Answer :
SECTION A
- Attempt all questions in brief.
a. What is Statistics?
Statistics is the study of collecting, analyzing, interpreting, presenting, and organizing data. It helps in making informed decisions based on data analysis.
b. What is Mean Absolute Deviation?
Mean Absolute Deviation (MAD) is a measure of the average distance between each data point and the mean of the data set. It provides a sense of variability within a data set.
c. What is Time Series?
A time series is a sequence of data points recorded at successive time intervals. It is used in different fields to track changes over time.
d. What is Time Reversal Test?
The time reversal test is a method that checks for consistency in index numbers by comparing the index number of forward time data with its backward time data.
e. What is Regression Coefficient?
A regression coefficient is a value that represents the relationship between variables in a regression model. It shows how much the dependent variable is expected to change when the independent variable changes by one unit.
f. Define Probability.
Probability is the measure of the likelihood that an event will occur. It ranges between 0 and 1, where 0 indicates an impossibility and 1 indicates certainty.
g. What is Null and Alternative Hypothesis?
The null hypothesis is a statement that there is no effect or no difference; it is the hypothesis that researchers aim to test. The alternative hypothesis is the statement that there is an effect or a difference.
SECTION B
- Attempt any three of the following:
b. What are the components of time Series? How would you find out the trend values in a time series by the method of least squares?
Components of a time series include:
- Trend: The long-term movement in the series.
- Seasonal variations: Regular movements repeating over periods.
- Cyclical variations: Long-term oscillations around the trend line.
- Irregular variations: Random, unpredictable movements.
To find the trend values using the least squares method:
- Fit the data into a linear equation: [tex]y = a + bt[/tex], where [tex]y[/tex] is the value at time [tex]t[/tex], [tex]a[/tex] is the y-intercept, and [tex]b[/tex] is the slope.
- Use formulas to calculate [tex]a[/tex] and [tex]b[/tex] based on minimizing the sum of squared differences between observed and predicted values.
c. Define correlation and explain how the coefficient of correlation is calculated by Karl Pearson's method. What are the limits between which the value of r is found?
Correlation is a statistical measure that expresses the extent to which two variables are linearly related. Karl Pearson's method calculates the Pearson correlation coefficient (r) as:
[tex]r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}}[/tex]
The value of [tex]r[/tex] ranges between -1 and 1. An [tex]r[/tex] of 1 implies a perfect positive linear correlation, -1 implies a perfect negative linear correlation, and 0 implies no linear correlation.
SECTION C
- Attempt any one part of the following:
(a) Find out Mean, Median and Mode from the following data:
Marks: 10-20 20-30 30-40 40-50 50-60
No. of students: 15 20 45 15 5
Mean:
Multiply the mid-point of each class interval by the number of students, sum these products, and divide by the total number of students to get the mean.
Median:
Find the cumulative frequencies, locate the median class, and use the median formula.
Mode:
Identify the class interval with the highest frequency.
(Specific numerical calculations would depend on the exact midpoint values used.)
