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Use these statistics and the Normal Chart to calculate the following probabilities for

each quartile:

a. P (Annual Average Temperature > 7.6°)

According to King’oriah, (76) the Z formula for a normal distribution Z =

Where X bar-sample mean and is the standard deviation.

Quartile 1

To get Z score for Annual Average Temperature > 7.6°; Z = =1.183

P (Z>1.183) =1- P (Z<1.183)

P (Z>1.183) =1-0.8816=0.1184

Quartile 2

To get Z score for Annual Average Temperature > 7.6°; Z = =0.07921

P (Z>0.07921) =1- P (Z<0.07921)

P (Z>0.07921) =1-0.5316=0.4684

Quartile 3

To get Z score for Annual Average Temperature > 7.6°; Z = =-1.536

P (Z>-1.536) =1- P (Z<-1.536)

P (Z>-1.536) =1-0.06224=0.9378

Quartile 4

To get Z score for Annual Average Temperature > 7.6°; Z = =-1.405

P (Z>-1.405) =1- P (Z<-1.405)

P (Z>-1.405) =1-0.0800=0.9200

b. P(7.3° < Annual Average Temperature < 8.5°)

Quartile 1

Z score for likelihood that the sample mean is 7.3; Z = =0.7606

Z score for likelihood that the sample mean is 8.5; Z = =2.451

P (Z<0.7606) =0.7765

P (Z<2.451) = 0.9929

P (7.3

Quartile 2

Z score for likelihood that the sample mean is 7.3; Z = =-0.2178

Z score for likelihood that the sample mean is 8.5; Z = =0.9703

P (Z<-0.2178) =0.4138

P (Z<0.9703) = 0.8341

P (7.3

Quartile 3

Z score for likelihood that the sample mean is 7.3; Z = =-1.971

Z score for likelihood that the sample mean is 8.5; Z = =-0.2319

P (Z<-1.971) =0.02436

P (Z<-0.2319) = 0.4083

P (7.3

Quartile 4

Z score for likelihood that the sample mean is 7.3; Z = =-1.785

Z score for likelihood that the sample mean is 8.5; Z = =-0.2658

P (Z<-1.785) =0.03715

P (Z<-0.2658) = 0.3952

P (7.3

c. Based on these results, has a trend emerged with respect to temperature change in Toronto over the last 170 years? Justify your answer (50–100 words)!

For the probability of temperature being greater than 7.6 degree Celsius over the years, the trend was 0.1184, 0.4684, 0.9378 and 0.9200. This shows that the there is a high likelihood of an increase in temperature over 7.6 degrees Celsius over the last 170 years. This is because of the rise in probabilities in the four quartiles examine over the 170 years. This, however, was experienced most between 1926 to 1968.

The likelihood of temperature occurring between 7.3 to 8.5 degrees Celsius was 0.2163, 0.4207, 0.3840 and 0.3580. Similarly, the temperature measured for the last 170 years has been changing and the probability of that temperature being between 7.3 and 8.5 degrees Celsius is increasing over the years. This, however, was experienced most between 1926 to 1968.

2) Separately, prepare a written report (minimum 300 words) outlining your analysis of:

i) the distributions in terms of symmetry, skews, centres, and spread for the variables that you chose to investigate. Comment on how these distributions give you insight into the ways in which resources and political systems exist worldwide

Urban population variable has a mean of 57.25 and median of 60.1. This shows the average urban population as a percent. The distribution is close to each other since the mean and median are not far apart. This variable has a negative value of kurtosis and skewness of -0.901 and -0.178This shows that the distribution is light tailed. This suggests that the median is greater than the mean variable. According to George & Mallery (10) symmetry and kurtosis values of between -2 and +2 are acceptable and can give a proof of normal distribution. The standard deviation of the urban population is 22.78. This shows the squared deviation from the mean. In general, the distribution is normally distributed with most of the frequency bars in the histogram forming a bell shape.

Income per person variable has a mean of 7455.1 and median of 2459. This shows the symmetry of the distribution. On average, income per person ranges at US$7455.1.The median is has a big interval to the mean. The variable has a Skewness and kurtosis values of 1.69 and 1.77 respectively. This shows that the distribution is positively skewed. It heavy-tailed and has a right tail. Most scores fall in the lower side and few in the higher side. The positive skewness also shows that the mean is greater than the median value. The standard deviation of the income variable is 10504.03. This is the squared deviation from the mean. In general, the distribution is not normally distributed. More values occur on the lower side.

From the above studies, the urban populations seem to have a normally distributed attitude on democracy. This could be attributed to the mix of people from all races and tribes living in towns. Most of them are also civilized hence they tend to respect laws and embrace democracy than rural population. Income per person is skewed on the democracy aspect. From the study, there seems to be a skewed distribution with the people with higher incomes having more sense on democracy aspects.

ii) and, the degree to which pairs of variables are correlated and whether there is any hint of “causation” between them.

Correlation in statistical analysis tests the relationship between variables. It shows the strength and direction of the association between any two variables. Correlation coefficient value is commonly denoted by letter r. This coefficient ranges from -1 to +1. The -1 coefficient shows that the two variables have a negative linear relationship. An r-value of 0 suggest that the variables involved have no linear relationship value of +1shows a positive association between variables. The extreme values of this coefficients (near +1 and -1) shows a strong relationship, otherwise a weak one. In this study, democracy and urban population were tested to find if they are related. A r=0.195 resulted. This shows that the two variables have a positive relationship. However, the relationship is a weak one. The urban population variable has some influence on the kind of democracy of the residents. Between democracy and income per person, a r=0.259 resulted. Similarly, this shows a positive relationship between the two variables. The relationship is also relatively weak. Causation is a cause and effect relationship between two events. In this case, there is no significant evidence to show that democracy is a direct result of urban population or income. Therefore no causation between the variables.

Works Cited

King’oriah, George K. ”Fundamentals of applied statistics.” Nairobi: The Jomo Kenyatta Foundation (2004).

George, Darren, and Paul Mallery. ”SPSS for Windows step by step. A simple study guide and reference (10. Baskı).” (2010).

September 25, 2023

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