For the data procured via the provided excel sheet on the New York survey, the observations had to independently be tabulated so as the results would then provide a contextualized narrative. That said, the first aspect of the paper would deal with the sample mean µ 1 (average mean value for Manhattan) along with ( µ 2 - µ 5 ), which are the sample mean values for Brooklyn, Queens, Bronx and The Staten Island. The notation used to calculating the said mean was as follows:
Additionally, the coefficient limits were also to be calculated along with the standard error which implicated the necessary use of standard deviation, the critical value t, and the use of a Z distribution chart. The expressions demonstrated subsequently are all used in calculating the below tabulated results.
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Manhattan | Brooklyn | Queens | The Bronx | Staten Island | Education | Marital Status | |
Mean | 2.67 | 4.53 | 2.47 | 2.33 | 4.09 | 3.48 | 1.75 |
Standard Deviation | 1.06 | 1.04 | 1.20 | 1.37 | 1.22 | 1.19 | 0.43 |
Standard Error | 0.05 | 0.05 | 0.06 | 0.06 | 0.06 | 0.06 | 0.02 |
Coefficient limits | 0.10 | 0.10 | 0.11 | 0.13 | 0.11 | 0.11 | 0.04 |
The most influential metric among all the tabulated entries above is the standard error which is calculated using the following equation:
Where is the standard deviation of the sample data set expressed as below
The confidence limits are then calculated using the confidence coefficient (Z α /2) after which, using a Z chart, the obtained value can then be multiplied with the above standard error. The result procured is displayed as the calculated mean ± confidence limits therefore, for all our variables mentioned in the provided data set we can calculate the confidence limits for each of them based on a 95 percent accuracy.
Variable | Confidence Limits |
Manhattan | 2.67 ± 0.10 |
Brooklyn | 4.53 ± 0.10 |
The Bronx | 2.47 ± 0.11 |
Queens | 2.33 ± 0.13 |
Staten Island | 4.09 ± 0.11 |
Education | 3.48 ± 0.11 |
Marital Status | 1.75 ± 0.04 |
Percentage Population Doing PHD | 16.18 |
Marital Status as ‘Single or Other’ | 74.83 |
Given that range for the variable ‘education’ maps between 1-5, with 1 being the lowest and 5 being the highest, the sample data set estimates that 16.18 percent of the population holds doctoral degrees while at the same and using the same parameters 74.83 percent of population is having a marital status as ‘single or other’.
Estimating Sample Size – Cochran’s Formula
In derivative statistics Cochran’s formula is used to determine the appropriate sample size needed in order for a successful statistical review. In this paper we will assume that the sample needed should estimate eighty percent of the total burrow population for each borrow. This can be achieved by plugging in the necessary values in the following expression:
Where ‘Z’ is the z value obtained from the z chart, ‘p’ being the percentage of population representing the data set having the attributes in question, ‘q’ = 1 – p, and finally ‘e’ being the precession or the stated margin of error. Since we are assuming that our data set represents eighty percent of the burrow population p = 0.8 which would imply q = 0.2 and consequently our will be equal to 246 sample observations.
