Msc
Msc program program in in Energy Energy Technology, Technology, Mechanical Mechanical Engineering Engineering Page Page 11 Answers to Questions
Answers to Questions Question 1.
Question 1. (a).
(a). The scatter plot The scatter plot of the two variables of the two variables that are variable crate and vthat are variable crate and variable educat treating crimeariable educat treating crime rate as dependent variable is shown below. [
rate as dependent variable is shown below. [ file from crime.sav file from crime.sav]]
(b). after superimposing a straight line on to the scatter plot that is using the linear fit method, the (b). after superimposing a straight line on to the scatter plot that is using the linear fit method, the relationship roughly looks linear
relationship roughly looks linear RR sq linear of .066. Although it may be curving up slightly orsq linear of .066. Although it may be curving up slightly or there may be an outlier. But if we use the cubic fit method the values are more fitted because the there may be an outlier. But if we use the cubic fit method the values are more fitted because the value of
(c) From the Pearson’s correlation value we can say that, there is a perfect positive correlation between these variables, which is statically significant at the 5% level. Because the perason’s coefficient r is 1.
Correlations
Correlations
violent crime rate pct hs graduates
violent crime rate[ Pearson Correlation 1 -.256
Sig. (2-tailed) .070
N 51 51
pct hs graduates Pearson Correlation -.256 1
Sig. (2-tailed) .070
Msc program in Energy Technology, Mechanical Engineering Page 3 (d) When the variable crate is correlate with variable educat the result of this regression is
Model Summary Model R R Square Adjusted R Square Std. Error of the Estimate 1 .256a .066 .046 430.724
a. Predictors: (Constant), pct hs graduates
Coefficientsa Model Unstandardized Coefficients Standardized Coefficients t Sig.
95% Confidence Interval for B
B Std. Error Beta Lower Bound Upper Bound
1 (Constant) 2152.347 832.477 2.585 .013 479.421 3825.273
pct hs
graduates -20.197 10.893 -.256 -1.854 .070 -42.087 1.693
a. Dependent Variable: violent crime rate
(E) plot of the standardized residual against the predicted values in order to detect any outliers and to assess whether the relationship is linear and whether the residual variance is constant
There is one residual greater than 4 and the trend indicates that there is approximately a linear relationship between crime rate and education. And from the scatter of points that tends to
increase a little as the predicted value increases which indicating that the assumption of constant variance may not be appropriate.
Question2.
The data file for question 2 is in the country.sav which contains the demographic information of 122 countries.
(a). Explore the relationship between the variable using a scatter plot. Dependent variable=lifeexpf
Independent variables=urban,docs,hospbed,gdp,andradio The result of the scatter plot matrix is shown below.
(b) The scatter plot matrix using the logarithm of the variables that don’t have a linear relationship is depicted below.
Logarithm of the variables are=lndocs,lnbeds,lngdp,and lnradio
Msc program in Energy Technology, Mechanical Engineering Page 5 (C) using the forward selection to find the subset of variables that best explain the dependent variable.
Dependent variable=lifeexp
Coefficientsa Model Unstandardized Coefficients Standardized Coefficients t Sig. B Std. Error Beta 1 (Constant) 57.232 .688 83.233 .000
Natural log of doctors per
10000 6.290 .318 .880 19.792 .000
2 (Constant) 42.138 3.206 13.143 .000
Natural log of doctors per
10000 4.261 .513 .596 8.307 .000
Natural log of GDP 2.493 .519 .345 4.802 .000
3 (Constant) 41.697 3.140 13.278 .000
Natural log of doctors per
10000 4.123 .505 .577 8.168 .000
Natural log of GDP 1.871 .566 .259 3.306 .001
Natural log of radios per 100
people 1.684 .679 .142 2.482 .015
a. Dependent Variable: Female life expectancy 1992
Number of doctors, GDP and number of radio are all positively related to life expectancy in females after controlling for the other variables.
(d) The cook’s distance against the variable sequence Dependent variable= lifeexp
Independent variable =lndocs,lngdp,lnradio
As we can see from the result of the plot of the matrix of the cook’s distance the most influential countries are Chad,Afghanistan, an d Guinea.
