Perform an appropriate test at the $5\%$ significance level to see if the mean marks achieved by the students in the school are higher than the national standard. It can be assumed that the marks come from a normal population.

Find the $p$-value for the test.

Write down the degrees of freedom.

Write down the expected frequency of rolling a $1$.

State the alternative hypothesis.

Calculate the $p$-value for this test.

State the alternative hypothesis.

Given that all assumptions for this test are satisfied, carry out an appropriate hypothesis test. State and justify your conclusion. Use a $5\%$ significance level.

Stating null and alternative hypotheses, carry out an appropriate test at the 5 % level to decide whether the farmer’s claim can be justified.

Find the p-value for this t-test.

State an assumption that the company is making, in order to use a t-test.

The new drug.

Calculate the number of volunteers in the sample under the age of 30.

Find unbiased estimates for the population Variance.

Give two reasons why the prediction in part (b)(ii) might be lower than 14.

$c$.

the number of new people infected on day 6.

Find the probability that a basketball player has a weight that is less than 61 kg.

Sketch a normal curve to represent this probability.

Write down the number of degrees of freedom for this test.

For this test, write down the ${\chi }^2$ statistic.

Find unbiased estimates for the population mean.

Write down the null hypothesis, H₀ , for this test.

Given that this flight was not heavily delayed, find the probability that it travelled between 500 km and 5000 km.

Use an exponential regression to find the value of $a$ and of $b$, correct to 4 decimal places.

Six coins are tossed simultaneously 320 times, with the following results.

At the 5% level of significance, test the hypothesis that all the coins are fair.

Perform the ${\chi }^2$ goodness of fit test and state, with reason, a conclusion.

State what your conclusion means in context.

Find the expected frequency for each of the numbers if the manufacturer’s claim is true.

State the hypotheses for this t-test.

State the conclusion for this test. Give a reason for your answer.

Test the hypothesis at the 5% level of significance.

State the conclusion of the test. Justify your answer.

Write down the number of degrees of freedom for her test.

By finding a critical value, perform this test at a $5 \%$ significance level.

Write down the null and alternative hypotheses.

State the conclusion of the test. Give a reason for your answer.

State one reason why the test might not be valid.

State the null and alternative hypotheses.

Perform a suitable test, at the 5% significance level, to determine if it is easier to achieve a distinction on the new exam. You should clearly state your hypotheses, the critical region and your conclusion.

a Type II error.

In order to test for the goodness of fit, the test statistic was calculated to be 1.0847. Show how this was done.

Find the p-value for this t-test.

Write down the associated p-value.

Write down, for $N$, the mid-interval value of the modal class.

Write down the number of degrees of freedom for this test.

Find the value of q.

State the null hypothesis, H₀, for this test.

Calculate the $p$-value for this test.

Use the normal distribution model to find the score required to pass.

The current drug.

the day when the total number of people infected will be greater than 1000.

$L$.

State an assumption that Anita is making, in order to use a t-test.

$k$.

Given that P(W > k) = 0.225 , find the value of k.

Write down the total number of people, from this group, who are pet owners.

Perform a suitable test, at the 5% significance level, to determine if the scores follow a normal distribution, with the mean and variance found in part (a). You should clearly state your hypotheses, the degrees of freedom, the p-value and your conclusion.

Write down the modal number of pets.

A different horse breeder collected data on the time and outcome of births. The data are summarized in the following table:

Carry out an appropriate test at the 5% significance level to decide whether there is an association between time and outcome.

State the conclusion of this test, in context, giving a reason.

Use your answer to part (a) to show that the model predicts 16.7 people will be infected on the first day.

the $p$-value.

State the conclusion of this test, in context, giving a reason.

Write down the χ² statistic.

State the number of degrees of freedom.

the expected frequency of female students who chose to take the Chinese class.

State whether or not H₀ should be rejected. Justify your statement.

State the alternative hypothesis.

Use your graphic display calculator to estimate the mean of $N$;

Write down the ${\chi }^2$ statistic.

Write down the value of $x$.

Write down the number of degrees of freedom for this test.

Calculate the expected number of teenagers that prefer cats.

Write down the number of degrees of freedom.

For these data, write down the upper quartile.

Write down the ratio of teenagers to non-teenagers in its simplest form.

State the null hypothesis.

Test this claim at the 5% level of significance.

State the conclusion for this test. Give a reason for your answer.

State the alternative hypothesis.

Find the standard deviation of the changes.

If you were to repeat the test performed in part (e) intending to compare the quality of the teaching between the two schools, suggest two ways in which you might choose your sample to improve the validity of the test.

State why comparing only the final IB points of the students from the two schools would not be a valid test for the effectiveness of the two different teaching methods.

