1.State whether the following statements are True or False. Give reasons in support of your answers :
(a) If two variables are related in the form Y=X^(2)\mathrm{Y}=\mathrm{X}^2, then the variables are highly linearly related.
Answer:
The statement "If two variables are related in the form Y=X^(2)Y = X^2, then the variables are highly linearly related" is false.
Justification:
Definition of Linear Relationship:
A linear relationship between two variables XX and YY implies that the relationship can be described by an equation of the form:
Y=aX+bY = aX + b
where aa and bb are constants. This means that for every unit change in XX, YY changes by a constant amount aa.
Nature of Y=X^(2)Y = X^2:
The equation Y=X^(2)Y = X^2 describes a quadratic relationship, not a linear one. This means that the relationship between XX and YY involves the square of XX, rather than a constant multiple of XX.
Non-linearity:
Plot Analysis:
If you plot Y=X^(2)Y = X^2 on a graph with XX on the x-axis and YY on the y-axis, you will get a parabolic curve that opens upwards. This is not a straight line, which would be characteristic of a linear relationship.
Rate of Change:
In a linear relationship, the rate of change of YY with respect to XX is constant. However, in Y=X^(2)Y = X^2, the rate of change (slope) of YY is given by the derivative (dY)/(dX)=2X\frac{dY}{dX} = 2X, which is not constant but varies with XX.
Example to Illustrate Non-linearity:
Consider two points:
When X=1X = 1, Y=1^(2)=1Y = 1^2 = 1
When X=2X = 2, Y=2^(2)=4Y = 2^2 = 4
When X=3X = 3, Y=3^(2)=9Y = 3^2 = 9
The differences in YY for equal changes in XX are not constant (from X=1X = 1 to X=2X = 2, YY changes by 3, but from X=2X = 2 to X=3X = 3, YY changes by 5), demonstrating a non-linear relationship.
Conclusion:
Since the relationship Y=X^(2)Y = X^2 does not meet the criteria for linearity (a constant rate of change and a linear plot), the statement is false.
(b) In regression analysis, the two regression coefficients are -2 and -2//3-2 / 3.
Answer:
The statement "In regression analysis, the two regression coefficients are -2 and -2//3-2 / 3" is false.
Justification:
In regression analysis involving two variables XX and YY, there are typically two regression equations:
The regression of YY on XX:
Y=a+bXY = a + bX
where bb is the regression coefficient of YY on XX.
The regression of XX on YY:
X=c+dYX = c + dY
where dd is the regression coefficient of XX on YY.
Relationship Between Regression Coefficients:
There is a specific relationship between these two regression coefficients, bb and dd:
b xx d=r^(2)b \times d = r^2
where rr is the correlation coefficient between XX and YY.
Proof Using Given Coefficients:
Given b=-2b = -2 and d=-(2)/(3)d = -\frac{2}{3}, let’s check the relationship:
b xx d=(-2)xx(-(2)/(3))=(4)/(3)b \times d = (-2) \times \left(-\frac{2}{3}\right) = \frac{4}{3}
For this product to be valid, it must equal r^(2)r^2, where rr is the correlation coefficient.
Constraints on r^(2)r^2:
The value of rr (and thus r^(2)r^2) must lie within the interval [-1,1][-1, 1]:
0 <= r^(2) <= 10 \leq r^2 \leq 1
However, in this case:
r^(2)=(4)/(3)r^2 = \frac{4}{3}
This is not possible because r^(2)r^2 cannot exceed 1.
Conclusion:
Since the product b xx db \times d should equal r^(2)r^2 and r^(2)r^2 cannot be greater than 1, the given regression coefficients b=-2b = -2 and d=-(2)/(3)d = -\frac{2}{3} cannot exist simultaneously in a valid regression model. Hence, the statement is false.
(c) Sum of deviations of the observations from their mean is zero.
Answer:
The statement "Sum of deviations of the observations from their mean is zero" is true.
Justification:
Definition of Mean:
The mean (average) of a set of observations x_(1),x_(2),dots,x_(n)x_1, x_2, \ldots, x_n is given by:
The sum of the deviations of the observations from their mean is indeed zero. Hence, the statement is true.