Msc program in Energy Technology, Mechanical Engineering Page 7 (e) The distribution of the standardized residuals is shown below
With some possible outliers we can say that the distribution is normally distributed with the normal distribution.
Question 3.
(a) Independent sample T test (b) Independent sample T test (c) Paired T test
(d) Paired T test
(e) Independent T test (f) Paired test
(g) Independent T test Question 4.
Msc program in Energy Technology, Mechanical Engineering Page 9 The first section of the Independent Samples Test output box gives us the results of Levene’s test for equality of variances. This tests whether the variance (variation) of ages for the two groups (populations) is the same. The outcome of this test determines which of the t-values that SPSS provides the correct one is.
Since the significance value from the output [.82] is larger than .05 it should be the first column of the out table to be used, which is Equal variance is assumed.
In the given the output from the question, the significance level for Levene’s test is .82. This is larger than the cut-off of .05.
This means that the assumption of equal variances has not been violated; therefore, when it is reported the t-value used is the one in the first column from the output.
From the out table the value of sig(2-tailed) in the first column is .000 less than .05 the required cut off there is a significance difference in the population’s mean ages of the two groups.
The value of t from the output table from the equal variance assumed column 3.9 and the values for N1 and N2 is the same from the output table 100.
Up on substituting the value of the Eta squared is .0713.
Then according to the guideline( proposed by Cohen,1998) for interoperating this value are .01=small effect
.06= moderate effect .14= large effect
For this particular question the, which have the effect size of .0713, effect is in the range of moderate and large.
An independent sample test was conducted to compare the average ages of people who buy and who don’t buy a product. There is a significance difference in buying the product [mean 29.45,SD 15.56 and mean 38, SD 15.49];t(198)=-3.9,p<.001) The magnitude of the difference in the means was large (eta squared=.0713).
Question 14. The data file for this question is school.sav and the aim is to check whether there are differences or not between the two groups that is above and below the median percentage of low income for all Chicago schools.
Group Statistics
above or below median loinc N Mean Std. Deviation Std. Error Mean Percent low income above the median for low inc
% 1993 32 73.219 8.7498 1.5468
below the median for low inc
% 1993 32 39.706 13.5002 2.3865
Independent Samples Test Levene's Test for
Equality of
Variances t-test for Equality of Means
F Sig. t df Sig. (2-tailed) Mean Difference Std. Error Difference 95% Confidence Interval of the Difference Lower Upper Percent low income Equal variances assumed 5.793 .019 11.784 62 .000 33.5125 2.8439 27.8276 39.1974 Equal variances not assumed 11.784 53.138 .000 33.5125 2.8439 27.8087 39.2163
From the Levene’s test for equality of variances we have sig. value of .019 which is less than below the cut off value .05. The means that the variance between the two group (below and above) are not the same. Therefore the value of t-test is used the one that is in the raw of variance not assumed. To find out whether there is a significance difference between the two groups, refer to the column labeled sig(2-tailed), which appears under the section labeled t-test for unequal means. In combination with the Leven’s test result this value is 000 which is below .05. Therefore there is a significant difference in the means of the two groups.
