# 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) British Scout height (cm) 147 153 151 142 155 149 154 156 143 152 149 158 142 146 155 145 149 148 152 143 147 150 149 154 150 144 146

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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