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Year 2 Statistics
STATISTICS Contents
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Solution:
b) Positive correlation.
A meteorologist believes that there is a relationship between the daily mean wind speed, w kn, and the daily mean temperature, t°C. A random sample of 9 consecutive days is taken from past records from a town in the UK in July and the relevant data is given in the table below.
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CORRECT SOLUTION:
A meteorologist believes that there is a relationship between the daily mean wind speed, w kn, and the daily mean temperature, t°C. A random sample of 9 consecutive days is taken from past records from a town in the UK in July and the relevant data is given in the table below.
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Tessa owns a small clothes shop in a seaside town. She records the weekly sales figures, £ w, and the average weekly temperature, t °C, for 8 weeks during the summer.
�The product moment correlation coefficient for these data is −0.915
(a) Stating your hypotheses clearly and using a 5% level of significance, test whether or not the correlation between sales figures and average weekly temperature is negative.
(3)
(b) Suggest a possible reason for this correlation.
(1)
Tessa suggests that a linear regression model could be used to model these data.
(c) State, giving a reason, whether or not the correlation coefficient is consistent with Tessa's suggestion.
(1)
(d) State, giving a reason, which variable would be the explanatory variable.
(1)
Tessa calculated the linear regression equation as w = 10 755 – 171t
(e) Give an interpretation of the gradient of this regression equation.
(1)
Question:
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Tessa owns a small clothes shop in a seaside town. She records the weekly sales figures, £ w, and the average weekly temperature, t °C, for 8 weeks during the summer.
�The product moment correlation coefficient for these data is −0.915
(a) Stating your hypotheses clearly and using a 5% level of significance, test whether or not the correlation between sales figures and average weekly temperature is negative.
(3)
(b) Suggest a possible reason for this correlation.
(1)
Tessa suggests that a linear regression model could be used to model these data.
(c) State, giving a reason, whether or not the correlation coefficient is consistent with Tessa's suggestion.
(1)
(d) State, giving a reason, which variable would be the explanatory variable.
(1)
Tessa calculated the linear regression equation as w = 10 755 – 171t
(e) Give an interpretation of the gradient of this regression equation.
(1)
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A company has three machines that produce a component. Machine A produces 28% of the components. Machine B produces 37% of the components and machine C produces the rest.
If a component is produced by machine A the chance that it will be faulty is 2%.
If a component is produced by machine B the chance that it will be faulty is 3%.
If a component is produced by machine C the chance that it will be faulty is 4%.
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Solution:
a)
b) P(faulty) = (0.28 X 0.02) + (0.37 X 0.03) + (0.35 X 0.04)
P(faulty) = 0.0307
c) 0.04
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A company has three machines that produce a component. Machine A produces 28% of the components. Machine B produces 37% of the components and machine C produces the rest.
If a component is produced by machine A the chance that it will be faulty is 2%.
If a component is produced by machine B the chance that it will be faulty is 3%.
If a component is produced by machine C the chance that it will be faulty is 4%.
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Cooking sauces are sold in jars containing a stated weight of 500 g of sauce.
The jars are filled by a machine.
The actual weight of sauce in each jar is normally distributed with mean 505 g and variance 100 g.
(b) Find the probability of a jar containing less than the stated weight.
(c) In a box of 30 jars, find the probability that more than 12 of the jars contain less than the stated weight.
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Cooking sauces are sold in jars containing a stated weight of 500 g of sauce.
The jars are filled by a machine.
The actual weight of sauce in each jar is normally distributed with mean 505 g and variance 100 g.
(b) Find the probability of a jar containing less than the stated weight.
(c) In a box of 30 jars, find the probability that more than 12 of the jars contain less than the stated weight.
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CORRECT SOLUTION:
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The heights of a group of people are normally distributed with a mean of 164 cm and 10% of the people have a height of more than 178 cm.
Find the standard deviation of the heights of the people.
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The heights of a group of people are normally distributed with a mean of 164 cm and 10% of the people have a height of more than 178 cm.
Find the standard deviation of the heights of the people.
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Solution:
X ~ B(30, 0.4)
(b) Use the normal distribution to calculate an approximation for P(X < 8)
(c) Calculate the percentage error in the approximation found in part b.
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X ~ B(30, 0.4)
(b) Use the normal distribution to calculate an approximation for P(X < 8)
(c) Calculate the percentage error in the approximation found in part b.
Question:
The length of the bus journey from Leeds to London is normally distributed with a mean of 220 minutes and a standard deviation of 8 minutes.
The bus company suspect that the mean bus time has changed. They take a sample of 10 bus journeys and find a mean time of 230 minutes.
b) Test at the 1% significance level whether there is evidence that the mean time has changed, clearly stating your hypotheses.
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Solution:
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The length of the bus journey from Leeds to London is normally distributed with a mean of 220 minutes and a standard deviation of 8 minutes.
The bus company suspect that the mean bus time has changed. They take a sample of 10 bus journeys and find a mean time of 230 minutes.
b) Test at the 1% significance level whether there is evidence that the mean time has changed, clearly stating your hypotheses.