MATHEMATICS 3D TESTTime Allowed20 minutes
Total Marks16
Calculator Free Section.
Equipment Allowedpencils, pens ruler
Question 1(6 marks)
(a)On a uniform probability distribution graph the height of the horizontal line is 0.05. What is the length of this line? (1 mark)
(b)What can be said about the three averages in a normal distribution?(1 mark)
(c)(i)Explain the meaning of the interval represented as µ 2.
(ii)How many scores might you expect to be in this interval?(2 marks)
(d)Consider a normal distribution where the height X of a single randomly chosen two year old varies with mean 78 cm and standard deviation 6.25 cm. A medical study investigated the height of 100 of these two year olds. What would be the mean and standard deviation for this distribution? (1 marks)
(e)Tom investigated the mean change in his share portfolio last year and found the mean and standard deviation to be $0 and $3.20 respectively. Comment on what might have occurred.
(1 marks)
Question 2(7 marks)
(a)Describe the difference between a normal distribution and a uniform distribution.(2 marks)
(b)From a normal distribution 3 scores are selected. Determine the probability that two of these scores are below the mean. (2 marks)
(c)A data set of 500 scores had 68% of the scores within one standard deviation of the mean. Could we assume the data could best be modeled by the normal distribution? Explain your answer. (1 mark)
(d)Two samples of size 50 and 70 are taken from the same uniform distribution. Which sample will have the larger 95% confidence interval? Explain you answer. (2 marks)
Question 3(3 marks)
Shown here is a calculator display from a question Stephanie was doing in her MAT 3D class. The question involved taking a sample from a large binomial distribution to find the probability a random sample point would be in an interval.
(a)What is the mean and standard deviation of the binomial distribution?(1 mark)(b)How many sample points are taken?(1 mark)
(c)What were the upper and lower bounds of the sample?(1 mark) /
MATHEMATICS 3D TESTTime Allowed30 minutes
Total Marks24
Calculator Section.
Equipment AllowedCAS Calculator, pencils, pens ruler
Question 4(9 marks)
An analysis of lead content in children five years and younger, follows a normal distribution with a standard deviation of 1.681 micrograms per deciliter (mcg/dL). An analysis of three children in this group gave concentrations of 8.403, 8.363 and 8.447 mcg/dL. (Work with values correct to 3 decimal places).
(a)Find the 99% confidence interval for this sample.(4 marks)
(b)Suppose that a single measurement gave a reading of 8.404. Find the 99% confidence interval for this single measurement. Comment on the difference of your answer to part (a).
(3 marks)
(c)The international recommended safe guideline, is 10 micrograms per decilitre. Comment on how an agency might reassure the people that there is minimal contamination in this town. Substantiate your comments using a mathematical argument. (2 marks)
Question 5(4 marks)
A manufacturer claims the cream XAB will reduce eczema (ie a skin disorder) in 40% of users. In an experiment to test this claim, the cream XAB will be given to 2000 people with the disorder. The researchers must be confident that the sample size is large enough to enable them to be 90% confident they will observe between 750 and 850 favourable results. Show mathematically whether this will be achieved in this experiment.
Question 6(5 marks)
The number of accidents at an intersection in Perth varies with a mean and standard deviation of 2.2 and 1.4 per week respectively.
(a)Is the distribution discrete? Explain your answer.(1 mark)
(b)Let be the mean number of accidents per week over a one-year period. Using the Central Limit Theorem determine an approximate distribution. (1 mark)
(c)Approximate the probability that is less than 2.(1 mark)
(d)Approximate the probability that fewer than 100 accidents will occur at the intersection in a year. (2 marks)
Question 7(3 marks)
X is a normally distributed random variable with population mean = µ and standard deviation==12. A random sample with mean will be taken from this population. If we want to be 95% confident that differs from µ by no more than 0.3 units, how large must the sample size be?
Question 8(3 marks)
A sample is taken from a normal population having a standard deviation of 3.87 units. The lower and upper limits of a 99% confidence interval were 43.18 and 46.82 respectively. Find the sample mean and sample size.
Insert School Logo / Insert School NameMATHEMATICS 3D TEST SolutionsTime Allowed20 minutes
Total Marks16
Calculator Free Section.
Equipment Allowedpencils, pens ruler
Question 1(6 marks)
(a)On a uniform probability distribution graph the height of the horizontal line is 0.05. What is the length of this line? (1 mark)
20 √
(b)What can be said about the three averages in a normal distribution?(1 mark)
They are in the same position at the centre of the distribution √
(c)(i)Explain the meaning of the interval represented as µ 2.
within two standard deviations ofthe mean √
(ii)How many scores might you expect to be in this interval?(2 marks)
95% √
(d)Consider a normal distribution where the height X of a single randomly chosen two year old varies with mean 78 cm and standard deviation 6.25 cm. A medical study investigated the height of 100 of these two year olds. What would be the mean and standard deviation for this distribution? (1 mark)
Mean = 78cm SD = 0.625cm √
(e)Tom investigated the mean change in his share portfolio last year and found the mean and standard deviation to be $0 and $3.20 respectively. Comment on what might have occurred.