(d) If the value of beta_(2) < 3\beta_2<3, then the curve is said to be leptokurtic.
Answer:
The statement "If the value of beta_(2) < 3\beta_2 < 3, then the curve is said to be leptokurtic" is false.
Justification:
Kurtosis Overview:
Kurtosis is a statistical measure that describes the distribution of data points in the tails relative to the overall shape of the distribution. It indicates the "tailedness" of the distribution. The excess kurtosis is often used to compare with the normal distribution’s kurtosis.
Beta Coefficient (beta_(2)\beta_2):
beta_(2)\beta_2 is a measure of kurtosis. For a normal distribution, beta_(2)=3\beta_2 = 3.
Types of Kurtosis:
Leptokurtic: Distributions with kurtosis greater than 3 (beta_(2) > 3\beta_2 > 3). These distributions have fatter tails and a sharper peak compared to the normal distribution.
Mesokurtic: Distributions with kurtosis equal to 3 (beta_(2)=3\beta_2 = 3). The normal distribution is an example of a mesokurtic distribution.
Platykurtic: Distributions with kurtosis less than 3 (beta_(2) < 3\beta_2 < 3). These distributions have thinner tails and a flatter peak compared to the normal distribution.
Analysis of the Given Statement:
The statement asserts that if beta_(2) < 3\beta_2 < 3, the curve is leptokurtic. However, by the definitions provided:
If beta_(2) < 3\beta_2 < 3, the curve is actually platykurtic, not leptokurtic.
A leptokurtic curve would require beta_(2) > 3\beta_2 > 3.
Conclusion:
Given the definitions, the correct classification for beta_(2) < 3\beta_2 < 3 is platykurtic, not leptokurtic. Therefore, the statement is false.
(e) In a company of 1000 persons, 750 were male out of whom 530 were married. Among females, the number of married ones was 350 , then the data is consistent.
Answer:
To determine whether the given data is consistent, we need to verify if all the numbers provided logically add up without any contradictions.
Given Data:
Total number of persons in the company: 1000
Number of males: 750
Number of married males: 530
Number of females: (Total persons – Number of males) = 1000 – 750 = 250
Number of married females: 350
Consistency Check:
The main point of inconsistency would be in the number of married persons. Specifically, if the number of married males and married females exceeds the total population, the data is inconsistent.
Number of Married Persons:
Married males: 530
Married females: 350
Total number of married persons = Married males + Married females = 530 + 350 = 880
Total Population:
Total persons in the company = 1000
The total number of married persons (880) is less than the total population (1000), which is logically possible. Thus, there is no immediate contradiction from this calculation alone. However, a closer look at the gender-wise distribution of the unmarried persons should be considered to fully confirm consistency.
Calculation of Unmarried Persons:
Unmarried Males:
Total males = 750
Married males = 530
Unmarried males = 750 – 530 = 220
Unmarried Females:
Total females = 250
Married females = 350
This presents an inconsistency because the number of married females (350) cannot exceed the total number of females (250).
Conclusion:
The data given is inconsistent because the number of married females (350) exceeds the total number of females (250). This contradiction indicates a clear error in the provided information. Therefore, the statement that the data is consistent is false.
Question:-02
2.(a) A candidate obtained the following percentage of marks in different courses of PGDAST programme :
MST-001-46%
MST-002-67%
MST-003-72%
MST-004-58%
MST-005-53%
It is agreed to give double weights to marks in MST-001 and MST-002 as compared to other courses. What is the simple mean and weighted mean?
Answer:
To calculate both the simple mean and the weighted mean of the percentages obtained by the candidate in the different courses, we need to follow these steps:
Step 1: List the percentages obtained
MST-001: 46%
MST-002: 67%
MST-003: 72%
MST-004: 58%
MST-005: 53%
Step 2: Calculate the Simple Mean
The simple mean is the average of all the percentages. It is calculated by summing up all the percentages and then dividing by the number of courses.
