SPS SPS SM (SPS SM) 2021 February

Which of the options below best describes the correlation shown in the diagram below? \includegraphics{figure_1} Tick \((\checkmark)\) one box. [1 mark] moderate positive \(\square\) strong positive \(\square\) moderate negative \(\square\) strong negative \(\square\)

Lenny is one of a team of people interviewing shoppers in a town centre. He is asked to survey 50 women between the ages of 18 and 29 Identify the name of this type of sampling. Circle your answer. [1 mark] simple random stratified quota systematic

The Venn diagram shows the probabilities associated with four events, \(A\), \(B\), \(C\) and \(D\) \includegraphics{figure_3}

1. Write down any pair of mutually exclusive events from \(A\), \(B\), \(C\) and \(D\) [1]
2. Given that \(P(B) = 0.4\) find the value of \(p\) [1]
3. Given also that \(A\) and \(B\) are independent find the value of \(q\) [2]
4. Given further that \(P(B'|C) = 0.64\) find
   1. the value of \(r\)
   2. the value of \(s\)
   [4]

Each member of a group of 27 people was timed when completing a puzzle. The time taken, \(x\) minutes, for each member of the group was recorded. These times are summarised in the following box and whisker plot. \includegraphics{figure_4}

1. Find the range of the times. [1]
2. Find the interquartile range of the times. [1]
3. For these 27 people \(\sum x = 607.5\) and \(\sum x^2 = 17623.25\) calculate the mean time taken to complete the puzzle. [1]
4. calculate the standard deviation of the times taken to complete the puzzle. [2]
5. Taruni defines an outlier as a value more than 3 standard deviations above the mean. State how many outliers Taruni would say there are in these data, giving a reason for your answer. [1]
6. Adam and Beth also completed the puzzle in \(a\) minutes and \(b\) minutes respectively, where \(a > b\). When their times are included with the data of the other 27 people
   Suggest a possible value for \(a\) and a possible value for \(b\), explaining how your values satisfy the above conditions. [3]
7. Without carrying out any further calculations, explain why the standard deviation of all 29 times will be lower than your answer to part (d). [1]

Patrick is practising his skateboarding skills. On each day, he has 30 attempts at performing a difficult trick. Every time he attempts the trick, there is a probability of 0.2 that he will fall off his skateboard. Assume that the number of times he falls off on any given day may be modelled by a binomial distribution.

1. 1. Find the mean number of times he falls off in a day. [1 mark]
   2. Find the variance of the number of times he falls off in a day. [1 mark]
2. 1. Find the probability that, on a particular day, he falls off exactly 10 times. [2 marks]
   2. Find the probability that, on a particular day, he falls off 5 or more times. [3 marks]
3. Patrick has 30 attempts to perform the trick on each of 5 consecutive days.
   1. Calculate the probability that he will fall off his skateboard at least 5 times on each of the 5 days. [2 marks]
   2. Explain why it may be unrealistic to use the same value of 0.2 for the probability of falling off for all 5 days. [1 mark]

The discrete random variable \(D\) has the following probability distribution where \(k\) is a constant.

1. Show that the value of \(k\) is \(\frac{600}{137}\) [2]
2. The random variables \(D_1\) and \(D_2\) are independent and each have the same distribution as \(D\). Find \(P(D_1 + D_2 = 80)\) Give your answer to 3 significant figures. [3]
3. A single observation of \(D\) is made. The value obtained, \(d\), is the common difference of an arithmetic sequence. The first 4 terms of this arithmetic sequence are the angles, measured in degrees, of quadrilateral \(Q\) Find the exact probability that the smallest angle of \(Q\) is more than \(50°\) [5]

A health centre claims that the time a doctor spends with a patient can be modelled by a normal distribution with a mean of 10 minutes and a standard deviation of 4 minutes.

1. Using this model, find the probability that the time spent with a randomly selected patient is more than 15 minutes. [1]
2. Some patients complain that the mean time the doctor spends with a patient is more than 10 minutes. The receptionist takes a random sample of 20 patients and finds that the mean time the doctor spends with a patient is 11.5 minutes. Stating your hypotheses clearly and using a 5% significance level, test whether or not there is evidence to support the patients' complaint. [4]
3. The health centre also claims that the time a dentist spends with a patient during a routine appointment, \(T\) minutes, can be modelled by the normal distribution where \(T \sim N(5, 3.5^2)\) Using this model,
   1. find the probability that a routine appointment with the dentist takes less than 2 minutes [1]
   2. find \(P(T < 2 | T > 0)\) [3]
   3. hence explain why this normal distribution may not be a good model for \(T\). [1]
4. The dentist believes that she cannot complete a routine appointment in less than 2 minutes. She suggests that the health centre should use a refined model only including values of \(T > 2\) Find the median time for a routine appointment using this new model, giving your answer correct to one decimal place. [5]

Tiana is a quality controller in a clothes factory. She checks for four possible types of defects in shirts. Of the shirts with defects, the proportion of each type of defect is as shown in the table below. Tiana wants to investigate the proportion, \(p\), of defective shirts with a fabric defect. She wishes to test the hypotheses $$H_0 : p = 0.3$$ $$H_1 : p < 0.3$$ She takes a random sample of 60 shirts with a defect and finds that \(x\) of them have a fabric defect.

1. Using a 5% level of significance, find the critical region for \(x\). [5 marks]
2. In her sample she finds 13 shirts with a fabric defect. Complete the test stating her conclusion in context. [2 marks]