(1 mark)
The combined change (ie negative and positive growth) resulted in no change to his shares.
√ √
Question 2(7 marks)
(a)Describe the difference between a normal distribution and a uniform distribution.(2 marks)
A uniform distribution has scores equally distributed throughout the interval √ while the normal distribution has more scores clustered about the centre √.
(b)From a normal distribution 3 scores are selected. Determine the probability that two of these scores are below the mean. (2 marks)
√ √
(c)A data set of 500 scores had 68% of the scores within one standard deviation of the mean. Could we assume the data could best be modeled by the normal distribution? Explain your answer. (1 mark)
No we would need more information about the other 32%. √
(d)Two samples of size 50 and 70 are taken from the same uniform distribution. Which sample will have the larger 95% confidence interval? Explain you answer. (2 marks)
The sample of size 50 √ will have the greater interval because … √
Question 3(3 marks)
Shown here is a calculator display from a question Stephanie was doing in her MAT 3D class. The question involved taking a sample from a large binomial distribution to find the probability a random sample point would be in an interval.
(a)What is the mean and standard deviation of the binomial distribution?(1 mark)mean = 50.2 and SD = 10.3 √
(b)How many sample points are taken?(1 mark)
n = 35 √
(c)What were the upper and lower bounds of the sample?(1 mark)
Upper was 65 and lower was 23 √ /
MATHEMATICS 3C TESTTime Allowed30 minutes
Total Marks24
Calculator Section.
Equipment AllowedCAS Calculator, pencils, pens ruler
Question 4(9 marks)
An analysis of lead content in children five years and younger, follows a normal distribution with a standard deviation of 1.681 micrograms per deciliter (mcg/dL). An analysis of three children in this group gave concentrations of 8.403, 8.363 and 8.447 mcg/dL. (Work with values correct to 3 decimal places).
(a)Find the 99% confidence interval for this sample.(4 marks)
Mean = √
The 99% confidence interval = 8.404 2.576 √ √
= (5.904, 10.904) √
(b)Suppose that a single measurement gave a reading of 8.404. Find the 99% confidence interval for this single measurement. Comment on the difference of your answer to part (a).
(3 marks)
8.404 2.576 √
(4.074, 12.734) √
The interval is reduced the larger the sample size. √
(c)The international recommended safe guideline, is 10 micrograms per decilitre. Comment on how an agency might reassure the people that there is minimal contamination in this town. Substantiate your comments using a mathematical argument. (2 marks)
By taking a larger sample, n = 100 (assuming the mean remains the same) the upper 99% confidence interval will be 8.662 mcg/dL. This is below the international safe guideline. √ √
Question 5(4 marks)
A manufacturer claims the cream XAB will reduce eczema (ie a skin disorder) in 40% of users. In an experiment to test this claim, the cream XAB will be given to 2000 people with the disorder. The researchers must be confident that the sample size is large enough to enable them to be 90% confident they will observe between 750 and 850 favourable results. Show mathematically whether this will be achieved in this experiment.
Mean = 2000 0.4 = 800
Standard deviation = = 21.9089 √
Lower limit of 90% confidence interval
= 799.19 √
Upper limit of 90% confidence interval
= 800.80 √
No it will not be achieved. √
Question 6(5 marks)
The number of accidents at an intersection in Perth varies with a mean and standard deviation of 2.2 and 1.4 per week respectively.
(a)Is the distribution discrete? Explain your answer.(1 mark)
Yes it is discrete because we are dealing with countable objects.
(b)Let be the mean number of accidents per week over a one-year period. Using the Central Limit Theorem determine an approximate distribution. (1 mark)
Normal with mean 2.2 and SD 0.1941 ie N(2.2, 0.1941) √ √
(c)Approximate the probability that is less than 2.(1 mark)
0.1515 accept 0.1514 (ie using values from part (b)) √
(d)Approximate the probability that fewer than 100 accidents will occur at the intersection in a year. (2 marks)
0.0769 accept 0.0768 √ √
Question 7(3 marks)
X is a normally distributed random variable with population, mean = µ and standard deviation==12. A random sample with mean will be taken from this population. If we want to be 95% confident that differs from µ by no more than 0.3 units, how large must the sample size be?
0.3 ≤ µ ≤ 0.3
ie 0.3 + ≤ µ ≤ + 0.3 √
√
n = 6146.56
Hence the sample size must be at least 6147 √
Question 8(3 marks)
A sample is taken from a normal population having a standard deviation of 3.87 units. The lower and upper limits of a 99% confidence interval were 43.18 and 46.82 respectively. Find the sample mean and sample size.
= 43.18 √
= 46.82 √
Solving Simultaneous equations
Sample size 30 with mean 45 units √
1