Significance Testing

Part 1:

1.        

	Interval Type	Confidence Level	n	Significance Level	Z or t?	Z or T Value
A	Two Tailed	90	45	0.1	Z test	1.64, -1.64
B	Two Tailed	95	12	0.05	t test	2.201, -2.201
C	One Tailed	95	36	0.05	Z test	±1.64
D	Two Tailed	99	180	0.01	Z test	2.57, -2.57
E	One Tailed	80	60	0.2	Z test	±0.84
F	One Tailed	99	23	0.01	t test	±2.508
G	Two Tailed	99	15	0.01	t test	2.977, -2.977

 

      

 

 

 

 

 

 

 

 

 

 

2.

A Department of the interior in Washington D.C. estimates that the number of particular

invasive species in a certain county (Bucks County) should number as follows (averages based on

data from the whole state of Pennsylvania) per acre: Asian-Long Horned Beetle, 4; Emerald Ash

Borer Beetle, 10; and Golden Nematode, 75.  A survey of 50 fields had the following results:

                                                                      μ               σ

• Asian-Long Horned Beetle          3.2          0.73
• Emerald Ash Borer Beetle           11.7        1.3
• Golden Nematode                        77           5.71

Asian Long Horned Beetle

·       Null Hypothesis: there is no difference between the average amount of Asian Long Horned Beetles in the 50 field sample in Bucks County and the average amount of Asian Long Horned Beetles in all of Pennsylvania

·         Alternative Hypothesis: There is a difference between the average amount of Asian Long Horned Beetles in 50 field sample in Bucks County and the average amount of Asian Long Horned Beetles in all of Pennsylvania.

·         A Z test will be used to determine whether or not there is a difference between the mean of the sample species compared to the entire state of Pennsylvania.

·         The significance level is 0.05 using a two tailed test and a critical value of ±1.96.

·         The Z value is -7.75. Thus rejecting the null hypothesis stating there is a difference between the average amount of Asian Long Horned Beetles in the 50 field sample in Bucks County compared to the average amount of Asian Long Horned Beetles in the entire state of Pennsylvania. There are far less Asian Long Horned Beetles in the Bucks County, compared to the entire state.  

Emerald Ash Borer Beetle

·         Null Hypothesis: there is no difference between the average amount of Emerald Ash Borer Beetles in the 50 field sample in Bucks County and the average amount of Emerald Ash Borer Beetles in all of Pennsylvania

·         Alternative Hypothesis: There is a difference between the average amount of Emerald Ash Borer Beetles in the 50 field sample in Bucks County and the average amount of Emerald Ash Borer Beetle in all of Pennsylvania.

·         A Z test will be used to determine whether or not there is a difference between the mean of the sample species compared to the entire state of Pennsylvania.

·         The significance level is 0.05 using a two tailed test and a critical value of ±1.96.

·         The Z value is 9.25. Thus rejecting the null hypothesis stating there is a difference between the average amount of Emerald Ash Borer Beetles in the 50 field sample in Bucks County compared to the average amount of Emerald Ash Borer Beetles in the entire state of Pennsylvania. There are far more Emerald Ash Borer Beetles in Buck County compared to the entire state.  

Golden Nematode

·         Null Hypothesis: there is no difference between the average amount of Golden Nematodes in the 50 field sample in Bucks County and the average amount of Golden Nematodes in all of Pennsylvania

·         Alternative Hypothesis: There is a difference between the average amount of Golden Nematode in the 50 field sample in Bucks County and the average amount of Golden Nematode in all of Pennsylvania.

·         A Z test will be used to determine whether or not there is a difference between the mean of the sample species compared to the entire state of Pennsylvania.

·         The significance level is 0.05 using a two tailed test and a critical value of ±1.96.

·         The Z value is 2.48. Thus rejecting the null hypothesis stating there is a difference between the average amount of Golden Nematodes in the 50 field sample in Bucks County compared to the average amount of Golden Nematodes in the entire state of Pennsylvania. There are more Golden Nematodes in Bucks County compared to the entire state. 

 

3.  

