7. The weights of tubs of margarine are known to be normally distributed. A random sample of 10 tubs of margarine were weighed, to the nearest gram, and the results were as follows.
$$\begin{array} { l l l l l l l l l l }
498 & 502 & 500 & 496 & 509 & 504 & 511 & 497 & 506 & 499
\end{array}$$
- Find unbiased estimates of the mean and the variance of the population from which this sample was taken.
Given that the population standard deviation is 5.0 g ,
- estimate limits, to 2 decimal places, between which \(90 \%\) of the weights of the tubs lie,
- find a \(95 \%\) confidence interval for the mean weight of the tubs.
A second random sample of 15 tubs was found to have a mean weight of 501.9 g .
- Stating your hypotheses clearly and using a \(1 \%\) level of significance, test whether or not the mean weight of these tubs is greater than 500 g .
\section*{END}
\section*{Items included with question papers Nil}
Answer Book (AB16)
Graph Paper (ASG2)
Mathematical Formulae (Lilac)
Candidates may use any calculator EXCEPT those with the facility for symbolic algebra, differentiation and/or integration. Thus candidates may NOT use calculators such as the Texas Instruments TI 89, TI 92, Casio CFX 9970G, Hewlett Packard HP 48G.
Paper Reference(s)
6685
\section*{Edexcel GCE
Statistics S3}
Advanced/Advanced Subsidiary
Thursday 5 June 2003 - Morning
Time: \(\mathbf { 1 }\) hour \(\mathbf { 3 0 }\) minutes
In the boxes on the answer book, write the name of the examining body (Edexcel), your centre number, candidate number, the unit title (Statistics S3), the paper reference (6685), your surname, other name and signature.
Values from the statistical tables should be quoted in full. When a calculator is used, the answer should be given to an appropriate degree of accuracy.
A booklet 'Mathematical Formulae and Statistical Tables' is provided.
Full marks may be obtained for answers to ALL questions.
This paper has seven questions.
You must ensure that your answers to parts of questions are clearly labelled.
You must show sufficient working to make your methods clear to the Examiner. Answers without working may gain no credit.
- Explain how to obtain a sample from a population using
- stratified sampling,
- quota sampling.
Give one advantage and one disadvantage of each sampling method.
2. A random sample of 30 apples was taken from a batch. The mean weight of the sample was 124 g with standard deviation 20 g . - Find a \(99 \%\) confidence interval for the mean weight \(\mu\) grams of the population of apples. Write down any assumptions you made in your calculations.
Given that the actual value of \(\mu\) is 140 ,
- state, with a reason, what you can conclude about the sample of 30 apples.
3. Given the random variables \(X \sim \mathrm {~N} ( 20,5 )\) and \(Y \sim \mathrm {~N} ( 10,4 )\) where \(X\) and \(Y\) are independent, find - \(\mathrm { E } ( X - Y )\),
- \(\operatorname { Var } ( X - Y )\),
- \(\mathrm { P } ( 13 < X - Y < 16 )\).
4. A new drug to treat the common cold was used with a randomly selected group of 100 volunteers. Each was given the drug and their health was monitored to see if they caught a cold. A randomly selected control group of 100 volunteers was treated with a dummy pill. The results are shown in the table below. - Write down a suitable model for \(X\).
- Test, at the \(1 \%\) level of significance, the suitability of your model for these data.
- Explain how the test would have been modified if it had not been assumed that the dice were fair.
7. The random variable \(D\) is defined as
$$D = A - 3 B + 4 C$$
where \(A \sim \mathrm {~N} \left( 5,2 ^ { 2 } \right) , B \sim \mathrm {~N} \left( 7,3 ^ { 2 } \right)\) and \(C \sim \mathrm {~N} \left( 9,4 ^ { 2 } \right)\), and \(A , B\) and \(C\) are independent. - Find \(\mathrm { P } ( \mathrm { D } < 44 )\).
The random variables \(B _ { 1 } , B _ { 2 }\) and \(B _ { 3 }\) are independent and each has the same distribution as \(B\). The random variable \(X\) is defined as
$$X = A - \sum _ { i = 1 } ^ { 3 } B _ { i } + 4 C$$
- Find \(\mathrm { P } ( X > 0 )\).
\section*{END}
\section*{6685/01 6691/01
Edexcel GCE}
\section*{Thursday 9 June 2005 - Morning}
Materials required for examination
Mathematical Formulae (Lilac)
Graph Paper (ASG2)
Candidates may use any calculator EXCEPT those with the facility for symbolic algebra, differentiation and/or integration. Thus candidates may NOT use calculators such as the Texas Instruments TI 89, TI 92, Casio CFX 9970G, Hewlett Packard HP 48G.
In the boxes on the answer book, write the name of the examining body (Edexcel), your centre number, candidate number, the unit title (Statistics S3), the paper reference (6685), your surname, other name and signature.
Values from the statistical tables should be quoted in full. When a calculator is used, the answer should be given to an appropriate degree of accuracy.
A booklet 'Mathematical Formulae and Statistical Tables' is provided.
Full marks may be obtained for answers to ALL questions.
This paper has seven questions.
The total mark for this paper is 75 .
Items included with question papers
Nil
Nil
You must ensure that your answers to parts of questions are clearly labelled.
You must show sufficient working to make your methods clear to the Examiner. Answers without working may gain no credit.
- A researcher carried out a survey of three treatments for a fruit tree disease. The contingency table below shows the results of a survey of a random sample of 60 diseased trees.
Using a \(5 \%\) significance level, test whether or not there is an association between gender and acceptance or rejection of an annual flu injection. State your hypotheses clearly.
5. Upon entering a school, a random sample of eight girls and an independent random sample of eighty boys were given the same examination in mathematics. The girls and boys were then taught in separate classes. After one year, they were all given another common examination in mathematics.
The means and standard deviations of the boys' and the girls' marks are shown in the table. - 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.
- 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. - Find the mean and variance of the total weight of the 15 people that enter the lift.
- Comment on any relationship you have assumed in part (a) between the two samples.
- Find the probability that the maximum load of the lift will be exceeded by the total weight of the 15 people.
6. A research worker studying colour preference and the age of a random sample of 50 children obtained the results shown below.
| Age in years | Red | Blue | Totals |
| 4 | 12 | 6 | 18 |
| 8 | 10 | 7 | 17 |
| 12 | 6 | 9 | 15 |
| Totals | 28 | 22 | 50 |
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
$$\begin{array} { l l l l l }
49.7 , & 50.3 , & 51.0 , & 49.5 , & 49.9
50.1 , & 50.2 , & 50.0 , & 49.6 , & 49.7 .
\end{array}$$ - 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 .
- Estimate the limits between which \(95 \%\) of the weights of metal containers lie.
- Determine the \(99 \%\) confidence interval for the mean weight of metal containers.