1. You are given the following course metrics from the 1000 students thatcompleted 'Introduction to the Matrix' for the Fall 2021 semester thatresulted in a perfectly normal distribution:Mean Course Grade - 75.0Standard Deviation of Course Grades - 5.0You are asked to evaluate the following Course Grades from an unidentifiedclass of students:#12345Grade # Grade # Grade # Grade# Grade75660 11 55 16 9021 9555770128017 6522 60608751385 188023 907596514 70 1975 24 7560 109015 9525 9520 80(a) Is this class 25 student class representative of an 'Introduction to theMatrix' class?***Provide statistical analysis (Table, Graphs, metrics, etc.) and anexplanation of the rationale used for comparison between this classof 25 students and the 1000 'Introduction to the Matrix' students.****

Given the data as#Grade#Grade#Grade#Grade#Grade1756601155169021952557701280176522603608751385188023904759651470197524755601090159520802595Let X denote scores metrics in the 'Introduction to the Matrix'.It is given that X follows a normal distribution with mean (μ), 75 and the standard deviation (σ), 5To determine whether the sample data also represents the given population data, the Kolmogorov-Smirnov test can be used.The null and alternative hypotheses can be stated as followsNull Hypothesis: The class of 25 students is representative of an 'Introduction to the Matrix' classH0:F(X)=F0(X)Alternative Hypothesis: The class of 25 students is not representative of an 'Introduction to the Matrix' class.H1:F(X)≠F0(X)Where, F0(X) is the CDF of Normal (75 ,5)The test statistic is given by, Dn=Sup|Fn(X)−Fo(X)|Here, Fn(X) is the empirical distribution function and is given as followsCalculations:It is given that X∼Normal (75,5), the CDF of X is as followsNow to compute the test statistics consider the following tableix(i)zi=xi−755F0(Xi)Fn(Xi)Di=|Fn(Xi)−F0(Xi)|155-43.1671E-051/ 25 = 0.040.039968255-43.16712E-052/ 25 = 0.080.079968360-30.0013498983/ 25 = 0.120.11865460-30.0013498984/ 25 = 0.160.15865560-30.0013498985/ 25 = 0.20.19865660-30.0013498986/ 25 = 0.240.23865765-20.0227501327/ 25 = 0.280.25725865-20.0227501328/ 25 = 0.320.29725970-10.1586552549/ 25 = 0.360.2013451070-10.15865525410/ 25 = 0.40.241345117500.511/ 25 = 0.440.06127500.512/ 25 = 0.480.02137500.513/ 25 = 0.520.02147500.514/ 25 = 0.560.06157500.515/ 25 = 0.60.1168010.84134474616/ 25 = 0.640.201345178010.84134474617/ 25 = 0.680.161345188010.84134474618/ 25 = 0.720.121345198520.97724986819/ 25 = 0.760.21725209030.99865010220/ 25 = 0.80.19865219030.99865010221/ 25 = 0.840.15865229030.99865010222/ 25 = 0.880.11865239540.99996832923/ 25 = 0.920.079968249540.99996832924/ 25 = 0.960.039968259540.99996832925/ 25 = 13.17E-05Note: In the above tablex(i) values are obtained by arranging the x values in increasing order.F0(Xi) values are obtained using the Excel function =NORM.S.DIST( z,1)Thus, from the above table, the supremum or the maximum of Di is 0.29725Rejection Criteria:The rejection region for the given test is Dn>C.Here the critical value C at a 10% level of significance can be known using the following table,The above conclusion can also be tested using the quantile-quantile (Q-Q) graph. If all the values of x lies on the straight line Q-Q plot, it suggests that x follows normal distribution otherwise it supports the alternative hypothesis.The Q-Q plot for the given data can be obtained using the following R code:1# Provided data 2grades <- c(55, 55, 60, 60, 60, 60, 65, 65, 70,70, 75, 75, 75, 75, 75, 80, 80, 80, 85, 90, 90, 90, 95, 95, 95) 3 4# Q-Q plot 5qqnorm(grades) 6qqline(grades)The R-output is Interpretation: Observe that the values of x are deviating from the straight line. Hence the sample data deviates from the normal distribution.Thus through the Q-Q plot, it is evident that The class of 25 students is not representative of an 'Introduction to the Matrix' class.Statistical analysis using Kolmogorov-Smirnov test:The test statistic value is Dn=0.29725The critical value at a 1% level of significance is C=0.270Since, Dn>C, reject the null hypothesis Conclusion: The class of 25 students is not representative of an 'Introduction to the Matrix' class.Visual inspection method using Q-Q plot:Interpretation: Observe that the values of x are deviating from the straight line. Hence the sample data deviates from the normal distribution.Thus through the Q-Q plot, it is evident that The class of 25 students is not representative of an 'Introduction to the Matrix' class.