“Simple Mean”=(sum”Percentages”)/(“Number of Courses”)\text{Simple Mean} = \frac{\sum \text{Percentages}}{\text{Number of Courses}}
Given that MST-001 and MST-002 have double weights compared to the other courses, we need to assign appropriate weights. Let’s denote the weight for MST-001 and MST-002 as ww and for the other courses as w//2w/2.
We can arbitrarily set w=2w = 2 for simplicity. Therefore:
Weight for MST-001: 2
Weight for MST-002: 2
Weight for MST-003: 1
Weight for MST-004: 1
Weight for MST-005: 1
Now, calculate the weighted mean using these weights:
(b) For two Firms A and B, the following details are available :
A
B
No. of employees
100
200
Average salary
16000
18000
SD of salary
16
18
A B
No. of employees 100 200
Average salary 16000 18000
SD of salary 16 18| | A | B |
| :— | :—: | :—: |
| No. of employees | 100 | 200 |
| Average salary | 16000 | 18000 |
| SD of salary | 16 | 18 |
Compute the following :
(i) Which Firm pays larger package of salary?
(ii) Which Firm shows greater variability in the distribution of salary?
(iii) Compute the combined average salary and combined variance of both firms.
Answer:
Let’s address each of the questions step by step:
Given Data:
Firm A:
Number of employees: n_(A)=100n_A = 100
Average salary: mu _(A)=16000\mu_A = 16000
Standard deviation of salary: sigma _(A)=16\sigma_A = 16
Firm B:
Number of employees: n_(B)=200n_B = 200
Average salary: mu _(B)=18000\mu_B = 18000
Standard deviation of salary: sigma _(B)=18\sigma_B = 18
(i) Which Firm Pays a Larger Package of Salary?
The larger package of salary refers to the average salary. Comparing the average salaries of both firms:
Firm A: 16000
Firm B: 18000
Answer: Firm B pays a larger package of salary because the average salary at Firm B (18000) is higher than that at Firm A (16000).
(ii) Which Firm Shows Greater Variability in the Distribution of Salary?
The variability in the distribution of salary can be compared using the standard deviation.
Firm A: 16
Firm B: 18
Answer: Firm B shows greater variability in the distribution of salary because the standard deviation of salaries at Firm B (18) is higher than that at Firm A (16).
(iii) Compute the Combined Average Salary and Combined Variance of Both Firms
Combined Average Salary
The combined average salary (mu\mu) for both firms can be calculated using a weighted average:
Greater variability in the distribution of salary: Firm B
Combined average salary: 17333.3317333.33
Combined variance: 889190.22889190.22
Question:-03
3.(a) Define coefficient of determination and correlation ratio.
Answer:
Coefficient of Determination (R^(2)R^2)
The coefficient of determination, denoted as R^(2)R^2, is a statistical measure used in the context of regression analysis to assess the goodness of fit of a model. It provides the proportion of the variance in the dependent variable that is predictable from the independent variable(s).
“SS”_(“res”)\text{SS}_{\text{res}} (Residual Sum of Squares): The sum of the squares of the residuals (the differences between the observed and predicted values).
“SS”_(“tot”)\text{SS}_{\text{tot}} (Total Sum of Squares): The sum of the squares of the differences between the observed values and the mean of the observed values.
Interpretation:
R^(2)R^2 ranges from 0 to 1.
R^(2)=0R^2 = 0: The independent variable does not explain any of the variation in the dependent variable.
R^(2)=1R^2 = 1: The independent variable perfectly explains all the variation in the dependent variable.
An R^(2)R^2 value close to 1 indicates a strong relationship, whereas a value close to 0 indicates a weak relationship.
Example:
If R^(2)=0.85R^2 = 0.85, it means that 85% of the variance in the dependent variable is explained by the independent variable(s).
Correlation Ratio (η or eta^(2)\eta^2)
The correlation ratio, denoted by the Greek letter eta (η), or eta^(2)\eta^2 when referring to the squared correlation ratio, is a measure of the strength of the relationship between a continuous dependent variable and a categorical independent variable. It is used in analysis of variance (ANOVA) and is particularly useful when the relationship is not strictly linear.
