It involves the distribution of various samples from bulk-like population data. It mainly explains the distribution of frequencies or statistics. It can be measured with the help of variance or standard deviation. It mainly depends on the number of observations of the population, sample, and the choice of a random sample. With a large population size compared to the sample size, the sampling distribution has the same standard error.
The probability of an event occurring measures the likelihood of that event. It provides a value between zero and one, depending upon the favorable outcome and sample space.
The value close to zero, classifies the less likelihood of the event, and the value close to one, shows the certainty of the event.
The random variable is a set of all possible outcomes in a sample space, represented by some number. In other words, it is a function that represents each point on a sample space by a number. For example, the following table represents the outcome of tossing a coin twice.
Sample Space (S) | TH | TT | HT | HH |
Random Variable (X) | 1 | 2 | 1 | 0 |
Let X represent the number of tails that can come up. We can represent that by assigning a number to each sample point. We can see, the number of tails is randomly changing, Thus X is a random variable.
The nonparametric test is used when the population size is small and has an unknown distribution.
In this type of statistical technique, the population data analyzed does not require to meet the population parameter.
However, this technique has some advantages, but this is less effective to analyze the relationship between the variable, if one exists.
Hypothesis testing is an analysis performed on the sample data. It is used to check the hypothesis regarding a population.
In order to test the idea regarding population parameters, we begin by setting a null hypothesis, and an alternate hypothesis. And, we try to reject the null hypothesis.
Statistical analysis refers to techniques of extracting useful pieces of information, such as trends, patterns, etc, from the collected data.
The collected data is represented using statistics, in a way that makes it easy to visualize the information.
For example, the analyzed data could be represented by a Pie Chart. It makes more sense out of given data.
The tree diagram is used to represent the sample space data of an event. It is used for a relatively small set of data.
It shows the probability of an event on the branch, and outcome at the end of the branch.
For example, the probability of an event A tossing a coin twice could be represented as:
An outlier is a variable, or data point, which is either very below from the mean of the data, or very far from the mean.
In other words, the outlier lies outside the concentrated data points. The outliers increase the standard deviation.
Bayes’ theorem gives a formula for determining conditional probability. It can calculate the probability of an event, based on certain previous likelihood of the event.
Where:
Q. If m things are distributed among ‘a’ men and ‘b’ women, find the chance that the number of things received by men is odd.
Sol. Probability that the man gets a things = a/(a+b) = p
Probability that the woman gets a things = b/(a+b) = q
Probability that r things are received by men = P(r) = .^m C_r p^r.q^(m-r)
Since men are to receive odd number of things i.e. 1 or 3 or 5 or … … …
So required probability is
=P(1)+P(3)+P(5)+⋯………
=^m C_1 p.q^(m-1)+.^m C_3 p^3.q^(m-3)+.^m C_5 p^5.q^(m-5)+⋯………
=1/2{(p+q)^m-(p-q)^m}
=1/2 {1-((b-a)/(b+a))^m }
Q. Find correlation coefficient between X and Y if covariance of X and Y is 100, variance of X is 10000, and variance of Y is 4.
Sol. Given,
Cov(X,Y)=100
Variance of X = 10000 ⇒ σ_x^2=10000
⇒σ_x=100
Variance of Y = 4 ⇒σ_y^2=4
⇒σ_y=2
So, r= Cov(X,Y)/(σ_x 〖 σ〗_y )
r= 100/(100×2)
r= 1/2
r=0.5
Q. Pumpkins were grown under two experimental conditions. Two random sample of 11 and 9 pumpkins show the sample standard deviations of their weights as 0.8 and 0.5 respectively. Assuming that the weight distributions are normal, test the hypothesis that the true variance are equal, against the alternative that they are not equal, at the 10% level
Sol. We want to test Null Hypothesis, H_0:σ_X^2=σ_Y^2 against the
Alternative Hypothesis, H_1:σ_X^2≠σ_Y^2 ( Two-Tailed)
We are given : n_1=11, n_2=9
s_X=0.8, S_Y=0.5
Under the null hypothesis we have statistics as
F= (s_X^( 2))/(s_Y^( 2) ) , follows F distribution with (n_1-1,n_2-1) degree of freedom
Also, n_1 s_X^( 2)=(n_1-1) S_X^( 2)
So, S_X^( 2)= n_1/(n_1-1) s_X^( 2 )= 11/10 ×(0.8)^2=0.704
Similarly, S_Y^( 2)=0.28125
So, F= 0.704/0.28125 =2.5
The significant values of F for the two-tailed test at the level of significance α=0.10 are:
F>F_10,8 (α/2) =F_10,8 (0.05)
F<F_10,8 (1-α/2) =F_10,8 (0.95)
We are given the tabulated values
P(F_10,8≥3.35)=0.05⇒F_10,8 (0.05)=3.35
P(F_8,10≥3.07)=0.05⇒P (1/F_8,10 ≤1/3.07) =0.05
⇒P(F_10,8≤0.326)=0.05⇒P(F_10,8≥0.326)=0.95
Hence from above values the critical values for testing null hypothesis against the alternative hypothesis at the level of significance α=0.10 are given by:
F>3.35 and F<0.326=0.33
since the calculated value of F=2.5 lies between 0.33 and 3.35, it is not significant and hence null hypothesis of equality of population variances may be accepted at level of significance α=0.10
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Statistics is an interdisciplinary field concerned with collecting data, analyzing, interpreting and presenting empirical facts by carrying out qualitative and quantitative research.
You will learn how to arrive at inference and learn techniques like sampling to summarize results. There are two types of Statistics: Descriptive and Inferential.
In the study of descriptive statistics, a given data set is summarized whereas in inferential statistics, you get to make predictions from a given sample or size of populations.
By studying Statistics, you can make intelligent assumptions and draw conclusions by analyzing data in any field.
The symbol 'n’ stands for the total number of observations or individuals in the sample. It is also referred to as sample size.
The symbol ‘r’ in Statistics refers to correlation coefficient. The values for it range from ‘-1’ to ‘+1’ and if it is ‘0’ it means there is no relationship between the variables. The variables are believed to be related if it is closer to +1 or -1.
The greek letter “mu” stands for population mean.
In statistics, N refers to population size.
In a statistical analysis, parameter stands for characteristics that are used to define a given population. Hence, it is a useful component to carry out an analysis as it points to a specific characteristic of an entire population.
The letter ‘s’ stands for standard deviation of a sample.
The p value in statistics helps you determine whether you should accept or reject a null hypothesis. The smaller the p-value, the stronger the evidence to go ahead and reject the null hypothesis.The value of ‘p’ is used as evidence against a null hypothesis.
Descriptive Statistics simply describes what the data shows. In other words, it describes the data and summarizes it. It is a simple summary about the given sample.
Inferential data stands for making inferences or drawing conclusions that extend beyond immediate data. It takes data from the sample and makes predictions about the entire population.
The greek letter sigma denoted as “σ” stands for the standard deviation of a population.
To put it simply, power is a probability to not make a Type 2 error and it increases when the sample size is increased. To detect a false null hypothesis, power must be close to 1 and if it is lower than 0.8, is considered too low to carry out research. Therefore, it is an important concept to understand in Statistics.
In Statistics, R squared is also known as the coefficient of determination.It determines how close the data is to the fitted regression line. It is always between 0-100%. In most cases the larger the R squared the better the regression model fits your observation.