Introduction to Hypothesis Testing

The acceptance region is other than the 

The acceptance region is basically complement of the 

How do you know if a $p$-value is statistically significant?

Explain $p$ - value in hypothesis testing.

It is claimed that $80\%$ of Americans believe in horoscopes. Madison doubts the claim is correct. Madison conducts a survey and asks $160$ Americans if they believe in horoscopes. Define the distribution. Using a suitable approximating distribution, write down two hypotheses and test the claim given that $116$ of the people asked believe in horoscopes. Take $5\%$ significance level.

To test the claim that a coin is biased towards tails, it is flipped nine times. Tails appears seven times. Test at the $5\%$ significance level whether the claim is justified.

A farmer finds that $30\%$ of his sheep are deficient in a particular mineral. He changes their feed and tests $80$ sheep to find out if the number has decreased.

Using a suitable approximating distribution, carry out a hypothesis test given that $19$ of the sheep are mineral deficient. Take $10\%$ significance level.

A claim is that $40\%$ of professional footballers cannot explain the offside rule. Alberto thinks the percentage is lower. To test this, he asks $15$ professional footballers to explain the offside rule. Carry out a hypothesis test given that two of the professional footballers asked cannot explain the rule. Take $5\%$ significance level.

A television channel claims that $25\%$ of its programmes are nature programmes. Narobi thinks the percentage claimed is too high. To test her hypothesis, she chooses $20$ programmes at random. If there are only two nature programmes in the chosen programmes, calculate the test statistic. What conclusion does Narobi reach if the significance level is $10\%$.

State the null and alternative hypotheses for the following tests.
The population mean differs from $41.$

What is meant by critical value in general procedure for testing of hypotheses.

What is the critical value at $\alpha =0.01$ for testing $H_0:\mu ={\mu }_0$ Against $H_1:\mu <{\mu }_0$.

The buttons for these shirts are produced by a company that claims only $1\%$ of the buttons it produces are defective. 

From a randomly selected batch of $120$ of these buttons, three are found to be defective. Test the claim at the $2\%$ significance level.

In Europe the diameters of women's rings have mean $18.5 \mathrm{mm}$. Researchers claim that women in Jakarta have smaller fingers than women in Europe. The researchers took a random sample of $20$ women in Jakarta and measured the diameters of their rings. The mean diameter was found to be $18.1 \mathrm{mm}$. Assuming that the diameters of women's rings in Jakarta have a normal distribution with standard deviation $1.1 \mathrm{mm}$, carry out hypothesis test at the $2.5\%$ level to determine whether the researchers' claim is justified.

Jeevan thinks that a six-sided die is biased in favour of six. To test this, Jeevan throws the die $10$ times. If the die shows a six on at least $4$ throws out of $10$, she will conclude that she is correct.

State appropriate null and alternative hypotheses.

An engineering test consists of $100$ multiple-choice questions. Each question has five suggested answers, only one of which is correct. Ashok knows nothing about engineering, but he claims that his general knowledge enables him to get more questions correct than just by guessing. Ashok actually gets $27$ answers correct. Use a suitable approximating distribution to test at the $5\%$ significance level whether his claim is justified.

A cereal manufacturer claims that $25\%$ of cereal packets contain a free gift. Lola suspects that the true proportion is less than $25\%$ To test the manufacturer's claim at the $5\%$ significance level, she checks a random sample of $20$ packets.

State, with a reason, the conclusion she should draw.

A cereal manufacturer claims that $25\%$ of cereal packets contain a free gift. Lola suspects that the true proportion is less than $25\%$ To test the manufacturer's claim at the $5\%$ significance level, she checks a random sample of $20$ packets.

Find the critical region for the test.

A machine is designed to generate random digits between $1$ and $5$ inclusive. Each digit is supposed to appear with the same probability as the others, but Max claims that the digit $5$ is appearing less often than it should. To test this claim the manufacturer uses the machine to generate $25$ digits and finds that exactly $1$ of these digits is a $5$.

Max carried out a similar hypothesis test by generating $1000$ digits between $1$ and $5$ inclusive. The digit $5$ appeared $180$ times. Without carrying out the test, state the distribution that Max should use, including the values of any parameters.

A machine is designed to generate random digits between $1$ and $5$ inclusive. Each digit is supposed to appear with the same probability as the others, but Max claims that the digit $5$ is appearing less often than it should. To test this claim the manufacturer uses the machine to generate $25$ digits and finds that exactly $1$ of these digits is a $5$.

Carry out a test of Max's claim at the $2.5\%$ significance level.