t-tests

A researcher set up a trial to assess the effect that a food supplement has on the increase in weight of Herdwick lambs. The researcher randomly selected 8 sets of twin lambs. One of each set of twins was given the food supplement and the other had no food supplement. The gain in weight, in kg, of each lamb over the period of the trial was recorded.

State why a two sample t-test is not suitable for use with these data.

Suggest 2 other factors about the lambs that the researcher may need to control when selecting the sample.

State one assumption, in context, that needs to be made for a paired t-test to be valid.

For a pair of twin lambs, the random variable W represents the weight gain of the lamb given the food supplement minus the weight gain of the lamb not given the food supplement.

Using the data in the table, calculate a 98% confidence interval for the mean of W Show your working clearly.

The researcher believes that the mean of W is greater than 200 g

Stating your hypotheses clearly, use your confidence interval to explain whether or not there is evidence to support the researcher’s belief.

A psychologist claims to have developed a technique to improve a person’s memory.

A random sample of 8 people are each given the same list of words to memorise and recall.

Each person then receives memory training from the psychologist. After the training, each person is given the same list of new words to memorise and recall.

The table shows the percentage of words recalled by each person before and after the training.

State why a paired t-test is suitable for these data.

State an assumption that needs to be made in order to carry out a paired t-test in this case.

Test, at the 5% level of significance, whether or not there is evidence of an increase in the percentage of words recalled after receiving the psychologist’s training. State your hypotheses, test statistic and critical value used for this test.

A doctor believes that a four-week exercise programme can reduce the resting heart rate of her patients. She takes a random sample of 7 patients and records their resting heart rate before the exercise programme and again after the exercise programme.

Using a 5% level of significance, carry out an appropriate test of the doctor’s belief. You should state your hypotheses, test statistic and critical value.

State the assumption made about the resting heart rates that was required to carry out the test.

A machine is set to fill pots with yoghurt such that the mean weight of yoghurt in a pot is 505 grams.

To check that the machine is working properly, a random sample of 8 pots is selected. The weight of yoghurt, in grams, in each pot is as follows

508 510 500 500 498 503 508 505

Given that the weights of the yoghurt delivered by the machine follow a normal distribution with standard deviation 5.4 grams,

find a 95% confidence interval for the mean weight, grams, of yoghurt in a pot. Give your answers to 2 decimal places.

Comment on whether or not the machine is working properly, giving a reason for your answer.

State the probability that a 95% confidence interval for will not contain grams.

Without carrying out any further calculations, explain the changes, if any, that would need to be made in calculating the confidence interval in part (a) if the standard deviation was unknown. Give a reason for your answer. You may assume that the weights of the yoghurt delivered by the machine still follow a normal distribution.

Alexa believes that students are equally likely to achieve the same percentage score on each of two tests, paper I and paper II. She randomly selects 8 students and gives them each paper I and paper II. The percentage scores for each paper are recorded.

The following paired data are collected.

Test, at the 1% significance level, whether or not there is evidence to support Alexa’s belief. State your hypotheses clearly and show your working.

A company manufactures bolts. The diameter of the bolts follows a normal distribution with a mean diameter of 5 mm.

Stan believes that the mean diameter of the bolts is less than 5 mm. He takes a random sample of 10 bolts and measures their diameters. He calculates some statistics but spills ink on his work before completing them. The only information he has left is as follows

99% confidence interval for the variance is = (0.01712, 0.23280)

Stating your hypotheses clearly, test, at the 5% level of significance, whether or not Stan’s belief is supported.