Methods

The report aims at analyzing how mortality rate is related with components of air pollution, weather and socioeconomic variables. The mortality rate is predicted using six predictor variables. The predictor variables are average annual precipitation rate, Average maximum temperature n January, Average maximum temperature in July, Average size of the household, Percentage of White population in urbanized areas in 1960s and relative Sulphur di oxide pollution potential  in 1960.

A random sample of 60 metropolitan areas in USA is collected. The sample contains information of relevant variables from the late 1950s to early 1960. In order to fit linear model for predicting morality rate as against each of the predictor variable six separate regression is run taking mortality as independent variable and other six as dependent variable. Each model is estimated separately.

For the first model, morality is predicted taking average precipitation rate as dependent variable. The R square value is 26.0. This means average precipitation rate can explain 26 percent variation in mortality rate. The p value of the co efficient is 0.0000. Therefore, the variable precipitation is statistically significant. In the second model, mortality rate is predicted against maximum temperature in January. The R square value implies only 0.3% variation in mortality can be explained by concerned variable. The P value is 0.660 meaning the variable is not statistically significant. The third model is estimated using mortality rate and maximum temperature in July. In the model, the dependent variable explains 7.8% variation in mortality. The P value is 0.031. As the p value is less than 0.031, the variable is statistically significant. The forth predictor variable is average household size. In this model, household size explains 12.8% variation of the dependent variable as implied from the R square. The P value is 0.005. This shows statistical significance of average household size. The next model is estimated using White population as independent variable and mortality as dependent variable. In this model the independent variable explains 41.4% variation in the mortality rate as the estimate R square measure is as a percentage is 41.4%. The P value equals 0.0000. This shows statistical significance of the variable White population. The last predictor variable is the presence of sulphur di oxide. This estimated R square value is 18.1%, implying it explains 18.1% variation in mortality rate. The variable is statistically significant as shown from the P value. The P value is 0.001 that is less than 0.05. Therefore, the variable is statistically significant at 5%  level of significance.

Conclusion 

Among the six predictor variables all except maximum temperature in January turns out as statistically significant. The highest R square value is obtained for White population in the urbanized areas. Henceforth, the White population is a strong predictor of mortality rate. The weakest predictor is maximum temperature in the month of July.

