equal to N). In other words, complete sample implies complete accuracy in measuring the parent&#039;s portion of an item.My review of the theoretical math presented in the cited source below around the selection of the required sample estimate m (here 500) as to meeting the criteria stated remains, however, a relative function of each cities actual size (see extended discussion and formula derivation in this educational source, for example).However, numerically assigning values from a rationale range of potential parent population city sizes may be insightful if, for example, a sample size equal to 500 (as the m) is at all possibly sufficient/insufficient to meet the required confidence level and error level specified.

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