6. Let be a random sample from the Exponential distribution with pdf
(i) Obtain the maximum likelihood estimator (MLE) of .
(ii) Hence, find the maximum likelihood estimate of for the observed sample:
2.1, 1.6, 3.4, 0.9, 2.5
The probability density function of the exponential distribution is [ f(x; \theta) = \theta e^{-\theta x}, \quad x>0, \theta>0. ] For a random sample (X_1, X_2, \dots, X_n), the likelihood function is [ L(\theta) = \prod_{i=1}^{n} f(X_i;\theta) = \prod_{i=1}^{n} \theta ______ ______ _____ _____ ___ ___.
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State whether the following statements are True or False. Give reason in support of your answer:
(a) The Wilcoxon rank-sum test is a non-parametric alternative to the two-sample t-test when normality is not assumed.
(b) The power of a statistical test is the probability of accepting the null hypothesis when it is true.
(c) If the sample size tends to infinity, the variance of a consistent estimator must tend to zero.
(d) In the method of moments, population moments are equated to corresponding sample moments.
(e) The chi-square distribution is symmetric for all degrees of freedom.
3. 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) Check whether it is more efficient than sample mean.
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.
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 |
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 Name | Age (in years) |
|---|---|
| Meena | 5 |
| Aarav | 4 |
| Nisha | 6 |
| Kabir | 4 |
| Pooja | 5 |
(i) What is the nature (type) of the population of the ages of children?
(ii) Construct the sampling distribution of the sample mean for all possible samples of size .
(iii) Comment on whether the sampling distribution is approximately normal.
(iv) Find the mean and standard error of the sampling distribution.
6. Let be a random sample from the Exponential distribution with pdf
(i) Obtain the maximum likelihood estimator (MLE) of .
(ii) Hence, find the maximum likelihood estimate of for the observed sample:
2.1, 1.6, 3.4, 0.9, 2.5