Comment on claim using CI

6 An anthropologist was studying the inhabitants of two islands, Raloa and Tangi. Part of the study involved the incidence of blood group type A. The blood of 80 randomly chosen inhabitants of Raloa and 85 randomly chosen inhabitants of Tangi was tested. The number of inhabitants with type A blood was 28 for the Raloa sample and 46 for the Tangi sample. The anthropologist calculated \(90 \%\) confidence intervals for the population proportions of inhabitants with type A blood. They were \(( 0.262,0.438 )\) for Raloa and \(( 0.452,0.630 )\) for Tangi, where each figure is correct to 3 decimal places. It is known that \(43 \%\) of the world's population have type A blood.

1. State, giving your reasons, whether there is evidence for the following assertions about the proportions of people with type A blood.
   1. The proportion in Raloa is different from the world proportion.
   2. The proportion in Tangi is different from the world proportion.
   3. Carry out a suitable test, at the \(2 \%\) significance level, of whether the proportions of people with type A blood differ on the two islands.

8 Two groups of Year 12 pupils, one at each of schools \(A\) and \(B\), are given the same mathematics test. The scores, \(x\) and \(y\), of pupils at schools \(A\) and \(B\) respectively are summarised as follows.

1. Assuming that the two groups are random samples from independent normal populations with means \(\mu _ { A }\) and \(\mu _ { B }\) respectively and a common, but unknown, variance, construct a \(98 \%\) confidence interval for \(\mu _ { A } - \mu _ { B }\).
2. Comment, with a reason, on any difference in ability between the two schools.

3 Small amounts of a potentially hazardous chemical are discharged into a river from a nearby industrial site. A random sample of size 6 was taken from the river and the concentration of the chemical present in each item was measured in grams per litre. The results are shown below. $$\begin{array} { l l l l l l } 1.64 & 1.53 & 1.78 & 1.60 & 1.73 & 1.77 \end{array}$$

1. Assuming that the sample was taken from a normal distribution with known variance 0.01 , construct a \(99 \%\) confidence interval for the mean concentration of the chemical present in the river.
2. If instead the sample was taken from a normal distribution, but with unknown variance, construct a revised \(99 \%\) confidence interval for the mean concentration of the chemical present in the river.
3. If the mean concentration of the chemical in the river exceeds 1.8 grams per litre, then remedial action needs to be taken. Comment briefly on the need for remedial action in the light of the results in parts (i) and (ii).

5 A random sample of 12 seventeen-year-old boys and a random sample of 14 seventeen-year-old girls were given a certain task. The times, \(t\) minutes, taken to complete the task by the members of the two samples are summarised as follows.

1. Stating any necessary assumption(s), find a \(95 \%\) symmetric confidence interval for the difference in the average times taken to complete the task by seventeen-year-old boys and seventeen-year-old girls.
2. State with a reason whether the confidence interval calculated in part (i) suggests that there may in fact be no difference in the average times taken by seventeen-year-old boys and by seventeen-year-old girls.

8 Specimens of rain were collected at random from the north and south sides of an island and analysed for sulphur content. The results (in suitable units) are given below. Assume that the sulphur contents have normal distributions with population means \(\mu _ { N }\) and \(\mu _ { S }\) and a common, but unknown, variance.

1. Calculate a symmetric \(95 \%\) confidence interval for the difference in population mean sulphur contents of the rain on the north and south sides of the island, \(\mu _ { S } - \mu _ { N }\).
2. Comment on a claim that the mean sulphur content is the same on both sides of the island.

The heights of a random sample of 10 imported orchids are measured. The mean height of the sample is found to be 20.1 cm. The heights of the orchids are normally distributed. Given that the population standard deviation is 0.5 cm,

1. estimate limits between which 95\% of the heights of the orchids lie, [3]
2. find a 98\% confidence interval for the mean height of the orchids. [4]

A grower claims that the mean height of this type of orchid is 19.5 cm.

1. Comment on the grower's claim. Give a reason for your answer. [2]

A restaurant states that its hamburgers contain 20\% fat. Paul claims that the mean fat content of their hamburgers is less than 20\%. Paul takes a random sample of 50 hamburgers from the restaurant and finds that they contain a mean fat content of 19.5\% with a standard deviation of 1.5\% You may assume that the fat content of hamburgers is normally distributed.

1. Find the 90\% confidence interval for the mean fat content of hamburgers from the restaurant. (4)
2. State, with a reason, what action Paul should recommend the restaurant takes over the stated fat content of their hamburgers. (2) The restaurant changes the mean fat content of their hamburgers to \(\mu\)\% and adjusts the standard deviation to 2\%. Paul takes a sample of size \(n\) from this new batch of hamburgers. He uses the sample mean \(\bar{X}\) as an estimator of \(\mu\).
3. Find the minimum value of \(n\) such that \(\mathrm{P}(|\bar{X} - \mu| < 0.5) \geq 0.9\) (5)

Observations have been made over many years of \(T\), the noon temperature in °C, on 21st March at Sunnymere. The records for a random sample of 12 years are given below. 5.2, 3.1, 10.6, 12.4, 4.6, 8.7, 2.5, 15.3, \(-1.5\), 1.8, 13.2, 9.3.

1. Find unbiased estimates of the mean and variance of \(T\). [5]

Over the years, the standard deviation of \(T\) has been found to be 5.1.

1. Assuming a normal distribution find a 90\% confidence interval for the mean of \(T\). [5]

A meteorologist claims that the mean temperature at noon in Sunnymere on 21st March is 4 °C.

1. Use your interval to comment on the meteorologist's claim. [2]