Sample Solution

1(a) State whether the following statements are True or False. Give reasons in support of your answer:
(i) If the form of the population is not known and data are in ordinal form then we apply the Wilcoxon Signed rank test for testing hypothesis about average.
(ii)The Neyman-Pearson lemma provides the most powerful test of size $\alpha$$\alpha$alpha\alpha$\alpha$ for testing a simple hypothesis against a simple alternative hypothesis.
(iii) The Rao-Blackwell theorem enables us to obtain a manimum variance unbiased estimator through complete statistic.
(iv) For testing goodness of fit when the data are in categorical form, we use K-S test.
(v) In the Bayesian approach, we treat the parameter as a constant.

(b) A radar system use radio waves to detect aircraft. The system receives a signal and based on the received signal, it needs to decide whether an aircraft is present or not. If $X$$X$XX$X$ denotes the received signal then

To test the hypothesis
${\mathrm{H}}_0$${\mathrm{H}}_0$H_(0)\mathrm{H}_0${\mathrm{H}}_0$ :No aircraft is present i.e. $\theta =0$$\theta =0$theta=0\theta=0$\theta =0$
$H_1:$$H_1:$H_(1):H_1:$H_1:$ An aircraft is present i.e. $\theta =1$$\theta =1$theta=1\theta=1$\theta =1$
Derive the test at $\alpha =0.05$$\alpha =0.05$alpha=0.05\alpha=0.05$\alpha =0.05$.

2 (a) State Lehmann-Scheffe theorem.
(b) If $X_1,X_2,\dots ,X_n$$X_1,X_2,\dots ,X_n$X_(1),X_(2),dots,X_(n)X_1, X_2, \ldots, X_n$X_1,X_2,\dots ,X_n$ is a random sample taken from a Poisson distribution whose pmf is given as

(i) Show that statistic $\sum ^{\mathrm{n}}{\mathrm{X}}_{\mathrm{i}}$$\sum ^{\mathrm{n}} {\mathrm{X}}_{\mathrm{i}}$sumnX_(i)\sum^{\mathrm{n}} \mathrm{X}_{\mathrm{i}}$\sum ^{\mathrm{n}}{\mathrm{X}}_{\mathrm{i}}$ is sufficient and complete.
(ii) Show that $\frac{1}{\mathrm{n}}\sum _{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{X}}_{\mathrm{i}}$$\frac{1}{\mathrm{n}}\sum _{\mathrm{i}=1}^{\mathrm{n}} {\mathrm{X}}_{\mathrm{i}}$(1)/(n)sum_(i=1)^(n)X_(i)\frac{1}{\mathrm{n}} \sum_{\mathrm{i}=1}^{\mathrm{n}} \mathrm{X}_{\mathrm{i}}$\frac{1}{\mathrm{n}}\sum _{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{X}}_{\mathrm{i}}$ is the uniformly minimum variance unbiased estimator.

3(a) A faculty member of a university receives a number of emails. If $X$$X$XX$X$ represents the number of spam emails in $n$$n$nn$n$ emails and follows a binomial distribution with parameters ( $\mathrm{n},\theta$$\mathrm{n},\theta$n,theta\mathrm{n}, \theta$\mathrm{n},\theta$ ) where $\theta$$\theta$theta\theta$\theta$ is the probability of getting spam email, then find the posterior distribution of $\theta$$\theta$theta\theta$\theta$ considering the following beta distribution.

Also, find the posterior mean of $\theta$$\theta$theta\theta$\theta$.

(b) A train is expected to arrive at a station at 8.00 AM. However, it has been observed that it reaches the station between 7.55 AM to 8.05 AM and the times are uniformly distributed between the 7.55 AM to 8.05 AM interval. Using the following $\mathrm{U}(0,1)$$\mathrm{U}(0,1)$U(0,1)\mathrm{U}(0,1)$\mathrm{U}(0,1)$ random numbers simulate time for arrival on ten days.

4 (a) A food processing company packages 10 gm of honey in small jars. Previous experience suggests that the volume of a randomly selected jar of the company’s honey is normally distributed with a known variance of 2 gm. Drive likelihood ratio test for testing

at a $\alpha$$\alpha$alpha\alpha$\alpha$ level of significance. Given $[\mathrm{P}[-\mathrm{\infty }<\mathrm{Z}<1.64]=0.95]$$[\mathrm{P}[-\mathrm{\infty }<\mathrm{Z}<1.64]=0.95]$[P[-oo < Z < 1.64]=0.95][\mathrm{P}[-\infty<\mathrm{Z}<1.64]=0.95]$[\mathrm{P}[-\mathrm{\infty }<\mathrm{Z}<1.64]=0.95]$

(b) If the number of weekly accidents occurring on a mile stretch of a particular road follows a Poisson distribution with parameter $\lambda$$\lambda$lambda\lambda$\lambda$. Then
(i) Find the Cramer-Rao lower bound for the variance.
(ii) Also, find the UMVUE of $\lambda$$\lambda$lambda\lambda$\lambda$.

5(a) The following data give the sales of 7 models of mobiles at four different stores. The sales of each mobile (in number of mobiles sold) from each store are given as follows:

Test whether there is a significant difference in the sales of the four stores by using the Kruskal Wallies test at $1\mathrm{\%}$$1\mathrm{\%}$1%1 \%$1\mathrm{\%}$ level of significance (Given $(x_{(3),0.01}^2=11.34)$$\left(x_{(3),0.01}^2=11.34\right)$(x_((3),0.01)^(2)=11.34)\left(x_{(3), 0.01}^2=11.34\right)$(x_{(3),0.01}^2=11.34)$

(b) The number of customer arrivals at a restaurant follows Poisson distribution whose pmf is given as follows.