Msc program in Energy Technology, Mechanical Engineering Page 11 Group Statistics
above or below median loinc N Mean Std. Deviation Std. Error Mean average ACT score 1994 above the median for low inc
% 1993 32 15.022 .8746 .1546
below the median for low inc
% 1993 32 16.700 2.1506 .3802
Independent Samples Test Levene's Test for
Equality of
Variances t-test for Equality of Means
F Sig. t df Sig. (2-tailed) Mean Difference Std. Error Difference 95% Confidence Interval of the Difference Lower Upper average ACT score 1994 Equal variances assumed 16.501 .000 -4.089 62 .000 -1.6781 .4104 -2.4985 -.8577 Equal variances not assumed -4.08940.982 .000 -1.6781 .4104 -2.5070 -.8493
In the similar approach there is a significance difference between the means of the two groups. Because the significance value under the labeled sig(2-tailed) is 000 which is below .05
Question 16. The data file for this question is buying.sav and the aim is to test the null hypothesis of the following
1. Family buying score is the same when pictures are shown and when they are not. The result is shown below with its interpretation
Group Statistics Picture
Accompanied
Question N Mean Std. Deviation Std. Error Mean
Family Buying Score Pictures 48 159.08 27.564 3.979
No Pictures 50 168.00 21.787 3.081
Independent Samples Test
Levene's Test for Equality of
Variances t-test for Equality of Means
F Sig. t df Sig. (2-tailed) Mean Difference Std. Error Difference 95% Confidence Interval of the Difference Lower Upper Family Buying Score Equal variances assumed 1.382 .243 -1.780 96 .078 -8.917 5.008 -18.858 1.025 Equal variances not assumed -1.772 89.42 9 .080 -8.917 5.032 -18.915 1.081
From independent sample test we have the value of the significance .243 from the assumption of equal variance that is the Levene’s test and it is greater than the cut off that is .243>.05. This tells us which t-test to be used weather the Equal variance or the unequal variance assumption. But for this case since the Levene’s test value is greater than .05 then the t-test value is from the Equal variance assumption i.e value of that in the raw of equal variance. Therefore once the
t-Msc program in Energy Technology, Mechanical Engineering Page 13 hypothesis is not rejected. Therefore it is concluded that the family buying score is the same
when the pictures are shown and when they are not.
An independent sample t-test is conducted to compare the family buying score with picture and without picture. There are no significance difference in scores with picture (M=159,SD=27.564) and without picture (M=159,SD=21.787;t(96)=-1.78,p=.078]. The magnitude of the difference in the mean is moderate (eta square=.03).
2. Ho: The average buying score for the husband is the same with and without pictures. Ho is the null hypothesis. Group Statistics Picture Accompani ed Question N Mean Std. Deviation Std. Error Mean Sum of husband's buying scores Pictures 48 80.12 14.258 2.058 No Pictures 50 83.98 14.329 2.026
Independent Samples Test
Levene's Test for Equality of
Variances t-test for Equality of Means
F Sig. t df Sig. (2-tailed) Mean Difference Std. Error Difference 95% Confidence Interval of the Difference Lower Upper Sum of husband's buying scores Equal variances assumed .036 .849 -1.335 96 .185 -3.855 2.889 -9.589 1.879 Equal variances not assumed -1.335 95.874 .185 -3.855 2.888 -9.588 1.878
With similar approach of the above, the significance for this part is .185 from [sig 2-tailed] column and it is greater than the cut off (.05), the null hypothesis is not rejected. Still the average buying score for husband is the same with and without picture.
An independent sample t-test is conducted to compare the family buying score with picture and without picture. There are no significance difference in scores with picture (M=80.12,SD=14.258) and without picture (M=83.98,SD=14.329;t(96)=-1.335,p=.185]. The magnitude of the difference in the mean is moderate (eta square=.018).
3.Ho : The average buying score for the wives is the same with and without pictures. Ho is the null hypothesis.
Group Statistics Picture
Accompanied
Question N Mean Std. Deviation Std. Error Mean
Sum of wife's buying scores Pictures 49 78.98 16.033 2.290
No Pictures 50 84.02 15.444 2.184
Independent Samples Test Levene's Test
for Equality of
Variances t-test for Equality of Means
F Sig. t df Sig. (2-tailed) Mean Difference Std. Error Difference 95% Confidence Interval of the Difference Lower Upper Sum of wife's buying scores Equal variances assumed .025 .876 -1.593 97 .114 -5.040 3.164 -11.319 1.239 Equal variances not assumed -1.593 96. 677 .115 -5.040 3.165 -11.322 1.241
Msc program in Energy Technology, Mechanical Engineering Page 15 An independent sample t-test is conducted to compare the family buying score with picture and without picture. There are no significance difference in scores with picture (M=78.98, SD=16.033) and without picture (M=84.02, SD=15.444;t(97)=-1.593,p=.114]. The magnitude of the difference in the mean is moderate (eta square=.025).
Question 18.
A manufacturer of high-performance automobiles produces disc brakes that must measure 322 millimeters in diameter. Quality control randomly draws 16 discs made by each of eight production machines and measures their diameters.