         An exhaustive survey of all users of a wilderness park taken in 1960 revealed that the average number of persons per party was 2.1.  In a random sample of 25 parties in 1985, the average was 3.4 persons with a standard deviation of 1.32

·         Null Hypothesis: there is no difference in the average number of persons per party in the 25 party sample in 1985 compared to the average number of persons per party in all parties in 1960.

·         Alternative Hypothesis: there is a difference in the average number of persons per party in the 25 party sample in 1985 compared to the average number of persons per party in all parties in 1960.

·         A t test will be used to determine whether or not there is a difference between the mean number of people of the sample of 1985 parties, compared to the mean number of people of the total parties in 1960.

·         The significance level is 0.05 using a one tailed test and a critical value of 1.177.

·         The t value is 4.92. Thus rejecting the null hypothesis stating there is a difference between the number of people per party in the sample of 25 people in 1985 and the number of people per party in all parties in 1960. There are significantly more people per party in 1985 compared to the number of people per party in 1960.

Part 2:

Introduction:

The perception of “Up North” that many individuals who live in Wisconsin have is an obscure concept for individuals outside of the state of Wisconsin. For many the idea of up north is simply viewed as the directional component, meaning north, south, east and west are merely a direction. However, individuals who inhabit that state of Wisconsin possess a conceptual idea of “Up North” attaching specific characteristics to the overall meaning. Certain characteristics connected to this perception may include forested land cover, cottages, and also deer hunting. To determine whether or not certain characteristics such as these truly capture the true Wisconsin meaning of “Up North” spatial representations are required. Mapping these variables and defining a boundary dividing north and south by Highway 29, creates a visuals perspective of the connection between the directionality and the characteristics that define “Up North”. Not only will spatial representations help define a connection between distinct variables and “Up North” but will also determine whether or not northern Wisconsin is distinctly different than southern Wisconsin overall. Therefore hypotheses testing becomes necessary to interpret noticeable differences. The null hypothesis stating that there is no difference between northern and southern Wisconsin according to the observed variables, compared to the alternative hypothesis stating that there is a difference between northern and southern Wisconsin according to the observed variables, must be considered.   

Methodology:

In order to determine distinct differences between northern and southern Wisconsin and establish a concept of “Up North”, a finite boundary dividing what is northern Wisconsin and southern Wisconsin must be created. Highway 29, running east and west through the state of Wisconsin fairly accurately divides that state into north and south segments. After a distinct boundary has been created, certain variables, many individuals associate with “Up North”, can be mapped in order to identify specific correlations between these characteristics and the actual northern part of Wisconsin. In addition to establishing these correlations, differences between the north and south can also be interpreted.

SCORP data, collected from the DNR, provides a variety of characteristics unique to Wisconsin. Several of these characteristics accurately reflect the Wisconsin perception of “Up North”. Specific data for variables such as amount of forested land cover, number of cottages, and number of deer gun licenses throughout the state can be mapped to display any spatial relationships which differentiate the north from the south. Data for each of the three variables were mapped into separate representations using a graduated color choropleth map. These maps show a division of four classes representing the saturation for each variable at the county level. The county level representation of the data provides the means to identify spatial patterns in terms of the amount of forest acreage, number of cottages, and number of deer gun licenses.   

To further identify differences between northern Wisconsin and southern Wisconsin, statistical chi squared testing of the data for each variable can be used in correlation with the mapped representations. Chi squared testing gives a numeric value for each variable comparing the observed distribution of each variable with the expected distribution. Calculating chi squared provides a statistical measure of how the observed variables are distributed throughout the state in respect to the expected distribution, with a significance level of 95 percent. This testing will indicate whether or not the observed variables for forest acreage, number of cottages, and number of deer gun licenses fit the expected distribution or if they are significantly different. Therefore, if there is a difference in the observed data compared to the expected distribution there is a difference between northern and southern Wisconsin in terms of these characteristics.     

