Answer to Question #349495 in Statistics and Probability for Aashna

Question

Give reason whether the following statements are True or False.

(a) If the probability of non rejection of H0 when H1 is true is 0.4 then power of the test will be 0.6.

(b) If T1 and T2 are two estimators of the parameter θ such that Var(T1) = 1/n and Var(T2) = n then T1 is more efficient than T2.

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

(d) If the level of significance is the same, the area of the rejection region in a two-tailed test is less than that in a one-tailed test.

(e) Non parametric tests are more powerful than the parametric tests.

Answer 1

Solution:

a) We reject the null hypothesis in favor of the alternative hypothesis if the sample evidence suggests so. If the sample does not contradict H0, we continue to believe it is true. Thus, the two possible conclusions from a hypothesis-testing analysis are reject H0 or fail to reject H0. The power of a test is the probability of rejecting the null hypothesis when it is false in other words, it is the probability of avoiding a type II error. The power may also be thought of as the likelihood that a particular study will detect a deviation from the null hypothesis given that one exists. So, if H1 true and probability of non rejection of H0 equal 0.4, it ture test power 0.6. Because test power is The power of a test is the probability of rejecting the null hypothesis when it is false.

Answer: True

b) Var(T1) = 1/n and Var(T2) = n, We need to find relative efficiency:

$eff(T1,T2)=\frac{Var(T2)}{Var(1)}=\frac{\frac{1}{n}}{n}=\frac{1}{n^{2}}<1;$ n not equal to 0;

It means variability of T1 greater than T2. So,

Answer: False.

c) The confidence interval represents how much certainty you have about a sample set falling within a range of values. These values support the confidence level and represent the probability of an entire population meeting the same outcomes or evaluation parameters as your statistical findings from a sample. So, formula of confidence interval:

$CI=X-/+z(\frac{\sigma}{\sqrt{n}});$

$\delta CI=2z(\frac{\sigma}{\sqrt{n}});$ $z_1(95\%)<z_2(99\%);$ So,

Answer: True.

d) One-tailed tests allow for the possibility of an effect in one direction. Two-tailed tests test for the possibility of an effect in two directions—positive and negative. For a one tailed test, the same significant level would be in one tail. For a two tailed test, the rejection region would be in two tails. So,

Answer: False.

e) Parametric tests are those that make assumptions about the parameters of the population distribution from which the sample is drawn. Non parametric tests are “distribution-free” and, as such, can be used for non-Normal variables. That's why parametric tests are in general more powerful than non parametric tests. Because, require a smaller sample size. Non parametric tests are used in cases where parametric tests are not appropriate. Most non parametric tests use some way of ranking the measurements and testing for weirdness of the distribution. So,

Answer: False.

Answers:

a) True;

b) False;

c) True;

d) False;

e) False.