Test using proportion

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.

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 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]

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. [1]
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]

The table shows information for England and Wales, taken from the UK 2011 census. A random sample of 10 000 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 \(m\). [5]