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Q. No 1 Ans T distribution The Studentās t-Distribution: The mathematical equation of the t-distribution is as follows:
This distribution has only one parameter Ī½, which is known as the degrees of freedom of the t- distribution. PROPERTIES OF STUDENTāS t-DISTRIBUTION The t-distribution has the following properties: i) The t-distribution is bell-shaped and symmetric about the value t = 0, ranging from ā ā to ā. ii) The number of degrees of freedom determines the shape of the t-distribution. Thus there is a different t-distribution for each number of degrees of freedom. As such, it is a whole family of distributions. The t-distribution, for small values of Ī½, is flatter than the standard normal distribution which means that the t distribution is more spread out in the tails than is the standard normal distribution.
As the degrees of freedom increase, the t- distribution becomes narrower and narrower, until, as n tends to infinity, it tends to coincide with the standard normal distribution. (The t-distribution can never become narrower than the standard normal distribution.) iii) The t-distribution has a mean of zero, when Ī½ ā„ 2. (The mean does not exist when Ī½ = 1.) iv)
The median of the t-distribution is also equal to zero. docsity.com
v) The t-distribution is unimodal. The density of the distribution reaches its maximum at t = 0 and thus the mode of the t- distribution is t = 0. (The students will recall that, for any hump-shaped symmetric distribution, the mean, median and mode are equal.)
Q. No 2 Husband and wife apply for same job and husband probability for job is like this 9/3 and wife job probability was something 3/5 like, what is the probability that husband could get job and wife not? Q. No 3 a boy sell umbrellas on rainy days and earn 30$ and in other days he can earn 24$ , if the probability of rain is 0.3% what will he expectation?
Q. No 4 CENTRAL LIMIT THEOREM The theorem states that: āIf a variable X from a population has mean Ī¼ and finite variance Ļ2, then the sampling distribution of the sample mean āÆ X approaches a normal distribution with mean Ī¼ and variance Ļ2/n as the sample size n approaches infinity.ā As n ā ā, the sampling distribution of āÆ X approaches normality.
Due to the Central Limit Theorem, the normal distribution has found a central place in the theory of statistical inference.(Since, in many situations, the sample is large enough for our sampling distribution to be approximately normal, therefore we can utilize the mathematical properties of the normal distribution to draw inferences about the variable of interest). The rule of thumb in
this regard is that if the sample size, n, is greater than or equal to 30, then we can assume that thedocsity.com
Q. No 7
Q. No 8 Degrees of freedom In statistics, the number of degrees of freedom is the number of values in the final calculation of a statistic that are free to vary. Mathematically, degrees of freedom is the dimension of the domain of a random vector, or essentially the number of 'free' components: how many components need to be known before the vector is fully determined.. In equations, the typical symbol for degrees of freedom is (lowercase Greek letter nu). In text and tables, the abbreviation "d.f." is commonly used.
. No 9 Statistical hypothesis testing A statistical hypothesis test is a method of making decisions using data, whether from a controlled experiment or an observational study (not controlled). In statistics, a result is called statistically significant if it is unlikely to have occurred by chance alone, according to a pre- determined threshold probability, the significance level. The phrase "test of significance" was coined by Ronald Fisher: "Critical tests of this kind may be called tests of significance, and when such tests are available we may discover whether a second sample is or is not significantly different from the first. Hypothesis testing is sometimes called confirmatory data analysis, in contrast to exploratory data analysis. In frequency probability, these decisions are almost always made using null-hypothesis tests. These are tests that answer the question assuming that the null hypothesis is true, what is the probability of observing a value for the test statistic that is at least as extreme as the value that was actually observed?) More formally, they represent answers to the question, posed before undertaking an experiment, of what outcomes of the experiment would lead to rejection of the null hypothesis for a pre-specified probability of an incorrect rejection.
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Q. No 10 DISTRIBUTION FUNCTION The distribution function of a random variable X, denoted by F(x), is defined by F(x) = P(X < x). The function F(x) gives the probability of the event that X takes a value LESS THAN OR EQUAL TO a specified value x. The distribution function is abbreviated to d.f. and is also called the cumulative distribution function (cdf) as it is the cumulative probability function of the random variable X from the smallest value up to a specific value x.
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Q. No 12
EXPERIMENTAL DESIGN By an experimental design, we mean a plan used to collect the data relevant to the problem under study in such a way as to provide a basis for valid and objective inference about the stated problem. The plan usually includes:
- The selection of treatments, whose effects are to be studied,
- The specification of the experimental layout, and
- The assignment of treatments to the experimental units. All these steps are accomplished before any experiment is performed. Experimental Design is a very vast area. In this course, we will be presenting only a very basic introduction of this area. There are two types of designs: Q. No 13 Define an independent and dependent variable in regression? 2 Marks
INDEPENDENT EVENTS Two events A and B in the same sample space S, are defined to be independent (or statistically independent) if the probability that one event occurs, is not affected by whether the other event
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Q. No 16 The 90% confidence interval for the population mean is 11 to 20, interpret this result? 3 Marks
Q. No 17 Define LSD test? 3 Marks THE LEAST SIGNIFICANT DIFFERENCE (LSD) TEST According to this procedure, we compute the smallest difference that would be judged significant, and compare the absolute values of all differences of means with it. This smallest difference is called the least significant difference or LSD, and is given by: LEAST SIGNIFICANT DIFFERENCE (LSD):
where MSE is the Mean Square for Error, r is the size of equal samples, and tĪ±/2 (Ī½) is the value of t at Ī±/2 level taken against the error degrees of freedom (Ī½).
