Test using proportion

2 An investigation in 2007 into the incidence of tuberculosis (TB) in badgers in a certain area found that 42 out of a random sample of 190 badgers tested positive for TB.
In 2010, 48 out of a random sample of 150 badgers tested positive for TB.

1. Assuming that the population proportions of badgers with TB are the same in 2007 and 2010, obtain the best estimate of this proportion.
2. Carry out a test at the \(2 \frac { 1 } { 2 } \%\) significance level of whether the population proportion of badgers with TB increased from 2007 to 2010.

6 Random samples of 200 'Alpha' and 150 'Beta' vacuum cleaners were monitored for reliability. It was found that 62 Alpha and 35 Beta cleaners required repair during the guarantee period of one year. The proportions of all Alpha and Beta cleaners that require repair during the guarantee period are \(p _ { \alpha }\) and \(p _ { \beta }\) respectively.

1. Find a \(95 \%\) confidence interval for \(p _ { \alpha }\).
2. Give a reason why, apart from rounding, the interval is approximate.
3. Test, at the \(5 \%\) significance level, whether \(p _ { \alpha }\) differs from \(p _ { \beta }\).

5 Each of a random sample of 200 steel bars taken from a production line was examined and 27 were found to be faulty.

1. Find an approximate \(90 \%\) confidence interval for the proportion of faulty bars produced. A change in the production method was introduced which, it was claimed, would reduce the proportion of faulty bars. After the change, each of a further random sample of 100 bars was examined and 8 were found to be faulty.
2. Test the claim, at the \(10 \%\) significance level.

11 Zac is planning to write a report on the music preferences of the students at his college. There is a large number of students at the college.

1. State one reason why Zac might wish to obtain information from a sample of students, rather than from all the students.
2. Amaya suggests that Zac should use a sample that is stratified by school year. Give one advantage of this method as compared with random sampling, in this context. Zac decides to take a random sample of 60 students from his college. He asks each student how many hours per week, on average, they spend listening to music during term. From his results he calculates the following statistics.

   Mean	Standard deviation	Median	Lower quartile	Upper quartile
   21.0	4.20	20.5	18.0	22.9

3. Sundip tells Zac that, during term, she spends on average 30 hours per week listening to music. Discuss briefly whether this value should be considered an outlier.
4. Layla claims that, during term, each student spends on average 20 hours per week listening to music. Zac believes that the true figure is higher than 20 hours. He uses his results to carry out a hypothesis test at the 5\% significance level. Assume that the time spent listening to music is normally distributed with standard deviation 4.20 hours. Carry out the test.

12 The table shows information for England and Wales, taken from the UK 2011 census. A random sample of 10000 people in another country was chosen in 2011 , and the number, \(m\), of children aged 5-17 was noted.
It was found that there was evidence at the \(2.5 \%\) level that the proportion of children aged 5-17 in the same year was higher than in the UK.
Unfortunately, when the results were recorded the value of \(m\) was omitted. Use an appropriate normal distribution to find an estimate of the smallest possible value of \(m\). TURN OVER FOR THE NEXT QUESTION

4. A teacher wants to investigate the sports played by students at her school in their free time. She decides to ask a random sample of 120 pupils to complete a short questionnaire.

1. Give two reasons why the teacher might choose to use a sample survey rather than a census.
2. Suggest a suitable sampling frame that she could use. The teacher believes that 1 in 20 of the students play tennis in their free time. She uses the data collected from her sample to test if the proportion is different from this.
3. Using a suitable approximation and stating the hypotheses that she should use, find the critical region for this test. The probability for each tail of the region should be as close as possible to 5\%.
4. State the significance level of this test.

4 An analysis of a sample of 250 patients visiting a medical centre showed that 38 per cent were aged over 65 years. An analysis of a sample of 100 patients visiting a dental practice showed that 21 per cent were aged over 65 years. Assume that each of these two samples has been randomly selected.
Investigate, at the \(5 \%\) level of significance, the hypothesis that the percentage of patients visiting the medical centre, who are aged over 65 years, exceeds that of patients visiting the dental practice, who are aged over 65 years, by more than 10 per cent.

4

1. A large survey in the USA establishes that 60 per cent of its residents own a smartphone. A survey of 250 UK residents reveals that 164 of them own a smartphone.
   Assuming that these 250 UK residents may be regarded as a random sample, investigate the claim that the percentage of UK residents owning a smartphone is the same as that in the USA. Use the 5\% level of significance.
2. A random sample of 40 residents in a market town reveals that 5 of them own a 4 G mobile phone. Use an exact test to investigate, at the \(5 \%\) level of significance, the belief that fewer than 25 per cent of the town's residents own a 4 G mobile phone.
3. A marketing company needs to estimate the proportion of residents in a large city who own a 4 G mobile phone. It wishes to estimate this proportion to within 0.05 with a confidence of 98\%. Given that the proportion is known to be at most 30 per cent, estimate the sample size necessary in order to meet the company's need.
   [0pt] [5 marks]

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256 2 The random variable \(T\) has an exponential distribution with mean 2 Find \(\mathrm { P } ( T \leq 1.4 )\) Circle your answer. \(\mathrm { e } ^ { - 2.8 }\) \(\mathrm { e } ^ { - 0.7 }\) \(1 - e ^ { - 0.7 }\) \(1 - \mathrm { e } ^ { - 2.8 }\) The continuous random variable \(Y\) has cumulative distribution function $$\mathrm { F } ( y ) = \left\{ \begin{array} { l r } 0 & y < 2 \\ - \frac { 1 } { 9 } y ^ { 2 } + \frac { 10 } { 9 } y - \frac { 16 } { 9 } & 2 \leq y < 5 \\ 1 & y \geq 5 \end{array} \right.$$ Find the median of \(Y\) Circle your answer. 2 \(\frac { 10 - 3 \sqrt { 2 } } { 2 }\) \(\frac { 7 } { 2 }\) \(\frac { 10 + 3 \sqrt { 2 } } { 2 }\) Turn over for the next question 4 Research has shown that the mean number of volcanic eruptions on Earth each day is 20 Sandra records 162 volcanic eruptions during a period of one week. Sandra claims that there has been an increase in the mean number of volcanic eruptions per week. Test Sandra's claim at the \(5 \%\) level of significance.
5 The continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} \frac { 1 } { 6 } e ^ { \frac { x } { 3 } } & 0 \leq x \leq \ln 27 \\ 0 & \text { otherwise } \end{cases}$$ Show that the mean of \(X\) is \(\frac { 3 } { 2 } ( \ln 27 - 2 )\) 6 Over time it has been accepted that the mean retirement age for professional baseball players is 29.5 years old. Imran claims that the mean retirement age is no longer 29.5 years old.
He takes a random sample of 5 recently retired professional baseball players and records their retirement ages, \(x\). The results are $$\sum x = 152.1 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 7.81$$ 6

1. State an assumption that you should make about the distribution of the retirement ages to investigate Imran's claim. 6
2. Investigate Imran's claim, using the 10\% level of significance.