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Exam-Style Questions.Problems adapted from questions set for previous Mathematics exams. |
1. | IB Applications and Interpretation |
The Scrumptious Sweet Company sell a variety pack of colourful, shaped sweets. The sweets are produced such that 60% are square and 40% are circular. It is known that 20% of the square shaped sweets and 40% of the circular sweets are coloured red.
(a) Show this information in a tree diagram.
A sweet is selected at random.
(b) Find the probability that the sweet is red.
(c) Given that the sweet is red, find the probability it is circular.
The Scrumptious Sweet Company also produce variety packs of Rainbow Gums. Their specifications state that the colours in each variety pack should be distributed as follows.
| Colour | Red | Orange | Yellow | Green | Blue |
| Percentage (%) | 15 | 25 | 15 | 25 | 20 |
Inspector Lou Spowels opens a pack of 90 sweets and records the frequency of each colour:
| Colour | Red | Orange | Yellow | Green | Blue |
| Observed Frequency | 12 | 21 | 15 | 20 | 22 |
To investigate if the sample is consistent with the company's specifications, Mr Spowels conducts a \(\chi^2\) goodness of fit test. The test is carried out at a 5% significance level.
(d) Write down the null hypothesis for this test.
(e) Copy and complete the following table giving the frequencies correct to one decimal place:
| Colour | Red | Orange | Yellow | Green | Blue |
| Expected Frequency |
(f) Write down the number of degrees of freedom.
(g) Find the p-value for the test.
(h) State the conclusion of the test. Give a reason for your answer.
2. | IB Applications and Interpretation |
The 2nd Rutherford American Scouts joined the 37th Wolverhampton British Scouts for an International Camp. Skipper Jones is interested to see if the mean height of American Scouts, \( \mu_1\), is the same as the mean height of British Scouts, \( \mu_2\). The information is recorded in the following table.
| American Scout height (cm) | 147 | 153 | 151 | 142 | 155 | 149 | 154 | 156 | 143 | 152 | 149 | 158 | |||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| British Scout height (cm) | 142 | 146 | 155 | 145 | 149 | 148 | 152 | 143 | 147 | 150 | 149 | 154 | 150 | 144 | 146 |
At the 10% level of significance, a t-test was used to compare the means of the two groups. The data is assumed to be normally distributed and the standard deviations are equal between the two groups.
(a) State the null hypothesis.
(b) State the alternative hypothesis.
(c) Calculate the p-value for this test.
(d) State, giving a reason, whether Skipper Jones should accept the null hypothesis.
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