“SS”_(“between”)\text{SS}_{\text{between}} (Between-Group Sum of Squares): The sum of the squared deviations of the group means from the overall mean, weighted by the number of observations in each group.
“SS”_(“total”)\text{SS}_{\text{total}} (Total Sum of Squares): The sum of the squared deviations of each observation from the overall mean.
Interpretation:
eta^(2)\eta^2 ranges from 0 to 1.
eta^(2)=0\eta^2 = 0: There is no relationship between the dependent variable and the categorical independent variable.
eta^(2)=1\eta^2 = 1: There is a perfect relationship between the dependent variable and the categorical independent variable.
Higher values of eta^(2)\eta^2 indicate a stronger relationship.
Example:
If eta^(2)=0.65\eta^2 = 0.65, it means that 65% of the variance in the dependent variable can be attributed to the differences between groups defined by the categorical independent variable.
Summary:
Coefficient of Determination (R^(2)R^2): Measures the proportion of variance in the dependent variable explained by the independent variable(s) in a regression model. It ranges from 0 to 1.
Correlation Ratio (η or eta^(2)\eta^2): Measures the strength of the relationship between a continuous dependent variable and a categorical independent variable, particularly useful for non-linear relationships. It also ranges from 0 to 1.
(b) Calculate the correlation coefficient from the following data :
Let now each value of X\mathrm{X} be multiplied by 2 and then 6 be added to it.
Similarly, multiply each value of Y\mathrm{Y} by 3 and subtract 2 from it. What will be the correlation coefficient between the new series of X\mathrm{X} and Y\mathrm{Y} ?
Answer:
To calculate the correlation coefficient (rr) for the given data and the transformed data, we will follow these steps:
Step 1: Calculate the Correlation Coefficient for the Original Data
Step 2: Calculate the Correlation Coefficient for the Transformed Data
The transformations are:
X^(‘)=2X+6X’ = 2X + 6
Y^(‘)=3Y-2Y’ = 3Y – 2
Transformation Impact:
The transformations are linear transformations of the original variables XX and YY.
Linear transformations do not affect the correlation coefficient between the two variables. The correlation coefficient remains unchanged.
Conclusion:
The correlation coefficient for the original data is approximately 0.920.92.
The correlation coefficient for the transformed data will be the same as for the original data, which is approximately 0.920.92.
Question:-04
4.(a) Differentiate between correlation and regression.
Answer:
Correlation and regression are both statistical tools used to examine relationships between variables, but they serve different purposes and provide different information. Here’s a detailed differentiation between the two:
Correlation:
Purpose:
Correlation measures the strength and direction of the linear relationship between two variables.
Nature of Analysis:
It quantifies the degree to which two variables are related, but does not imply causation.
Output:
The result is a correlation coefficient, typically denoted by rr.
rr ranges from -1 to 1:
r=1r = 1 indicates a perfect positive linear relationship.
r=-1r = -1 indicates a perfect negative linear relationship.
r=0r = 0 indicates no linear relationship.
Symmetry:
Correlation is symmetric: “corr”(X,Y)=”corr”(Y,X)\text{corr}(X, Y) = \text{corr}(Y, X).
Units:
Correlation is a unitless measure, meaning it does not depend on the scale of the variables.
Types:
Pearson correlation (measures linear relationship).
Spearman correlation (measures monotonic relationship, suitable for non-parametric data).
Interpretation:
Correlation only indicates the degree of association, not the exact nature or causality.
Regression:
Purpose:
Regression assesses the relationship between a dependent variable and one or more independent variables, and models how the dependent variable changes when the independent variable(s) are varied.
Nature of Analysis:
It explains the nature of the relationship and provides a predictive model.
The focus is on predicting the value of the dependent variable based on the independent variable(s).
Output:
The result is an equation that describes the relationship between the variables.
In simple linear regression: Y=a+bXY = a + bX
YY is the dependent variable.
XX is the independent variable.
aa is the intercept.
bb is the slope (regression coefficient).
Symmetry:
Regression is not symmetric: the regression of YY on XX is not the same as the regression of XX on YY.
Units:
The regression coefficients have units and are interpreted as the change in the dependent variable for a one-unit change in the independent variable.