Results:

Chi squared testing for forest acreage, number of cottages, and deer gun licenses with a significance value of 95 percent produced similar results. The outcome of the chi squared testing for all three variables was less than 0.5. The chi squared for deer gun licenses is 0.445, the number of cottages it is 0.482, and the amount of forest acreage it is 0.445. All three of these values are less than 0.5 or 5 percent, indicating the observed variables fall outside the significance value of 95 percent. Because they fall outside the 95 percent significance value, it is clear that these factors are not reflective of the expected distribution and there is a clear difference between northern and southern Wisconsin in terms of these characteristics. Not only does the chi squared testing of the data for each variable identify a difference between the north and south, but the spatial representation of the mapped variables portrays an identifiable difference as well.

The Map Representing the number of cottages per county strongly reflects a greater number of cottages in the northern part of the state. With the exception of three counties in southern Wisconsin there are less than 2519 cottages in each county, which is significantly less than the northern part of the state. More than 10 counties in Northern Wisconsin have 2519 cottages or more, and in Vilas and Oneida County there are over 10,000 cottages in each.  The map reveals that the known difference between the north and south is reflective of a greater number of cottages in the north.  

In addition to the number of cottages, the amount of forest acreage in northern Wisconsin compared to southern Wisconsin is also significantly different according to the map. Almost the entire northern part of Wisconsin has over 226,000 acres of forest and three distinct counties have over 781,000 acres each. The vast majority of southern Wisconsin has less than 226,000 acres and there are only a few counties in which have slightly more. Thus, the amount of forest acreage is predominantly in the northern part of the state which establishes a distinct difference between northern and southern Wisconsin in terms of this specific characteristic.    

The number of cottages and forest acreage have a distinct concentration in the north, however, the map for the number of deer gun licenses is unclear when trying to interpret where a difference occurs. Chi Square testing for the deer gun license variable indicates there is a difference between the amounts in northern and southern Wisconsin, however the difference is not as apparent when the amount of deer gun licensees are mapped.  According to the map there is a similar distribution of the amount of deer gun licenses throughout the entire state of Wisconsin. In fact there are more counties in southern Wisconsin that have over 5000 deer gun licenses than there are in northern Wisconsin. Furthermore there are several counties in southern Wisconsin that have more than 16,500 deer gun licenses, including Dane, Waukesha, and Brown County, where Marathon county is the only county in Northern Wisconsin that has more than 16,500. The map represents a fairly even distribution in the amount of deer gun licenses throughout Wisconsin, and one could even argue there are more deer gun licenses in Southern Wisconsin. Considering the chi squared results, indicating a difference in deer gun licenses between the north and south, with respect to the map the difference may be in fact there are more deer gun licenses in southern Wisconsin.

Conclusion:
Based on analysis of the results for the chi squared testing in relation to the maps of the data the Null hypothesis is rejected, stating that there is difference between the observed and expected distribution of each variables and establishing a distinct difference between northern and southern Wisconsin. The chi square test of the data for each variable has indicated a difference in the observed and expected frequency for each variable, while the spatial representation allowed for a clarification on where differences are located. Because the chi squared testing identified that there was a difference in northern and southern Wisconsin, the maps could then be easily used to interpret where in Wisconsin each variable has a greater influence.

The amount of forest acreage, number of cottages, and number of deer gun licenses were used as variables considering their associated perception with what many in Wisconsin refer to as “Up North”. After chi square established an identifiable difference in these factors between northern and southern Wisconsin, the maps displayed whether or not these factors differed in terms of a significantly greater concentration in the north compared to the south. Two out of the three variables displayed a map with distinctly greater concentrations in the north. Forest acreage and number of cottages were the two variables that had significantly greater concentrations in northern Wisconsin compared to southern Wisconsin. The relationship between forests and cottages with the concept of “Up North” appears to be accurate according to the results. However the number of deer gun licenses, another characteristic that can be associated with “Up North”, did not reflect an accurate perception.

Unlike the accurate link the other variables have with the idea of “Up North”, the number of deer gun licenses is a characteristic of “Up North” that did not display its statistical difference based on greater concentrations in the north. Even though chi squared testing specified a known difference, the map for this variable portrays a fairly even distribution of the deer gun licenses with a slightly higher concentration in the south. When considering both the chi square results in relation to the map, it appears that more individuals in the southern part of Wisconsin have deer gun licenses compared to the north. Although there are slightly more deer gun licenses in the southern part of Wisconsin, the hunting characteristic is still an accurate perception of “Up North” when considering the fact that often individuals travel north to hunt. Even with the assumption that many travel from southern Wisconsin to the northern Wisconsin to hunt, the map does not accurately reflect the association of hunting to “Up North”.

