A television channel claims that $25\%$ of its programmes are nature programmes. Yolande thinks the percentage claimed is too high. To test her hypothesis, she chooses $20$ programmes at random.

If Yolande carries out the hypothesis test at the $10\%$ significance level, define the random variable, state its distribution, including parameters, and define the hypotheses.

Chief Educational Officer wanted to study the performance of$XII$ Standard students in Mathematics subject. The following are the information obtained from randomly selected students from two Educational Districts.

Examine at$5\%$ Level of significance whether students in District A perform better compared to students in District B.

If ${\mu }_{}$ denotes the population mean, then find the critical value to be used for testing $H_0:\mu =100$ against $H_1:\mu <100$ based on $250$ observations at $5\%$ level of significance.

Carry out hypothesis testing exercise for testing $H_0:{\mu }_X={\mu }_Y$ against $H_1:{\mu }_X\ne {\mu }_Y$ with usual notations, when $\overline{x}= 7$ and $\overline{y}=8$ ,$\sigma { }_x=3$ and ${\sigma }_y =2$and $m=40$ and $n=40.$ Use $\alpha =0.01.$

A study was conducted among randomly selected families who are living in two locations of a district, and parents were asked “Whether watching TV programs by parents affects the studies of their children?” Details are presented here under:

Test, at $5\%$ level of significance, whether the difference between the proportions of families in the two localities agreed the statement.

Preference of school students, who participate in Sports events, to do physical exercises in Modern Gymnasium rather than doing aerobic exercises was analyzed. The number of students randomly selected from two States and their preference for Modern Gymnasium are given below.

Test whether the difference between proportions of school students who prefer Modern Gymnasium to do their exercises in the two States is significant at $5\%$ level of significance.

 Formulate the null and alternative hypotheses for testing whether the average time required to $XI$ standard students to complete a Chemistry laboratory exercise is less than $30 \text{minutes}$

Represent the union of two sets by Venn diagram for each of the following.

$X=\{x | x$ is a prime number between $80$ and $100\}$

$Y=\{y | y$ is an odd number between $90$ and $100\}$

In test for two population proportions, if $m=100$, $n=150$, $m{p}_X=78$, $n{p}_Y=100$ and then calculate the value of the test statistic under $H_0:P_X=P_Y$.

A study was conducted to compare the performance of athletes of two States in InterState Athlete Meets. Details of the number of successes achieved by the athletes of the two States are given here under:

Does the above information ensure at $1\%$ level of significance that the difference between the performances of the athletes of the two States is significant?