Consider the following hypothesis test. $H_0: \mu \leq 12$ $H_a: \mu > 12$ If a sample of 25 is taken with sample mean 15 and a sample standard deviation of 6, then the value of t-test statistic is:

A small start-up started making wafers and distributing them to the retailers. After a week, average sales per week were found to be 150 packets. So, to increase the sales, a strategy was used to change the packaging and add a chocolate worth Rs. 5 as a free gift with the pack. After this, a sample of 17 shops was taken, which showed that sales went up with mean 165 and a standard deviation of 25. Check whether the strategy was effective @5%, level of significance ? [Given $t_{16} (0.05) =2.12$]

Which of the following are true ?

(A) If the mean of a normal variate is 12 and standard deviation is 4, then the Z-score of data point 20 is 2.

(B) Mean of the numbers obtained on throwing a die having '1' on three faces, '2' on two faces and '5' on one face is 3.

(C) If the variance of a Poisson's distribution is 2, then $P(X=0)=\frac{1}{e^2}$.

(D) For a binomial distribution, mean = 9 and the standard deviation is $\frac{3}{2}$, then the number of trials is 12.

Choose the correct answer from the options given below:

Which of the following are the assumptions underlying the use of t-distribution?

(A) The variance of population is known.

(B) The samples are drawn from a normally distributed population.

(C) Sample standard deviation is an unbiased estimate of the population variance.

(D) It depends on a parameter known as degree of freedom.

Choose the correct answer from the options given below:

If X is normally distributed with mean 30 and standard deviation 5, the value of $P(26 \le X \le 34)$ is: (Where $P(0 \le Z \le 0.8) = 0.2881$)

A random sample consists of five observations: 2,4,5,6,8. The point estimate of the population mean is :

If the sample size ($n$) is 32 in testing of hypothesis by t-test, then the degree of freedom ($\gamma$) is equal to :