Types:
Simple linear regression (one independent variable).
Multiple linear regression (more than one independent variable).
Non-linear regression (non-linear relationships).
Interpretation:
Regression provides insights into the nature of the relationship, including magnitude and direction of influence.
It also helps in making predictions and understanding causality, to an extent.
Summary:
Correlation: Measures strength and direction of linear relationship, symmetric, unitless, no causation.
Regression: Models relationship, provides predictive equation, not symmetric, coefficients have units, indicates causation to some extent.
Understanding these distinctions helps in choosing the appropriate method for analyzing data and interpreting the results accurately.
(b) In order to find the correlation between two variables X\mathrm{X} and Y\mathrm{Y} from 12 pairs of observations, the following calculations were obtained :
On subsequent verification, it was discovered that the pair (X=11,Y=4)(\mathrm{X}=11, \mathrm{Y}=4) was copied wrongly, the correct values being (X=10,Y=14)(\mathrm{X}=10, \mathrm{Y}=14). After making necessary correction, find :
(i) regression coefficients,
(ii) two regression equations, and
(iii) correlation coefficient.
Answer:
To address the given problem, we need to first correct the calculations with the correct pair of observations (X=10,Y=14)(X=10, Y=14) instead of (X=11,Y=4)(X=11, Y=4). Then, we will proceed to find the regression coefficients, the regression equations, and the correlation coefficient.
Step 1: Correct the Calculations
Given:
Original Sigma X=30\Sigma X = 30
Original Sigma Y=5\Sigma Y = 5
Original SigmaX^(2)=670\Sigma X^2 = 670
Original SigmaY^(2)=285\Sigma Y^2 = 285
Original Sigma XY=344\Sigma XY = 344
Incorrect pair: (X=11,Y=4)(X=11, Y=4)
Correct pair: (X=10,Y=14)(X=10, Y=14)
First, remove the contributions of the incorrect pair and then add the contributions of the correct pair.
Regression equation of YY on XX: Y=0.697 X-0.437Y = 0.697X – 0.437
Regression equation of XX on YY: X=0.905 Y+1.286X = 0.905Y + 1.286
Correlation Coefficient:
r~~0.793r \approx 0.793
Question:-05
5.(a) In a musical contest, 168 contestants participated. The competition comprised three different stages. It was found that 57 contestants cleared first stage; 45 second stage and 72 third stage. The number of contestants who cleared all the stages, who did not clear any stage, who cleared only first two stages and who cleared only third stage were 17,29,1117,29,11 and 20 , respectively. With the given information, find how many contestants cleared at least two stages.
Answer:
To solve this problem, we will use the principle of inclusion-exclusion and the given data to find the number of contestants who cleared at least two stages. We denote the following:
AA: Contestants who cleared the first stage
BB: Contestants who cleared the second stage
CC: Contestants who cleared the third stage
We are given:
|A|=57|A| = 57
|B|=45|B| = 45
|C|=72|C| = 72
|A nn B nn C|=17|A \cap B \cap C| = 17 (cleared all stages)
|A^(c)nnB^(c)nnC^(c)|=29|A^c \cap B^c \cap C^c| = 29 (did not clear any stage)
|A nn B nnC^(c)|=11|A \cap B \cap C^c| = 11 (cleared only the first two stages)
|A^(c)nnB^(c)nn C|=20|A^c \cap B^c \cap C| = 20 (cleared only the third stage)
Finding the number of contestants who cleared at least two stages:
Contestants who cleared at least one stage:
Total number of contestants =168= 168
Contestants who did not clear any stage =29= 29
Contestants who cleared at least one stage =168-29=139= 168 – 29 = 139
