Edexcel S3 (Statistics 3) 2006 June

\begin{enumerate} \item Describe one advantage and one disadvantage of

1. quota sampling,
2. simple random sampling. \item A report on the health and nutrition of a population stated that the mean height of three-year old children is 90 cm and the standard deviation is 5 cm . A sample of 100 three-year old children was chosen from the population.

1. Write down the approximate distribution of the sample mean height. Give a reason for your answer.
2. Hence find the probability that the sample mean height is at least 91 cm . \item A biologist investigated whether or not the diet of chickens influenced the amount of cholesterol in their eggs. The cholesterol content of 70 eggs selected at random from chickens fed diet \(A\) had a mean value of 198 mg and a standard deviation of 47 mg . A random sample of 90 eggs from chickens fed diet \(B\) had a mean cholesterol content of 201 mg and a standard deviation of 23 mg .
3. Stating your hypotheses clearly and using a \(5 \%\) level of significance, test whether or not there is a difference between the mean cholesterol content of eggs laid by chickens fed on these two diets.
4. State, in the context of this question, an assumption you have made in carrying out the test in part (a). \item The table below shows the price of an ice cream and the distance of the shop where it was purchased from a particular tourist attraction. \end{enumerate}

   Shop	Distance from tourist attraction (m)	Price (£)
   A	50	1.75
   B	175	1.20
   C	270	2.00
   D	375	1.05
   E	425	0.95
   F	580	1.25
   G	710	0.80
   \(H\)	790	0.75
   I	890	1.00
   J	980	0.85

5. Find, to 3 decimal places, the Spearman rank correlation coefficient between the distance of the shop from the tourist attraction and the price of an ice cream.
6. Stating your hypotheses clearly and using a \(5 \%\) one-tailed test, interpret your rank correlation coefficient.

5. The workers in a large office block use a lift that can carry a maximum load of 1090 kg . The weights of the male workers are normally distributed with mean 78.5 kg and standard deviation 12.6 kg . The weights of the female workers are normally distributed with mean 62.0 kg and standard deviation 9.8 kg . Random samples of 7 males and 8 females can enter the lift.

1. Find the mean and variance of the total weight of the 15 people that enter the lift.
2. Comment on any relationship you have assumed in part (a) between the two samples.
3. Find the probability that the maximum load of the lift will be exceeded by the total weight of the 15 people.
   (4)

6. A research worker studying colour preference and the age of a random sample of 50 children obtained the results shown below. Using a \(5 \%\) significance level, carry out a test to decide whether or not there is an association between age and colour preference. State your hypotheses clearly.

7. A machine produces metal containers. The weights of the containers are normally distributed. A random sample of 10 containers from the production line was weighed, to the nearest 0.1 kg , and gave the following results $$\begin{array} { l l l l l } 49.7 , & 50.3 , & 51.0 , & 49.5 , & 49.9
50.1 , & 50.2 , & 50.0 , & 49.6 , & 49.7 . \end{array}$$

1. Find unbiased estimates of the mean and variance of the weights of the population of metal containers. The machine is set to produce metal containers whose weights have a population standard deviation of 0.5 kg .
2. Estimate the limits between which \(95 \%\) of the weights of metal containers lie.
3. Determine the \(99 \%\) confidence interval for the mean weight of metal containers.

8. Five coins were tossed 100 times and the number of heads recorded. The results are shown in the table below.

1. Suggest a suitable distribution to model the number of heads when five unbiased coins are tossed.
2. Test, at the \(10 \%\) level of significance, whether or not the five coins are unbiased. State your hypotheses clearly.