State whether the following statements are True or False. Give reasons in support of your answer:
(i) The mean square error is calculated when the estimator is unbiased.
(ii) If the sample size tends to infinity, the variance of a consistent estimator must tend to zero.
(iii) In the method of moments, population moments are equated to corresponding sample moments.
(iv) A 95% confidence interval is smaller than a 99% confidence interval.
(v) The chi-square distribution is symmetric for all degrees of freedom.
(i) False. Mean square error (MSE) is defined for any estimator, whether biased or unbiased, as it includes both variance and the square of bias; it is not restricted only to unbiased estimators. (ii) ____ _____ __________ __________ _____ ____ ________ __________ ____ _____ ___ _________.
__________ _____ _______ __________ ___ _________ ________ _____ _______.
___ ________ ____ _______ ___ __________ _____ _________ ____ ____ ____ ________.
__________ _________ _______ _______ ___ _______ ___ _____.
_______ _______ ______ ___ ____ _________.
____ ________ ___ _______ __________ _____ _____ __________ ____.
_______ __________ ______ _________ _________ _______ ___ ________.
______ ____ __________ ____ _______ ______ ___.
_____ ____ __________ __________ ____.
____ _______ ___ _______ ________ ___ _________ ________ _________ __________ ________ __________.
__________ ____ ________ ____ __________ ________ _______ ______ ________.
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Let be a random sample from a population with mean
and variance
. Consider the estimator
$
(i) Find the bias of the estimator .
(ii) Find the variance of .
(iii) Hence, calculate the Mean Square Error (MSE) of .
State whether the following statements are True or False. Give reasons in support of your answer:
(i) The mean square error is calculated when the estimator is unbiased.
(ii) If the sample size tends to infinity, the variance of a consistent estimator must tend to zero.
(iii) In the method of moments, population moments are equated to corresponding sample moments.
(iv) A 95% confidence interval is smaller than a 99% confidence interval.
(v) The chi-square distribution is symmetric for all degrees of freedom.
Describe the following:
(i) t-distribution
(ii) General procedure of Testing of hypothesis
A school caretaker looks after 6 children. The ages (in years) of the children are given below:
| Child | Age (in years) |
|---|---|
| Meena | 5 |
| Aarav | 4 |
| Nisha | 6 |
| Kabir | 4 |
| Pooja | 5 |
Let X1, X2, ..., Xbe a random sample from the Exponential distribution with pdf f(x; 0) = 0e-ex, x > 0, 0 > 0
(i) Obtain the maximum likelihood estimator (MLE) of 0.
(ii) Hence, find the maximum likelihood estimate of for the observed sample: 2.1, 1.6, 3.4, 0.9, 2.5
A researcher wants to compare the average test scores of students taught using Method A and Method B. A random sample of students was selected from each group, and their scores are given below:
| Student | Method A | Method B |
|---|---|---|
| 1 | 68 | 72 |
| 2 | 74 | 75 |
| 3 | 71 | 78 |
| 4 | 69 | 70 |
| 5 | 73 | 76 |
| 6 | 70 | 74 |
| 7 | 72 | 77 |
| 8 | 75 | 79 |
A survey was conducted to study the preference for online learning among students from urban and rural areas. Out of 300 urban students, 210 preferred online learning, while out of 250 rural students, 150 preferred online learning. Construct a 95% confidence interval for the difference between the population proportions of students preferring online learning in urban and rural areas.