Using Inclusion-Exclusion Principle:
We have the formula for the number of contestants who cleared at least one stage:
|A uu B uu C|=|A|+|B|+|C|-|A nn B|-|A nn C|-|B nn C|+|A nn B nn C||A \cup B \cup C| = |A| + |B| + |C| – |A \cap B| – |A \cap C| – |B \cap C| + |A \cap B \cap C|
We know:
|A uu B uu C|=139|A \cup B \cup C| = 139
Substituting the known values:
139=57+45+72-|A nn B|-|A nn C|-|B nn C|+17139 = 57 + 45 + 72 – |A \cap B| – |A \cap C| – |B \cap C| + 17
Simplifying:
139=191-(|A nn B|+|A nn C|+|B nn C|)+17139 = 191 – (|A \cap B| + |A \cap C| + |B \cap C|) + 17
139=208-(|A nn B|+|A nn C|+|B nn C|)139 = 208 – (|A \cap B| + |A \cap C| + |B \cap C|)
|A nn B|+|A nn C|+|B nn C|=208-139=69|A \cap B| + |A \cap C| + |B \cap C| = 208 – 139 = 69
Finding the individual intersections:
We know:
|A nn B nnC^(c)|=11quad(cleared only first two stages)|A \cap B \cap C^c| = 11 \quad \text{(cleared only first two stages)}
So:
|A nn B|=|A nn B nn C|+|A nn B nnC^(c)|=17+11=28|A \cap B| = |A \cap B \cap C| + |A \cap B \cap C^c| = 17 + 11 = 28
We also know:
|A^(c)nnB^(c)nn C|=20quad(cleared only third stage)|A^c \cap B^c \cap C| = 20 \quad \text{(cleared only third stage)}
Using the total of intersections:
|A nn B|+|A nn C|+|B nn C|=69|A \cap B| + |A \cap C| + |B \cap C| = 69
We have:
|A nn B|=28|A \cap B| = 28
|A nn C|+|B nn C|=69-28=41|A \cap C| + |B \cap C| = 69 – 28 = 41
Contestants who cleared only the third stage:
|A^(c)nnB^(c)nn C|=20|A^c \cap B^c \cap C| = 20
Contestants who cleared at least two stages:
Let’s break this into:
|A nn B||A \cap B| includes those who cleared both first and second stages
|A nn C||A \cap C| includes those who cleared both first and third stages
|B nn C||B \cap C| includes those who cleared both second and third stages
|A nn B nn C||A \cap B \cap C| is already counted thrice in the above summation
Using the intersection counts:
|A nn B nn C|=17|A \cap B \cap C| = 17
Therefore, the contestants who cleared at least two stages are:
|A nn B|+|A nn C|+|B nn C|-2xx|A nn B nn C||A \cap B| + |A \cap C| + |B \cap C| – 2 \times |A \cap B \cap C|
We know:
|A nn B|=28,quad|A nn C|=x,quad|B nn C|=y,quad x+y=41|A \cap B| = 28, \quad |A \cap C| = x, \quad |B \cap C| = y, \quad x + y = 41
And including all stages:
“Total contestants cleared at least two stages”=28+x+y-2xx17\text{Total contestants cleared at least two stages} = 28 + x + y – 2 \times 17
Since x+y=41x + y = 41:
“Total contestants cleared at least two stages”=28+41-34=35\text{Total contestants cleared at least two stages} = 28 + 41 – 34 = 35
Therefore, the number of contestants who cleared at least two stages is 3535.
(b) For a distribution, Bowley’s coefficient of Skewness is -0.56,Q_(1)=16.4-0.56, \mathrm{Q}_1=16.4 and median =24.2=24.2. What is its coefficient of quartile deviation.
Answer:
Bowley’s coefficient of skewness (S_(k)S_k) is given by the formula:
Step 2: Calculating the Coefficient of Quartile Deviation
The coefficient of quartile deviation is given by:
“Coefficient of Quartile Deviation”=(Q_(3)-Q_(1))/(Q_(3)+Q_(1))\text{Coefficient of Quartile Deviation} = \frac{Q_3 – Q_1}{Q_3 + Q_1}
Substitute the known values:
“Coefficient of Quartile Deviation”=(26.4-16.4)/(26.4+16.4)\text{Coefficient of Quartile Deviation} = \frac{26.4 – 16.4}{26.4 + 16.4}
“Coefficient of Quartile Deviation”=(10)/(42.8)\text{Coefficient of Quartile Deviation} = \frac{10}{42.8}
“Coefficient of Quartile Deviation”~~0.2336\text{Coefficient of Quartile Deviation} \approx 0.2336
Summary:
The coefficient of quartile deviation is approximately 0.23360.2336.
