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Numerical Answers (workings/explanations are after the numerical answers.)
Question 2: 12 seconds; m
Question 7: Sample;
Question 11: ; do not reject
When , then .
When , then .
(a) Using GC, .
The modulus is and the argument is
(b) Please do label the axes appropriately.
Let , for .
Using GC, .
Thus, and .
Note: must write answer in single fraction as requested.
We want .
Using GC, .
(b) Let .
Using GC, since .
(c) We have to consider a shape smaller. In this case, it will be smaller than hemisphere. We take the volume removed from the bottom and minus the volume found above. Other words, and
(a) Observe that we can simplify it to probability of getting the same as immediately before as and probability of getting a different as immediately before as .
The probability that A wins is
(b) The probability that B wins in his second round is .
The probability that B wins is .
The required conditional probability that B wins in his second round given that he won is
(a) These 53 employees form a sample as they are observations of the population of 75 employees in the Staffing department.
(b) She needs to collect a sample that is random such that each employee has an equal chance of being selected, independently of each other. She can do so by numbering the employees and using a random number generator to select the corresponding employee. By doing so, she can avoid bias and obtain a representative sample.
(c) We have mainly three cases.
Case 1: 5 from administration.
Case 2: 4 from administration.
Case 3: 3 from administration.
Total number of ways
(b) Observe that the mean is given by . Thus,
(b) This is conditional probability, we are interested in the probability that they both take exactly same route given that both arrive at B.
The probability that both arrive at B is given by .
The probability that they both take exactly same route is given by
(c) The probability of arriving at C is given by .
Using GC, since .
(a) The product moment correlation coefficient of suggests there is a moderately strong negative linear correlation between distance and price. From the scatter diagram, we observe that as the distance increases, the price decreases decreasingly which suggests a non-linear relationship between distance and price.
(b) The product moment correlation coefficient will have no change as it has no units and is independent of all scale of measurement.
(c) The car is clearly not a fair and representative observation compared to the other data point since this car has different features from teh other 6 cars. Thus, it is an outlier and should be omitted.
(d) Please do label the axes appropriately.
(e) These distances (residuals) may be positive or negative, which may cancel out when we add them up. By squaring these distance (residuals), we ensure that they are all positive and the sum of this residuals squared will be more meaningful as they will not cancel out. The method of least squares seeks to produce a best-fit line when the sum of these residuals squared are minimised.
(f) Using the GC, the least squares regression line is given by
Product moment correlation coefficient
(g) Let pounds sterling
The estimate is not reliable as lies outside of the data range of and this is extrapolation.
Let be Zhou’s time, in seconds.
Let be Tan’s time, in seconds.
Unbiased estimate of population mean seconds
Unbiased estimate of population variance
Let be the population mean time Zhou takes to swim 100 metres freestyle, in seconds.
Let be the null hypothesis and be the alternative hypothesis.
Test against at 5% level of significance
At 5% level of significance,
Thus, we do not reject at 5% level of significance and conclude with insufficient evidence that the mean time taken to swim 100 metres freestyle has improved.
(d) Tan should be using a 2-tailed test as he wants to find out if the mean time has changed, i.e., differs from the 79 seconds.
(e) We assume that while he is away, his time he takes to complete a 100 metres freestyle is still normally distributed, i.e., . We also assume that the population variance for Tan’s swim times is known, or has not changed from before.