The appropriate test to determine whether or not the mean diameters of the brakes in each sample significantly differ from 322 millimeters is One Sample T Test to determine. The file for this question is brake.sav and its confidence interval is 90%
The descriptive statics which displays the sample size, mean, standard deviation, and standard error for each of the eight samples
The sample means disperse around the 322mm standard by what appears to be a small amount of variation.
One-Sample Statistics
Machine Number N Mean Std. Deviation Std. Error Mean
1 Disc Brake Diameter (mm)
16 3.219985E
2 .0111568 .0027892
2 Disc Brake Diameter (mm)
16 3.220143E
2 .0106913 .0026728
3 Disc Brake Diameter (mm)
16 3.219983E
2 .0104812 .0026203
4 Disc Brake Diameter (mm)
16 3.219954E
2 .0069883 .0017471
5 Disc Brake Diameter (mm)
16 3.220042E
2 .0092022 .0023005
6 Disc Brake Diameter (mm)
16 3.220025E
2 .0086440 .0021610
7 Disc Brake Diameter (mm)
16 3.220062E
2 .0093303 .0023326
8 Disc Brake Diameter (mm)
16 3.219967E
Since their confidence intervals lie entirely above 0.0, it is possible to say that machines 2, 5 and 7 are producing discs that are significantly wider than 322mm on the average. And similarly, because its confidence interval lies entirely below 0.0, machine 4 is producing discs that are not wide enough.
Question19.
A physician is evaluating a new diet for her patients with a family history of heart disease. To test the effectiveness of this diet, 16 patients are placed on the diet for 6 months. Their weights and triglyceride levels are measured before and after the study, and the physician wants to know if either set of measurements has changed. The data are found in dietstudy.sav of SPSS sample files. Use appropriate test to determine whether there is a statistically significant difference between the pre- and post-diet weights and triglyceride levels of these patients
Paired Samples Statistics
Mean N Std. Deviation Std. Error Mean
Pair 1 Weight 198.38 16 33.472 8.368
Final weight 190.31 16 33.508 8.377
Pair 2 Triglyceride 138.44 16 29.040 7.260
Final triglyceride 124.38 16 29.412 7.353
As the study was made to know if there is a statistically significant difference between the pre-and post-diet weights pre-and triglyceride levels of these patients, a paired-samples t-test was appropriate test
1. There is a statistically significant decrease in weight from pre-diet ( M = 198.38) to post-diet ( M = 190.31), t(15)=11.175. Since the probability value p (0.000) <. 0005 (two-tailed) which is substantially smaller than our specified alpha value of .05, there is a significant difference in weight of the patients between the pre- and post-diet measurements. The mean decrease in weight is 8.062 with a 95% confidence interval ranging from 6.525 to 9.600. The t value is used to calculate the effect size statistic
Paired Samples Correlations
N Correlation Sig.
Pair 1 Weight & Final weight 16 .996 .000
Pair 2 Triglyceride & Final
Msc program in Energy Technology, Mechanical Engineering Page 17 Eta squared = .
.() = 0.893
According to Cohen 1988, pp. 284–7 guidlines, the Eta squared value .01=small effect, .06=moderate effect, .14=large effect.
Since the Eta squared value obtained 0.893 is greater than 0.14, there is a large effect with a significant difference in weight of the patients between the pre- and post-diet measurements.
2. There is a statistically significant decrease in triglyceride from pre-diet ( M = 138.44) to post-diet ( M = 124.38), t(15)=1.200. Since the probability value p (0.249) <. 0005 (two-tailed) which is substantially smaller than our specified alpha value of .05, there is a significant difference in triglyceride of the patients between the pre- and post-diet measurements. The mean decrease in triglyceride is 14.062 with a 95% confidence interval ranging from -10.915 to 39.040.
The effect size statistic squared (eta squared statistic): Eta squared = .
.() = 0.0876
Since the Eta squared value obtained 0.0876 is in between 0.06 and 0.14, there is moderate effect with a significant difference in triglyceride of the patients between the pre- and post-diet measurements.
