Two-sample t-test with summary statistics

6. A researcher collects data on the height of boys aged between nine and nine and-a-half years and their diet. The data on the height, \(V\) cm, of the 80 boys who had always eaten a vegetarian diet is summarised by $$\Sigma V = 10367 , \quad \Sigma V ^ { 2 } = 1350314 .$$

1. Calculate unbiased estimates of the mean and variance of \(V\). The researcher calculates unbiased estimates of the mean and variance of the height of boys whose diet has included meat from a sample of size 280, giving values of 130.5 cm and \(96.24 \mathrm {~cm} ^ { 2 }\) respectively.
2. Stating your hypotheses clearly, test at the \(1 \%\) level whether or not there is a significant difference in the heights of boys of this age according to whether or not they have a vegetarian diet.
   (8 marks)

5. For a project, a student is investigating whether more athletic individuals have better hand-eye coordination. He records the time it takes a number of students to complete a task testing coordination skills and notes whether or not they play for a school sports team. His results are as follows: Stating your hypotheses clearly, test at the \(5 \%\) level of significance whether or not there is evidence that those who play in a school team complete the task more quickly on average.
(8 marks)

7. A grocer receives deliveries of cauliflowers from two different growers, \(A\) and \(B\). The grocer takes random samples of cauliflowers from those supplied by each grower. He measures the weight \(x\), in grams, of each cauliflower. The results are summarised in the table below.

1. Show, at the \(10 \%\) significance level, that the variances of the populations from which the samples are drawn can be assumed to be equal by testing the hypothesis \(\mathrm { H } _ { 0 } : \sigma _ { A } ^ { 2 } = \sigma _ { B } ^ { 2 }\) against hypothesis \(\mathrm { H } _ { 1 } : \sigma _ { A } ^ { 2 } \neq \sigma _ { B } ^ { 2 }\).
   (You may assume that the two samples come from normal populations.)
   (6) The grocer believes that the mean weight of cauliflowers provided by \(B\) is at least 150 g more than the mean weight of cauliflowers provided by \(A\).
2. Use a \(5 \%\) significance level to test the grocer's belief.
3. Justify your choice of test.

3. The lengths, \(x \mathrm {~mm}\), of the forewings of a random sample of male and female adult butterflies are measured. The following statistics are obtained from the data.

1. Assuming the lengths of the forewings are normally distributed test, at the \(10 \%\) level of significance, whether or not the variances of the two distributions are the same. State your hypotheses clearly.
2. Stating your hypotheses clearly test, at the \(5 \%\) level of significance, whether the mean length of the forewings of the female butterflies is less than the mean length of the forewings of the male butterflies.
   (6)

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.

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.
2. Find a 99\% confidence interval for the difference in the mean times taken to complete the task with and without background music. Experiments like this are often performed using the same people in each group.
3. Explain why this would not be appropriate in this case.

1. The weights of the contents of breakfast cereal boxes are normally distributed.

A manufacturer changes the style of the boxes but claims that the weight of the contents remains the same.
A random sample of 6 old style boxes had contents with the following weights (in grams). $$\begin{array} { l l l l l l } 512 & 503 & 514 & 506 & 509 & 515 \end{array}$$ The weights, \(y\) grams, of the contents of an independent random sample of 5 new style boxes gave $$\bar { y } = 504.8 \text { and } s _ { y } = 3.420$$

1. Use a two-tail test to show, at the \(10 \%\) level of significance, that the variances of the weights of the contents of the old and new style boxes can be assumed to be equal. State your hypotheses clearly.
2. Showing your working clearly, find a \(90 \%\) confidence interval for \(\mu _ { x } - \mu _ { y }\), where \(\mu _ { x }\) and \(\mu _ { y }\) are the mean weights of the contents of old and new style boxes respectively.
3. With reference to your confidence interval comment on the manufacturer's claim.

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

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.
2. Given that the true values for the variance of the population of adult male turtles and adult female turtles are both \(0.9 \mathrm {~cm} ^ { 2 }\),
   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.