Use an appropriate test to determine whether showing an improvement is independent of gender.

State why it was important to test that both sets of points were normally distributed.

State the null hypothesis.

Showing all steps clearly, test whether the die is fair

(i)   at the 5% level of significance;

(ii)  at the 1% level of significance.

Copy and complete the table, showing how you arrived at your answers.

Explain what is meant by the 5% level of significance.

Write down appropriate hypotheses.

Carry out a test at the 1% significance level, and state your conclusion.

Test, at the 5% level of significance, whether or not the data can be modelled by a Poisson distribution.

The number of cars passing a certain point in a road was recorded during 80 equal time intervals and summarized in the table below.

Carry out a ${\chi }^2$ goodness of fit test at the 5% significance level to decide if the above data can be modelled by a Poisson distribution.

Calculate the value of the ${\chi }^2$ statistic and state your conclusion using a 10% level of significance.

The data from the two samples above are combined to form a single set of data. The following frequency table gives the observed frequencies for the combined sample. The data has been divided into five intervals.

Test, at the 5% level, whether the combined data can be considered to be a sample from a normal population with a mean of 380.

Write down the number of degrees of freedom.

A horse breeder records the number of births for each of 100 horses during the past eight years. The results are summarized in the following table:

Stating null and alternative hypotheses carry out an appropriate test at the 5% significance level to decide whether the results can be modelled by B (6, 0.5).

Calculate an appropriate value of ${\chi }^2$ and state your conclusion, using a 1% significance level.

In a training session there are 40 basketball players.

Find the expected number of players with a weight less than 61 kg in this training session.

State suitable hypotheses.

State suitable hypotheses for testing this belief.

the ${\chi }_2$ statistic.

Write down the number of degrees of freedom.

Identify a test that might have been used to verify the null hypothesis that the predictions from the standardized test can be modelled by a normal distribution.

the level of significance of a hypothesis test.

Write down, for $N$, the modal class;

State ${\text{H}}_0$, the null hypothesis for this test.

State whether $N$ is a discrete or a continuous variable.

Use your graphic display calculator to estimate the standard deviation of $N$.

a Type I error.

Find the mean and standard deviation of the sample data in the table above. Show how you arrived at your answers.

Find the mean change.

Calculate the probability that the customer is an adult or that the customer chose shrimp.

Given that the customer is a child, calculate the probability that they chose pasta or fish.

If the teacher chooses a response at random, find the probability that it is a response to the Calculus question;

State the name for this type of sampling technique.

Use the logistic model to find the day when the rate of increase of people infected is greatest.

Find the probability that at least one of the two students is female.

State the alternative hypothesis.

Hence estimate $p$, the probability that a randomly chosen egg is brown.

Hence, find the expected improvement between predicted and final points for an increase of one unit in effort grades, giving your answer to one decimal place.

Perform a ${\chi }^2$ goodness of fit test at the 5% significance level. You should clearly state your hypotheses, the p-value, and your conclusion.

Use a chi-squared goodness of fit test to investigate whether or not, at the 5 % level of significance, the N(0, 1) distribution can be used to model these results.

Show that the expected frequency of satisfactory Calculus responses is 12.

State, giving a reason, whether the null hypothesis should be accepted.

Explain why the number of degrees of freedom is 2.

State the null hypothesis for this test.

Show that the expected frequency for 20 < $x$ ≤ 4 is 31.5 correct to 1 decimal place.

State the conclusion for this test. Give a reason for your answer.

Find the probability that the student does not take the Spanish class.

Show that the expected number of children who chose shrimp is $31$, correct to two significant figures.

Hence predict the total number of people infected by this disease after several months.

By calculating an appropriate ${\chi }^2$ statistic, test, at the 5% significance level, whether or not the binomial distribution gives a good fit to these data.

Find the probability of making a Type I error.

Without doing any further calculations, explain briefly how you would carry out a test, at the 5% significance level, to decide if the data can be modelled by B(6, $p$), where $p$ is unspecified.

Write down the $p$-value for the test;

Write down the null hypothesis for this test;

Calculate the $p$-value for this test.

State the result of the test. Give a reason for your answer.

Use an appropriate test to determine whether there is evidence, at the 5 % significance level, that the students in school B have improved more than those in school A.

Find the probability that neither of the two students take the Spanish class.

Calculate unbiased estimates of the population mean and the population variance.

Calculate the probability that the customer is an adult.

Find the expected frequency of students choosing the Science category and obtaining 31 to 40 correct answers.

Find the p-value for this test.

Find the expected number of short trips when it rained.

The $p$-value for this test is $0.0206$.

State the conclusion to Isaac’s test. Justify your reasoning.

The test is performed at the $5\%$ significance level.

State the conclusion of the test, giving a reason for your answer.

