Tailed Tests and Errors in Hypothesis Testing

____ is a method in which the critical area of a distribution is two sided

A two-tailed test, in statistics, is a method in which the critical area of a distribution is

In one-tailed test the sample mean is

A one-tailed test is a test set up to show that the sample mean be

What is the difference between Type I and Type II Errors in hypothesis testing.

Type II error in hypothesis testing is:

 What is level of significance with resect to error type

Rejecting $H_0$ when it is true falls under which error type.

Every month Susan enters a particular lottery. The lottery company states that the probability, $p$, of winning a prize is $0.0017$ each month. Susan thinks that the probability of winning is higher than this, and carries out a test based on her $12$ lottery results in a one-year period. She accepts null hypothesis $p=0.0017$ if she has no wins in the year and accepts the alternative hypothesis $p>0.0017$ if she wins a prize in at least one of the $12$ months.

If in fact the probability of winning a prize each month is $0.0024$, find the probability of the test resulting in a Type $\mathrm{II}$ error 

Every month Susan enters a particular lottery. The lottery company states that the probability, $p$, of winning a prize is $0.0017$ each month. Susan thinks that the probability of winning is higher than this, and carries out a test based on her $12$ lottery results in a one-year period. She accepts null hypothesis $p=0.0017$ if she has no wins in the year and accepts the alternative hypothesis $p>0.0017$ if she wins a prize in at least one of the $12$ months.

Find the probability of the test resulting in a Type $\mathrm{I}$ error.

At the last election $70\%$ of people in the Apoli supported the president. Luigi believes that the same proportion support the president now. Maria believes that the proportion who support the president now is $35\%$. In order to test who is right, they agree on a hypothesis test, taking Luigi's belief as the null hypothesis. They will ask $6$ people from Apoli, chosen at random, and if more than $3$ support the president they will accept Luigi's belief.

In fact $2$ of the $6$ people say that they support the president. State which error, Type $\mathrm{I}$ or Type $\mathrm{II}$. Might be made. Explain your answer. (Take $10\%$ significance value)

At the last election $70\%$ of people in the Apoli supported the president. Luigi believes that the same proportion support the president now. Maria believes that the proportion who support the president now is $35\%$. In order to test who is right, they agree on a hypothesis test, taking Luigi's belief as the null hypothesis. They will ask $6$ people from Apoli, chosen at random, and if more than $3$ support the president they will accept Luigi's belief.

If Maria's belief is true, calculate the probability of a Type $\mathrm{II}$ error.

At the last election $70\%$ of people in the Apoli supported the president. Luigi believes that the same proportion support the president now. Maria believes that the proportion who support the president now is $35\%$. In order to test who is right, they agree on a hypothesis test, taking Luigi's belief as the null hypothesis. They will ask $6$ people from Apoli, chosen at random, and if more than $3$ support the president they will accept Luigi's belief.

Calculate the probability of a Type $\mathrm{I}$ error.

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.

Explain what is meant by a Type II error in this situation.

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.

If the die is actually biased so that the probability of throwing a six is $\frac{1}{2}$, calculate the probability of a Type II error.

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.

Calculate the probability of a Type I error.

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.

Hence, find the probability of a Type I error.

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 $5$.

State what is meant by a Type II error in this context.

Marie claims that she can predict the winning horse at the local races. There are eight horses in each race. Nadine thinks that Marie is just guessing, so she proposes a test. She asks Marie to predict the winners of the next ten races and, if she is correct in three or more, Nadine will accept Marie's claim.

State the significance level of the test.

Marie claims that she can predict the winning horse at the local races. There are eight horses in each race. Nadine thinks that Marie is just guessing, so she proposes a test. She asks Marie to predict the winners of the next ten races and, if she is correct in three or more. Nadine will accept Marie's claim.

Calculate the probability of a Type I error.