Two-sample t-test equal variance

A random sample of 8 elephants from region \(A\) is taken and their weights, \(x\) tonnes, are recorded. (1 tonne = 1000 kg.) The results are summarised as follows. $$\Sigma x = 32.4 \quad \Sigma x^2 = 131.82$$ A random sample of 10 elephants from region \(B\) is taken. Their weights give a sample mean of 3.78 tonnes and an unbiased variance estimate of 0.1555 tonnes\(^2\). The distributions of the weights of elephants in regions \(A\) and \(B\) are both assumed to be normal with the same population variance. Test at the 10% significance level whether the mean weight of elephants in region \(A\) is the same as the mean weight of elephants in region \(B\). [9]

Two methods of extracting juice from an orange are to be compared. Eight oranges are halved. One half of each orange is chosen at random and allocated to Method \(A\) and the other half is allocated to Method \(B\). The amounts of juice extracted, in ml, are given in the table. \includegraphics{figure_7} One statistician suggests performing a two-sample \(t\)-test to investigate whether or not there is a difference between the mean amounts of juice extracted by the two methods.

1. Stating your hypotheses clearly and using a 5\% significance level, carry out this test. (You may assume \(\bar{x}_A = 26.125\), \(s_A^2 = 7.84\), \(\bar{x}_B = 25\), \(s_B^2 = 4\) and \(\sigma_A^2 = \sigma_B^2\).) [7]

Another statistician suggests analysing these data using a paired \(t\)-test.

1. [(b)] Using a 5\% significance level, carry out this test. [9]
2. State which of these two tests you consider to be more appropriate. Give a reason for your choice. [1]

A teacher wishes to test whether playing background music enables students to complete a task more quickly. The same task was completed by 15 students, divided at random into two groups. The first group had background music playing during the task and the second group had no background music playing. The times taken, in minutes, to complete the task are summarised below.

	Sample size \(n\)	Standard deviation \(s\)	Mean \(\bar{x}\)
With background music	8	4.1	15.9
Without background music	7	5.2	17.9

You may assume that the times taken to complete the task by the students are two independent random samples from normal distributions.

1. Stating your hypotheses clearly, test, at the 10\% level of significance, whether or not the variances of the times taken to complete the task with and without background music are equal. [5]
2. Find a 99\% confidence interval for the difference in the mean times taken to complete the task with and without background music. [7]

Experiments like this are often performed using the same people in each group.

1. Explain why this would not be appropriate in this case. [1]

The times, \(x\) seconds, taken by the competitors in the 100 m freestyle events at a school swimming gala are recorded. The following statistics are obtained from the data.

	No. of competitors	Sample Mean \(\overline{x}\)	\(\Sigma x^2\)
Girls	8	83.10	55746
Boys	7	88.90	56130

Following the gala a proud parent claims that girls are faster swimmers than boys. Assuming that the times taken by the competitors are two independent random samples from normal distributions,

1. test, at the 10% level of significance, whether or not the variances of the two distributions are the same. State your hypotheses clearly. [7]
2. Stating your hypotheses clearly, test the parent's claim. Use a 5% level of significance. [6]

Two methods of extracting juice from an orange are to be compared. Eight oranges are halved. One half of each orange is chosen at random and allocated to Method \(A\) and the other half is allocated to Method \(B\). The amounts of juice extracted, in ml, are given in the table. One statistician suggests performing a two-sample \(t\)-test to investigate whether or not there is a difference between the mean amounts of juice extracted by the two methods.

1. Stating your hypotheses clearly and using a 5\% significance level, carry out this test. (You may assume \(\bar{x}_A = 26.125\), \(s_A^2 = 7.84\), \(\bar{x}_B = 25\), \(s_B^2 = 4\) and \(\sigma_A^2 = \sigma_B^2\)) [7]

Another statistician suggests analysing these data using a paired \(t\)-test.

1. Using a 5\% significance level, carry out this test. [9]
2. State which of these two tests you consider to be more appropriate. Give a reason for your choice. [1]

A biologist investigating the shell size of turtles takes random samples of adult female and adult male turtles and records the length, \(x\) cm, of the shell. The results are summarised below.

	Number in sample	Sample mean \(\bar{x}\)	\(\sum x^2\)
Female	6	19.6	2308.01
Male	12	13.7	2262.57

You may assume that the samples come from independent normal distributions with the same variance. The biologist claims that the mean shell length of adult female turtles is 5 cm longer than the mean shell length of adult male turtles.

1. Test the biologist's claim at the 5\% level of significance. [10]
2. Given that the true values for the variance of the population of adult male turtles and adult female turtles are both 0.9 cm\(^2\),
   1. show that when samples of size 6 and 12 are used with a 5\% level of significance, the biologist's claim will be accepted if \(4.07 < \bar{X}_F - \bar{X}_M < 5.93\) where \(\bar{X}_F\) and \(\bar{X}_M\) are the mean shell lengths of females and males respectively.
   2. Hence find the probability of a type II error for this test if in fact the true mean shell length of adult female turtles is 6 cm more than the mean shell length of adult male turtles. [6]

A new species of animal has been found on an uninhabited island. A zoologist wishes to investigate whether or not there is a difference in the mean weights of males and females of the species. She traps some of the animals and weighs them with the following results. \begin{align} \text{Males (kg)} &\quad 5.3, 4.6, 5.2, 4.5, 4.3, 5.5, 5.0, 4.8
\text{Females (kg)} &\quad 4.9, 5.0, 4.1, 4.6, 4.3, 5.3, 4.2, 4.5, 4.8, 4.9 \end{align} You may assume that these are random samples from normal populations with a common standard deviation of 0.5 kg.

1. State suitable hypotheses for this investigation. [1]
2. Determine the \(p\)-value of these results and state your conclusion in context. [9]