Question:-06
6.A researcher wants to study the association between temperament of husband and wife. She examined 5120 pairs and made the following contingency tables :
Temperament
of Husband
Temperament
of Husband| Temperament |
| :—: |
| of Husband |
Determine and interpret the association between the temperament of husband and wife.
Answer:
To determine the association between the temperament of husband and wife, we can use the chi-square test for independence. This test will help us determine whether there is a significant association between the temperament categories of husbands and wives.
Step 6: Compare the Chi-Square Statistic with the Critical Value
Since chi^(2)=24.22\chi^2 = 24.22 is greater than chi_(critical)^(2)=9.488\chi^2_{critical} = 9.488, we reject the null hypothesis.
Interpretation
There is a significant association between the temperament of husbands and wives. The observed frequencies differ significantly from the expected frequencies, suggesting that the temperament of one partner is related to the temperament of the other.
Question:-07
7.(a) Suppose a student of PGDAST calculated r_(12)=0.90,r_(13)=0.30r_{12}=0.90, r_{13}=0.30 and r_(23)=0.70r_{23}=0.70 from a data set. Examine whether these computations are error free.
Answer:
To examine whether the computations of the correlation coefficients r_(12)=0.90r_{12} = 0.90, r_(13)=0.30r_{13} = 0.30, and r_(23)=0.70r_{23} = 0.70 are error-free, we can use the property of correlation coefficients and the concept of the determinant of the correlation matrix.
The determinant of the correlation matrix should be non-negative for the matrix to be positive semidefinite, which is a requirement for a valid correlation matrix.
The determinant of the correlation matrix RR is -0.012-0.012, which is negative. For a valid correlation matrix, the determinant should be non-negative (i.e., >= 0\geq 0).
Conclusion
Since the determinant of the correlation matrix is negative, the given correlation coefficients r_(12)=0.90r_{12} = 0.90, r_(13)=0.30r_{13} = 0.30, and r_(23)=0.70r_{23} = 0.70 cannot all be correct simultaneously. Thus, the computations are not error-free. At least one of the given correlation coefficients must be incorrect.
(b) (i) Explain the method of least squares.
(ii) Fit an equation of the form y=ab^(X)y=a b^{\mathrm{X}} on the following data using the method of least squares.
The method of least squares is a standard approach in regression analysis to approximate the solution of overdetermined systems (sets of equations in which there are more equations than unknowns). It minimizes the sum of the squares of the residuals (the differences between observed and calculated values).
Steps in the Method of Least Squares:
Model Specification:
Define the mathematical form of the relationship between the dependent variable yy and the independent variable xx. Common forms include linear y=a+bxy = a + bx, exponential y=ab^(x)y = ab^x, etc.
Formulate the Objective Function:
For a given set of data points (x_(i),y_(i))(x_i, y_i), the residual for each point is the difference between the observed value and the value predicted by the model.
The sum of the squares of these residuals is the objective function that needs to be minimized:S=sum_(i=1)^(n)(y_(i)- hat(y)_(i))^(2)S = \sum_{i=1}^{n} (y_i – \hat{y}_i)^2
Derive the Normal Equations:
Calculate the partial derivatives of SS with respect to the model parameters and set them to zero. This yields a system of normal equations.
Solve the Normal Equations:
Solve these equations to obtain estimates of the model parameters.
(ii) Fitting an Equation of the Form y=ab^(X)y = a b^X
To fit the equation y=ab^(X)y = ab^X using the method of least squares, we first transform it into a linear form by taking the natural logarithm on both sides:
ln(y)=ln(a)+X ln(b)\ln(y) = \ln(a) + X \ln(b)
Let Y^(‘)=ln(y)Y’ = \ln(y), A=ln(a)A = \ln(a), and B=ln(b)B = \ln(b). Then the equation becomes:
Y^(‘)=A+BXY’ = A + BX
We can now use the method of least squares to fit a linear equation to the transformed data.