State whether the following statements are True or False. Give reason

Question

State whether the following statements are True or False. Give reason in support of your answer:

(i) if $X_1,X_2,X_3,X_4$ and $X_5$ is a random sample of size 5 taken from an Exponential distribution, then estimator T1 is more efficient than $T_2$ . $T_1=\frac{X_1+X_2+X_3+X_4+X_5}{5},\, T_2=\frac{X_1+2X_2+3X_3+4X_4+5X_5}{15}$

(ii) if $T_1$ and $T_2$ are two estimators of the parameter $\theta$ such thet Var $\left (T_1 \right )=1/n$ and Var $\left (T_2 \right )$ then $T_1$ is more efficient then $T_2.$

(iii) A 95% confidence interval is smaller than 99% confidence interval.

(iv) If the probability density function of a random variable X follows F-distribution is

$f(x)=\frac{1}{(1+x)},x\geq 0$

then degrees of freedom of the distribution will be (2,2).

(v) A patient suffering from fever reaches to a doctor and suppose the doctor formulate the hypotheses as

$H_0:$ The patient is a chikunguniya patient

$H_1:$ The patient is not a chikunguniya patient

If the doctor rejects H0 when the patient is actually a chikunguniya patient, then the doctor commits type II error.

02 Apr 2024

Answer :

Word Count : 255

(i) **True**. The efficiency of an estimator is related to its variance; lower variance implies higher efficiency. For $T_1$ and $T_2$, $T_1$ involves taking the average of the sample, while $T_2$ gives more weight to larger values in the sample. Generally, the sample mean is a more efficient estimator than a weighted sum of observations. Therefore, $T_1$ will have a smaller variance than _______ __________ _________ _______ ___.
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