Kansas and Oklahoma Tornado Analysis

Introduction:
The States of Kansas and Oklahoma are facing a dilemma regarding tornado activity, and whether or not building tornado shelters would be beneficial in certain areas. There are arguments concerning tornado patterns have been repetitive over a series of years, supporting the idea that building shelters in areas of high frequency to be necessary. Others oppose the idea, arguing that there are many areas that have no need for shelters and consider it to be a waste of money. In order to develop a solution to the problem, tornado data has been collected and displayed spatially to determine which citizens have a stronger argument. After providing spatial evidence to either prove or disprove tornado shelter as a need, it becomes easier to identify which group of citizens are more warranted in their complaints and actions can then be taken to solve to the problem.

Methodology:
In order to come to a consensus on the issue and thus address the problem, it is necessary to locate tornado locations and determine whether or not a spatial pattern exists. To do this, data was collected regarding tornadoes that have occurred between 1995 and 2006 as well as tornadoes from 2007 to 2012, along with their width in feet. By tracking the location of specific tornadoes that have occurred over a series of 17 years, it becomes possible to identify if there is a pattern occurring in terms of location. If a pattern is identified, then citizens who argue for the building of shelters have support for their argument.

Not only is it necessary to plot locations and widths of tornadoes throughout Kansas and Oklahoma, it is essential to determine the location of central tendency. To find this location the mean center of the all the tornadoes is calculated. Once this is calculated and plotted it becomes evident where the center of tornado activity is located. Although the mean center is effective in determining spatial patterns, the weighted mean center can be calculated for a more reflective representation of tornado activity based of severity. By calculating and plotting a mean center, and weighting it by the width of each specific tornado the initial mean center shifts. The direction in which the mean center shifts after weighting it indicates the areas which experience wider, more dangerous tornadoes. This becomes significantly helpful in not only identifying a pattern in tornado activity but areas which have experienced more severe tornado activity.

In addition to plotting the mean centers of activity, standard distance is also helpful in spatially representing patterns. The measurement shows tornadoes which are concentrated around the weighted mean center, and is spatially equivalent to the standard deviation. A standard distance of one standard deviation, weighted by the width, was calculated and displayed. This standard distance indicates 68 percent of the tornado locations which are located around the weighted mean center. This meaning that over half the tornadoes which occurred fall somewhere between the weighted mean and the edge of the weighted standard distance. The smaller the radial distance between the weighted mean and weighted standard distance means the more concentrated tornado activity is in a given area. The larger the radial distance represents a large dispersion of activity throughout.

By developing several spatial representations of the data, conclusions can then be made concerning tornado activity in Kansas and Oklahoma. Displaying the data in correspondence to mean center, weighed mean centers, and weighted standard distances helps indicated if any specific patterns exist in tornado activity. After analyzing each spatial representation, hypotheses can be made in terms of identifying areas with higher probability of tornado activity, thus concluded the necessity of tornado shelters.

Results:

The First few maps created display tornado locations for both 1995 to 2006 data as well as the data from 2007 to 2012 represented by their width. In addition to the spatial dispersion of tornado activity, both the mean center and weighted mean center were plotted in order to portray central tendencies in the data. Figure three displays data from both tornado location data sets along with the mean center and weighted mean center for both.

This map is most helpful in determining shifts in tornado activity in regards to both location overtime and severity. The first observation that can be made from this figure is that mean center for all tornadoes 1995 through 2012 are fairly centralized in the area of study. This indicates that there is a fairly even dispersion of tornado activity throughout the entire area of study. This even dispersion seems to have been consistent over time considering the very small shift in activity to the north. After weighting the mean center on width, the point shifts to the south in 1995 to 2006 data, and to the south east in the 2007 to 2012 data. This indicates there is a slight pattern showing more severe tornadoes occur further south in the area of study.