3. An archaeologist is studying the compression strength of bricks at some ancient European sites. He took random samples from two sites \(A\) and \(B\) and recorded the compression strength of these bricks in appropriate units. The results are summarised below. It can be assumed that the compression strength of bricks is normally distributed.

1. Test, at the \(2 \%\) level of significance, whether or not there is evidence of a difference in the variances of compression strength of the bricks between these two sites. State your hypotheses clearly.
   (5) Site \(A\) is older than site \(B\) and the archaeologist claims that the mean compression strength of the bricks was greater at the younger site.
2. Stating your hypotheses clearly and using a \(1 \%\) level of significance, test the archaeologist's claim.
3. Explain briefly the importance of the test in part (a) to the test in part (b).

3. A farmer is investigating the milk yields of two breeds of cow. He takes a random sample of 9 cows of breed \(A\) and an independent random sample of 12 cows of breed \(B\). For a 5 day period he measures the amount of milk, \(x\) gallons, produced by each cow. The results are summarised in the table below. The amount of milk produced by each cow can be assumed to follow a normal distribution.

1. Use a two-tail test to show, at the \(10 \%\) level of significance, that the variances of the yields of the two breeds can be assumed to be equal. State your hypotheses clearly.
2. Stating your hypotheses clearly, test, at the \(5 \%\) level of significance, whether or not there is a difference in the mean yields of the two breeds of cow.
3. Explain briefly the importance of the test in part (a) for the test in part (b).

7. Two groups of students take the same examination. A random sample of students is taken from each of the groups. The marks of the 9 students from Group 1 are as follows $$\begin{array} { l l l l l l l l l } 30 & 29 & 35 & 27 & 23 & 33 & 33 & 35 & 28 \end{array}$$ The marks, \(x\), of the 7 students from Group 2 gave the following statistics $$\bar { x } = 31.29 \quad s ^ { 2 } = 12.9$$ A test is to be carried out to see whether or not there is a difference between the mean marks of the two groups of students. You may assume that the samples are taken from normally distributed populations and that they are independent.

1. State one other assumption that must be made in order to apply this test and show that this assumption is reasonable by testing it at a \(10 \%\) level of significance. State your hypotheses clearly.
2. Stating your hypotheses clearly, test, using a significance level of \(5 \%\), whether or not there is a difference between the mean marks of the two groups of students.

1. As part of their research two sports science students, Ali and Bea, select a random sample of 10 adult male swimmers and a random sample of 13 adult male athletes from local sports clubs. They measure the arm span, \(x \mathrm {~cm}\), of each person selected.
   The data are summarised in the table below

The students know that the arm spans of adult male swimmers and of adult male athletes may each be assumed to be normally distributed.
They decide to share out the data analysis, with Ali investigating the means of the two distributions and Bea investigating the variances of the two distributions. Ali assumes that the variances of the two distributions are equal. She calculates the pooled estimate of variance, \(s _ { p } { } ^ { 2 }\)

1. Show that \(s _ { p } { } ^ { 2 } = 112.6\) to 1 decimal place. Ali claims that there is no difference in the mean arm spans of adult male swimmers and of adult male athletes.
2. Stating your hypotheses clearly, test this claim at the \(10 \%\) level of significance.
   (5) Bea believes that the variances of the arm spans of adult male swimmers and adult male athletes are not equal.
3. Show that, at the \(10 \%\) level of significance, the data support Bea's belief. State your hypotheses and show your working clearly. Ali and Bea combine their work and present their results to their tutor, Clive.
4. Explain why Clive is not happy with their research and state, with a reason, which of the tests in parts (b) and (c) is not valid.

5. Fire brigades in cities \(X\) and \(Y\) are in similar locations. The response times, in minutes, during a particular month, for randomly selected calls are summarised in the table below. You may assume that the response times are from independent normal distributions.
Stating your hypotheses and showing your working clearly

1. test, at the \(10 \%\) level of significance, whether or not the variances of the populations from which the response times are drawn are the same,
   (5)
2. test, at the \(5 \%\) level of significance, whether or not the mean response time for the fire brigade in city \(X\) is more than 5 minutes longer than the mean response time for the fire brigade in city \(Y\).
3. Explain why your result in part (a) enables you to carry out the test in part (b).