It is assumed that the arrival rate $(\lambda )$$(\lambda )$(lambda)(\lambda)$(\lambda )$ of the customers is a random variable and has gamma prior whose pdf is given as follows:

If ${\mathrm{X}}_1,{\mathrm{X}}_2,\dots ,{\mathrm{X}}_{\mathrm{n}}$${\mathrm{X}}_1,{\mathrm{X}}_2,\dots ,{\mathrm{X}}_{\mathrm{n}}$X_(1),X_(2),dots,X_(n)\mathrm{X}_1, \mathrm{X}_2, \ldots, \mathrm{X}_{\mathrm{n}}${\mathrm{X}}_1,{\mathrm{X}}_2,\dots ,{\mathrm{X}}_{\mathrm{n}}$ repersent the number of customer arrivals, then show that gamma is a conjugate distribution for the Poisson distribution.

(i) If the form of the population is not known and data are in ordinal form then we apply the Wilcoxon Signed rank test for testing hypothesis about average.
(ii) The Neyman-Pearson lemma provides the most powerful test of size $\alpha$$\alpha$alpha\alpha$\alpha$ for testing a simple hypothesis against a simple alternative hypothesis.
(iii) The Rao-Blackwell theorem enables us to obtain a minimum variance unbiased estimator through complete statistic.
(iv) For testing goodness of fit when the data are in categorical form, we use K-S test.
(v) In the Bayesian approach, we treat the parameter as a constant.

(i) If the form of the population is not known and data are in ordinal form then we apply the Wilcoxon Signed rank test for testing hypothesis about average.

The statement is true.

The Wilcoxon Signed Rank Test is a non-parametric test that is used when the population distribution is not assumed to be normal and data are in an ordinal or continuous form. It tests the hypothesis about the median (or average) of a single sample or the difference between paired samples. This test is particularly useful when the data do not meet the assumptions of the t-test, such as normality, and when the data are at least ordinal, as it relies on ranking rather than actual data values.

Thus, if the population form is unknown and data are in ordinal form, the Wilcoxon Signed Rank Test is appropriate for hypothesis testing about the average or median.

(ii) The Neyman-Pearson lemma provides the most powerful test of size $\alpha$$\alpha$alpha\alpha$\alpha$ for testing a simple hypothesis against a simple alternative hypothesis.

The statement is true.

The Neyman-Pearson Lemma states that for testing a simple null hypothesis $H_0:\theta ={\theta }_0$$H_0:\theta ={\theta }_0$H_(0):theta=theta_(0)H_0: \theta = \theta_0$H_0:\theta ={\theta }_0$ against a simple alternative hypothesis $H_1:\theta ={\theta }_1$$H_1:\theta ={\theta }_1$H_(1):theta=theta_(1)H_1: \theta = \theta_1$H_1:\theta ={\theta }_1$, the likelihood ratio test provides the most powerful test of a given size $\alpha$$\alpha$alpha\alpha$\alpha$. In other words, among all possible tests of size $\alpha$$\alpha$alpha\alpha$\alpha$, the likelihood ratio test maximizes the probability of correctly rejecting $H_0$$H_0$H_(0)H_0$H_0$ when $H_1$$H_1$H_(1)H_1$H_1$ is true. This is specifically applicable to simple hypotheses (i.e., hypotheses that specify a single value for the parameter), and the lemma guarantees the most powerful test only in this scenario.

(iii) The Rao-Blackwell theorem enables us to obtain a minimum variance unbiased estimator through complete statistic.

The statement is true.

The Rao-Blackwell theorem states that given an unbiased estimator $T$$T$TT$T$ of a parameter $\theta$$\theta$theta\theta$\theta$ and a sufficient statistic $S$$S$SS$S$ for $\theta$$\theta$theta\theta$\theta$, the conditional expectation $E(T|S)$$E(T|S)$E(T|S)E(T | S)$E(T|S)$ is an estimator of $\theta$$\theta$theta\theta$\theta$ that has a variance less than or equal to that of $T$$T$TT$T$. If $S$$S$SS$S$ is also a complete statistic, then $E(T|S)$$E(T|S)$E(T|S)E(T | S)$E(T|S)$ is the unique minimum variance unbiased estimator (MVUE) of $\theta$$\theta$theta\theta$\theta$. Therefore, the theorem provides a way to improve an unbiased estimator by conditioning on a complete, sufficient statistic, thus obtaining a minimum variance unbiased estimator.

(iv) For testing goodness of fit when the data are in categorical form, we use K-S test.

The statement is false.

The Kolmogorov-Smirnov (K-S) test is generally used to compare a sample with a continuous reference distribution or to compare two continuous sample distributions. It is not suitable for categorical data, as it relies on cumulative distribution functions, which are defined for continuous data. For categorical data, the Chi-square goodness-of-fit test is typically used, as it assesses how well the observed frequencies in categories match the expected frequencies based on a hypothesized distribution.

(v) In the Bayesian approach, we treat the parameter as a constant.

The statement is false.

In the Bayesian approach, we do not treat the parameter as a constant. Instead, we treat the parameter as a random variable with a probability distribution that reflects our prior beliefs about the parameter’s possible values. This prior distribution is then updated with the observed data through Bayes’ theorem to obtain the posterior distribution. Unlike the frequentist approach, where the parameter is considered a fixed but unknown constant, the Bayesian approach inherently involves treating the parameter as a variable with uncertainty.

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