The test-criterion that uses the least significant difference is called the LSD test. docsity.com
Q. No 18 CORRELATION CO-EFFICIENT OF TWO RANDOM VARIABLES Let X and Y be two r.v.ās with non-zero variances Ļ2X and Ļ2Y. Then the correlation coefficient which is a measure of linear relationship between X and Y, denoted by ĻXY (the Greek letter rho) or Corr(X, Y), is defined as:
If X and Y are independent r.v.ās, then ĻXY will be zero but zero correlation does not necessarily imply independence.
Q. No 19 Confidence Interval? 5 Marks A confidence interval is an interval computed from the sample observations x1, x2ā¦.xn, with a statement of how confident we are that the interval does contain the population parameter.
Confidence interval In statistics, a confidence interval (CI) is a kind of interval estimate of a population parameter and is used to indicate the reliability of an estimate. It is an observed interval (i.e. it is calculated from the observations), in principle different from sample to sample, that frequently includes the parameter of interest, if the experiment is repeated. How frequently the observed interval contains the parameter is determined by the confidence level or confidence coefficient Confidence intervals consist of a range of values (interval) that act as good estimates of the unknown population parameter.
Q. No 20 Calculate Harmonic Mean of given data? 5 Marks
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The mean of a Poisson distribution is 5 while its standard deviation is 4. Comment on it POISSON DISTRIBUTION The Poisson distribution is named after the French mathematician Simeāon Denis Poisson (1781-
- who published its derivation in the year 1837.THE POISSON DISTRIBUTION ARISES IN THE FOLLOWING TWO SITUATIONS:
- It is a limiting approximation to the binomial distribution, when p, the probability of success is
very small but n, the number of trials is so large that the product np = Ī¼ is of a moderate size;
- a distribution in its own right by considering a POISSON PROCESS where events occur
randomly over a specified interval of time or space or length. EXAMPLE Two hundred passengers have made reservations for an airplane flight. If the probability that a passenger who has a reservation will not show up is 0.01, what is the probability that exactly three will not show up? SOLUTION Let us regard a āno showā as success. Then this is essentially a binomial experiment with n = 200 and p = 0.01. Since p is very small and n is considerably large, we shall apply the Poisson distribution, using Ī¼= np = (200) (0.01) = 2. Therefore, if X represents the number of successes (not showing up), we have
Q. No 22 If an automobile is driven on the average no more than 16000 Km per year then formulate the null and alternative hypothesis (2) NULL AND ALTERNATIVE HYPOTHESES
NULL HYPOTHESIS docsity.com
A null hypothesis, generally denoted by the symbol H0, is any hypothesis which is to be tested for possible rejection or nullification under the assumption that it is true. A null hypothesis should always be precise such as āthe given coin is unbiasedā or āa drug is ineffective in curing a particular diseaseā or āthere is no difference between the two teaching methodsā. The hypothesis is usually assigned a numerical value. For example, suppose we think that the average height of students in all colleges is 62ā³. This statement
is taken as a hypothesis and is written symbolically as H0 : Ī¼ = 62ā³. In other words, we
hypothesize that Ī¼ = 62ā³. ALTERNATIVE HYPOTHESIS An alternative hypothesis is any other hypothesis which we are willing to accept when the null hypothesis H0 is rejected. It is customarily denoted by H1 or HA. A null hypothesis H0 is thus tested against an alternative hypothesis H1. For example, if our null hypothesis is H0 : Ī¼ = 62ā³, then our alternative hypothesis may be
H1 : Ī¼ ā 62 ā³ or H1 : Ī¼ < 62ā³.
Q. No 23 If the population proportions are gives as: P1 = 0.4, P2 = 0.20 find sigma^2 P-hat 1 - P-hat 2 , where n = 12.
Q. No 24 How many parameters are associated with F- distribution and what is the range of the distribution? (3) This distribution has two parameters Ī½1 and Ī½2, which are known as the degrees of freedom of
the F-distribution.The Fdistribution having the above equation have Ī½1 degrees of freedom in
the numerator and Ī½2 degrees of freedom in the denominator. It is usually abbreviated as F
(Ī½1, Ī½2). PROPERTIES OF F-DISTRIBUTION
- The F-distribution is a continuous distribution ranging from zero to plus infinity.
- The curve of the F-distribution is positively skewed. Q. No 25
Which of the following statement represents continuous data and discrete data? (5) docsity.com
The number of success, X in a hyper geometric experiment is called a hyper geometric random variable and its probability distribution is called the hyper geometric distribution. When the hyper geometric random variable X assumes a value x, the hyper geometric probability distribution is given by the formula range of chi-square distribution. ans(0 to infinity) given x- x(bar)=3,-3,4,-4,1,-1 find out mead deviation. ans is o what are steps involved in statistical research find out upper quartile of standard normal distribution
ANS Calculate sampling error if sample is 102 and population mean is 100 SAMPLING & NON-SAMPLING ERRORS
- SAMPLING ERROR The difference between the estimate derived from the sample (i.e. the statistic) and the true population value (i.e. the parameter) is technically called the sampling error. For example,
ANS Quartile deviation QUARTILE DEVIATION The quartile deviation is defined as half of the difference between the third and first quartiles i.e.
ANS
Pearson co-efficient of skewness:
A candidate for mayor in a large city hires the services of a poll-taking organization, and they found that 62 of 100 educated voters interviewed support the candidate, and 69 of 150
uneducated voters support him. docsity.com
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