Figure 1

Figure 2



Figure 3

The next few maps created once again displayed the tornado for both sets of data in the same manner as the previous maps. With these maps, however, the standard distance was taken into account along with the weighted mean centers. The tornado locations were plotted along with the weighted mean center surrounded by a weighted standard distance, of one standard deviation. Figure six displays tornado locations for both data sets in comparison to each other along with both weighted mean centers, and weighted standard distances.

This map is helpful in the same sense that figure three was helpful, because all the data is displayed together for easy comparison. The patterns this map will help identify are those of tornado concentration. The weighted mean center still indicates a central tendency located in the center of the study area reflecting fairly even activity throughout the study area. However, the addition of the weighted standard distance provides a more reflective representation of occurring patterns. It is clear that more than half of the study area has experienced 68 percent of the entire amount of tornadoes from 1995 to 2012. The concentration of tornadoes are widely dispersed throughout the study area, meaning a wide variety of areas throughout experience tornadoes. Even though the concentration of tornado activity has appeared to centralize in the more recent set of data, it is still apparent that tornado activity has maintained a wide dispersion pattern affecting many locations throughout the study area. 

Figure 4

Figure 5









Figure 6

Not only are these maps helpful in determining patterns in order to prove whether or not tornado shelters are needed, but when analyzed along with the actual data indicates the specific probable risk each location has in terms of experiencing tornadoes. For instance if patterns continue to hold true for the following years, the probability of a tornado can be calculated to predict the risk of tornado activity. All counties have a probable chance of experiencing 7.61 tornadoes 20 percent of the time. Despite the low chance of encountering at least 7 tornadoes it is possible for it to occur. Furthermore, all counties have a probably risk that they will experience 1.76 tornadoes at least 70 percent of the time. This means there is a significant chance that each county will most likely encounter at least one tornado. Although counties like Alfalfa County, Oklahoma have only experience five tornadoes from 2007 to 2012, in comparison to Russell County, Kansas that experienced 25 tornadoes, both counties have the same amount of probably risk for tornado activity. 

Figure 7

Conclusion:
Based on the overall results gained from analyzing each spatial representation in correspondence with the raw data and probability several observations can be made. It is apparent that the more recent set of data portraying tornado activity shows there has been a decrease in the number of tornadoes overall. Not only did the 2007 to 2012 data show a decrease in tornadoes, it also showed specific areas with higher frequencies in tornado activity, in comparison to the 1995 to 2006 data which showed a more even dispersion of activity throughout the entire area of study. Even with a decrease in tornado activity and more noticeable affected areas in recent data, the weighted mean center and standard distance show that majority of the study area has similar risk for tornadoes.

The weighted mean is located primarily in the center of the study area indicating there is a fairly even dispersion of tornadoes throughout both states. The standard distance also represents the same idea that most of the study area maintains a similar risk. This idea is supported because a great majority of the standard distance encloses the study area, and there is no clear evidence that there is a more specific area that has a greater frequency in tornado activity. Not only do the maps indicate a similar risk for tornadoes throughout the entire area of study, but probability represents the same idea. There is a 70 percent chance that there were will be at least one tornado in each county, and a 20 percent chance there could be 7. Each county faces the same probability. When putting probability and spatial data of tornado activity into perspective, and a fairly evenly distributed chance of tornadoes is represented, it becomes difficult to predict which areas are the most at risk.


Data does not specifically reflects any noticeable patterns in tornado activity throughout Kansas and Oklahoma, making  it difficult to conclude whether or not building tornado shelters would be beneficial in specific areas. There is no strong indication of specific areas which would benefit more from shelters over others. Unless specific clusters of tornados, portrayed by the tornado location points, are used as the only deciding factor to determine where to build shelters, there is no other obvious way to indicate the most beneficial areas. However, the tornado locations alone are not a substantial deciding factor, considering the significant changes in locations over time. Those who argue that some areas have higher frequencies need shelters are inaccurate because high frequency areas tend to change over time.  Thus, there are not identifiable areas where tornado shelters would benefit over other areas.  In order to address the issue, shelters would needed to be built at proportional locations throughout Kansas and Oklahoma to accommodate the same possibility every county faces.