Regression Analysis: Mort versus Precip

The regression equation is

Mort = 822 + 3.17 Precip

Predictor    Coef  SECoef      T      P

Constant   821.75    27.21  30.20  0.000

Precip     3.1743   0.7039   4.51  0.000

S = 53.9862   R-Sq = 26.0%   R-Sq(adj) = 24.7%

Results

Analysis of Variance

Source          DF      SS     MS      F      P

Regression       1   59266  59266  20.33  0.000

Residual Error  58  169041   2915

Total           59  228308

Regression Analysis: Mort versus Jan

The regression equation is

Mort = 951 – 0.301 Jan

Predictor     Coef  SECoef      T      P

Constant    950.84    25.05  37.96  0.000

Jan        -0.3011   0.6809  -0.44  0.660

S = 62.6348   R-Sq = 0.3%   R-Sq(adj) = 0.0%

Analysis of Variance

Source          DF      SS    MS     F      P

Regression       1     767   767  0.20  0.660

Residual Error  58  227541  3923

Total           59  228308

Regression Analysis: Mort versus July

 The regression equation is

Mort = 669 + 3.63 July

Predictor   Coef  SECoef     T      P

Constant   669.3    123.0  5.44  0.000

July       3.634    1.646  2.21  0.031

S = 60.2590   R-Sq = 7.8%   R-Sq(adj) = 6.2%

Analysis of Variance 

Source          DF      SS     MS     F      P

Regression       1   17701  17701  4.87  0.031

Residual Error  58  210607   3631

Total           59  228308 

Regression Analysis: Mort versus Hhsize 

The regression equation is

Mort = 404 + 164 Hhsize  

Predictor    Coef  SECoef     T      P

Constant    404.1    184.2  2.19  0.032

Hhsize     164.34    56.40  2.91  0.005  

S = 58.5984   R-Sq = 12.8%   R-Sq(adj) = 11.3% 

Analysis of Variance 

Source          DF      SS     MS     F      P

Regression       1   29149  29149  8.49  0.005

Residual Error  58  199159   3434

Total           59  228308 

Regression Analysis: Mort versus White 

The regression equation is

Mort = 1336 – 4.49 White 

Predictor     Coef  SECoef      T      P

Constant   1336.01    62.06  21.53  0.000

White      -4.4896   0.7007  -6.41  0.000 

S = 48.0099   R-Sq = 41.4%   R-Sq(adj) = 40.4% 

Analysis of Variance 

Source          DF      SS     MS      F      P

Regression       1   94621  94621  41.05  0.000

Residual Error  58  133687   2305

Total           59  228308 

Regression Analysis: Mort versus SO2 

The regression equation is

Mort = 918 + 0.418 SO2  

Predictor     Coef  SECoef      T      P

Constant   917.887    9.644  95.18  0.000

SO2         0.4179   0.1166   3.58  0.001 

S = 56.7657   R-Sq = 18.1%   R-Sq(adj) = 16.7%  

Analysis of Variance 

Source          DF      SS     MS      F      P

Regression       1   41411  41411  12.85  0.001

Residual Error  58  186896   3222

Total           59  228308 

The website Statistica.com claims that average household size in 1960 was equal to 3.67. The report evaluates this statement in light of specific statistical tests. However, except this there are two other important questions that are answered in this report. The questions are whether there is difference of average death rate between areas with high nitrous oxide potential and low nitrous oxide potential in 1960 and whether there was change in the average maximum temperature of July in the collected data between 1960 and 2000.

In order to answer the three questions the same sample as that used in report 1 is used.  The first question corresponds to the test of mean for population of average household size. The population standard deviation is unknown. Therefore, one sample t test is used where the null hypothesis is average household size equals 3.67. The alternative hypothesis is average is different from 3.67. In order to test whether there is any significant difference in the average death rate between areas with high nitrous potential and that with low nitrous potential in 1960 two sample t test for equality of means are used. The null hypothesis here is there is no significant difference in the mean values of two groups and the alternative hypothesis is there is significant difference in average values. For the last question of finding significant difference in average of maximum temperature in July two samples t test is used. The null and alternative hypotheses are same as in the previous case.

The one sample t test of average household size shows the t value as -23.00 and the corresponding probability value (P value) is 0.000. The null hypothesis that average household size in 1960 was 3.67 is rejected and the alternative hypothesis is accepted. The result of two sample t test for testing the difference the of average death rate of high and low nitrous oxide potential areas shows the estimated t value as -2.07 and the corresponding p value is 0.045. The p value is less than 0.05. Therefore, the null hypothesis is rejected and the alternative hypothesis is accepted. Another two sample t test is performed to analyze whether there is any significant difference in the mean values of maximum temperature in July between 1960 and 2000. The value of t statistics is 0.70 and the p value is 0.488. This implies acceptance of null hypothesis that there is no significant difference average value of maximum temperature in July between 1960 and 2000.

Conclusion 

The analysis shows the average household size in 1960 is significantly different from zero as against the claim of Statistica.com. The average mortality rate is different between areas with high potential of nitrous oxide and that with low nitrous oxide potential. Finally, it can be concluded that the average temperature (maximum) in July is different between 1960 and 2000.

One-Sample T: Hhsize 

Test of mu = 3.67 vs not = 3.67  

Variable   N    Mean   StDev  SE Mean       95% CI            T      P

Hhsize60  3.2632  0.1353   0.0175  (3.2282, 3.2981)  -23.30  0.000 

Two-Sample T-Test and CI: Mort, High_NOx 

Two-sample T for Mort 

High_NOx   N   Mean  StDev  SE Mean

0         36  926.4   51.8      8.6

1         24  961.4   71.2       15 

Difference = mu (0) – mu (1)

Estimate for difference:  -35.0

95% CI for difference:  (-69.2, -0.8)

T-Test of difference = 0 (vs not =): T-Value = -2.07  P-Value = 0.045  DF = 38 

Two-Sample T-Test and CI: July_2000, July 

Two-sample T for July_2000 vs July 

July_2000  60  75.25   5.45     0.70

July       60  74.60   4.77     0.62 

Difference = mu (July_2000) – mu (July)

Estimate for difference:  0.650

95% CI for difference:  (-1.201, 2.501)

T-Test of difference = 0 (vs not =): T-Value = 0.70  P-Value = 0.488  DF = 115