4 A council provides waste paper recycling services for local businesses. Some businesses use the standard service for recycling paper, others use a special service for dealing with confidential documents, and others use both. Businesses are classified as small or large. A survey of a random sample of 285 businesses gives the following data for size of business and recycling service.
| Recycling Service |
| \cline { 3 - 5 }
\multicolumn{2}{|c|}{} | Standard | Special | Both |
| \multirow{2}{*}{} | Small | 35 | 26 | 44 |
| Large | 55 | 52 | 73 |
- Write down null and alternative hypotheses for a test to examine whether there is any association between size of business and recycling service used.
The contributions to the test statistic for the usual \(\chi ^ { 2 }\) test are shown in the table below.
| Recycling Service |
| \cline { 3 - 5 }
\multicolumn{2}{|c|}{} | Standard | Special | Both |
| \multirow{2}{*}{} | Small | 0.1023 | 0.2607 | 0.0186 |
| Large | 0.0597 | 0.1520 | 0.0108 |
- Calculate the expected frequency for large businesses using the special service. Verify the corresponding contribution 0.1520 to the test statistic.
- Carry out the test at the \(5 \%\) level of significance, stating your conclusion clearly.
The council is also investigating the weight of rubbish in domestic dustbins. In 2008 the average weight of rubbish in bins was 32.8 kg . The council has now started a recycling initiative and wishes to determine whether there has been a reduction in the weight of rubbish in bins. A random sample of 50 domestic dustbins is selected and it is found that the mean weight of rubbish per bin is now 30.9 kg , and the standard deviation is 3.4 kg .
- Carry out a test at the \(5 \%\) level to investigate whether the mean weight of rubbish has been reduced in comparison with 2008 . State carefully your null and alternative hypotheses.
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