Find the $p$-value for the owner’s test.

Explain what is meant by “level of significance” in part (a).

Find the significance level of this procedure.

Find the expected frequency of the patients who became infected whilst in Nightingale ward.

The teacher groups the responses by topic, and chooses two responses to the Logic question. Find the probability that both are not satisfactory.

State the degrees of freedom for the test.

For this test state the null hypothesis.

State, giving a reason, whether the null hypothesis should be rejected.

Abhinav rejects H₀.

State a reason why Abhinav is incorrect in doing so.

Find the exact value of the mean of this distribution.

Some time later, the actual value of $\mu$ is 503. Find the probability of a Type II error.

State the null hypothesis.

Show that this data leads to an estimated value of $p=0.4$.

Write down ${\text{H}}_0$, the null hypothesis for this test.

State, with a reason, whether you would reject the null hypothesis.

Calculate the $p$-value for this test.

For these data, write down the median number of pets.

For these data, write down the lower quartile.

Use a paired $t$-test to determine whether there is significant evidence that the students in school A have improved their IB points since the start of the course.

Use your graphic display calculator to find the ${\chi }^2$ statistic for this test.

If the teacher chooses a response at random, find the probability that it is a satisfactory response, given that it is a response to the Calculus question.

State the conclusion of this ${\chi }^2$ test. Give a reason for your answer.

Use your graphic display calculator to find the ${\chi }^2$ statistic for this data.

State, in words, the null hypothesis.

State whether the result of the test supports Arriane’s claim. Justify your reasoning.

The scientists wish to investigate the claim that Group B gain weight faster than Group A. Test this claim at the 5% level of significance, noting which hypothesis test you are using. You may assume that the weight gain for each group is normally distributed, with the same variance, and independent from each other.

Calculate the expected frequency of flights travelling at most 500 km and arriving slightly delayed.

Calculate the mean of these data and hence estimate the value of $p$.

State the null and alternative hypotheses.

Find the $p$-value for the test.

For this test find the p-value.

Calculate the expected number of male engineers.

a goodness of fit test (a complete explanation required);

Write down the conclusion to the test. Give a reason for your answer.

Write down the probability that this flight arrived on time.

Two flights are chosen at random from those which were slightly delayed.

Find the probability that each of these flights travelled at least 5000 km.

Write down the degrees of freedom for this test.

Find an estimate for how many copies the vendor expects to sell each day.

Write down the conclusion to the test. Give a reason for your answer.

Find the $p$-value for this test.

Calculate the mean number of eggs laid by these birds.

Most of the expected frequencies have been calculated in the third column. (Frequencies have been rounded to the nearest integer, and frequencies in the first and last classes have been extended to include the rest of the data beyond 15 and 225. Find the values of $a$, $b$ and $c$ and show how you arrived at your answers.

Find an estimate for how many copies the vendor expects to sell each day.

Write down the degrees of freedom for this test.

Write down the conclusion to the test. Give a reason for your answer.

If the teacher chooses a response at random, find the probability that it is a satisfactory response to the Calculus question;

State the null and alternative hypotheses for this test.

Calculate the mean number of brown eggs in a box.

State the hypotheses for this t-test.

Write down the null and alternative hypotheses.

State a conclusion for this test. Justify your answer.

Perform a test on the data from school A to show it is reasonable to assume a linear relationship between effort scores and improvements in IB points. You may assume effort scores follow a normal distribution.

State your hypotheses, critical number, decision rule and conclusion (using a 5% level of significance).

Perform a suitable test, at the 5% significance level, to determine if there is a difference between the mean scores of males and females. You should clearly state your hypotheses, the p-value and your conclusion.

Given that $p=0.2$ find the probability of making a Type II error.

State the conclusion of the test. Give a reason for your answer.

Use an appropriate test, at the $5\%$ significance level, to determine whether a new employee staying with the firm is independent of their interview rating. State the null and alternative hypotheses, the $p$-value and the conclusion of the test.

Write down the null and alternative hypotheses.

Determine the conclusion of the test, clearly justifying your answer.

prefers a tablet.

is $11\text{–}13$ years old and prefers a mobile phone.

prefers a laptop given that they are $17\text{–}18$ years old.

prefers a tablet or is $14\text{–}16$ years old.

Write down the number of degrees of freedom.

Write down the ${\chi }^2$ test statistic.

Write down the $p$-value.

State the conclusion for the test in context. Give a reason for your answer.

Write down null and alternative hypotheses for Loreto’s test.

Using the data from Loreto’s sample, perform the hypothesis test at a $5\%$ significance level to determine if Loreto should employ extra staff.

Write down null and alternative hypotheses for this test.

Perform the test, clearly stating the conclusion in context.