Paired sample confidence interval

6 A company has a large fleet of cars. It is claimed that use of a fuel additive will reduce fuel consumption. In order to test this claim a researcher at the company randomly selects 40 of the cars. The fuel consumption of each of the cars is measured, both with and without the fuel additive. The researcher then calculates the difference \(d\) litres per kilometre between the two figures for each car, where \(d\) is the fuel consumption without the additive minus the fuel consumption with the additive. The sample mean of \(d\) is 0.29 and the sample standard deviation is 1.64 .

1. Showing your working, find a 95\% confidence interval for the population mean difference.
2. Explain whether the confidence interval suggests that, on average, the fuel additive does reduce fuel consumption.
3. Explain why you can construct the interval in part (i) despite not having any information about the distribution of the population of differences.
4. Explain why the sample used was random.

5 A researcher is investigating whether doing yoga has any effect on quality of sleep in older people. The researcher selects a random sample of 40 older people, who then complete a yoga course. Before they start the course and again at the end, the 40 people fill in a questionnaire which measures their perceived sleep quality. The higher the score, the better is the perceived quality of sleep. The researcher uses software to produce a 90\% confidence interval for the difference in mean sleep quality (sleep quality after the course minus sleep quality before the course). The output from the software is shown below. Z Estimate of a Mean Confidence level □ 0.9 Sample Result
Z Estimate of a Mean

1. Explain why the confidence interval is based on the Normal distribution even though the distribution of the population of differences is not known.
2. Explain whether the confidence interval suggests that the mean sleep qualities before and after completing a yoga course are different.
3. In the output from the software, SE stands for 'standard error'.
   1. Explain what standard error is.
   2. Show how the standard error was calculated in this case.
4. A colleague of the researcher suggests that the confidence level should have been \(95 \%\) rather than \(90 \%\). Determine whether this would have made a difference to your answer to part (b).

4 An amateur meteorologist records the total rainfall at her home each day using a traditional rain gauge. This means that she has to go out each day at 9 am to read the rain gauge and then to empty it. She wants to save time by using a digital rain gauge, but she also wants to ensure that the readings from the digital gauge are similar to those of her traditional gauge. Over a period of 100 days, she uses both gauges to measure the rainfall. The meteorologist uses software to produce a 95\% confidence interval for the difference between the two readings (the traditional gauge reading minus the digital gauge reading). The output from the software is shown in Fig. 4. Although rainfall was measured over a period of 100 days, there was no rain on 40 of those days and so the sample size in the software output is 60 rather than 100. \begin{table}[h] \end{table}

1. Explain why this confidence interval can be calculated even though nothing is known about the distribution of the population of differences.
2. State the confidence interval which the software gives in the form \(a < \mu < b\).
3. Show how the value 0.07444 (labelled SE) was calculated.
4. Comment on whether you think that the confidence interval suggests that the two different methods of measurement are broadly in agreement.

7 A physiotherapist is investigating hand grip strength in adult women under 30 years old. She thinks that the grip strength of the dominant hand will be on average 2 kg higher than the grip strength of the non-dominant hand. The physiotherapist selects a random sample of 12 adult women under 30 years old and measures the grip strength of each of their hands. She then uses software to produce a \(95 \%\) confidence interval for the mean difference in grip strength between the two hands (dominant minus nondominant), as shown in Fig. 7. \begin{table}[h]