The turnover (in millions of pounds) for 8 train companies and 8 coach companies are given below.

Train 4.3 13.2 134 88 92 66.5 122.1 102
Coach 2.2 31 44 72 19 89.3 55.9 43.1

(a)Using an appropriate statistical test determine whether train companies or coach companies have higher turnovers. State the test used, your null hypothesis, your conclusion and the evidence you base that conclusion upon.
(b)Find the 99% confidence interval for the train companies’ variation in turnover.
(c) Perform an F test (using α=0.05) to determine whether the difference in variation between train and coach company turnover is significant.

(a) To determine whether train companies or coach companies have higher turnovers, we can use the two-sample t-test. The null hypothesis (H0) is that there is no difference in the mean turnovers of the two types of companies. The alternative hypothesis (Ha) is that there is a significant difference in mean turnovers between the two types of companies.

To perform the test, you need to follow these steps:
1. Calculate the sample means for both train and coach companies.
2. Calculate the sample standard deviations for both train and coach companies.
3. Calculate the t-statistic using the formula:
t = (mean_train - mean_coach) / sqrt((sd_train^2/n_train) + (sd_coach^2/n_coach))
where mean_train and mean_coach are the sample means, sd_train and sd_coach are the sample standard deviations, and n_train and n_coach are the sample sizes.
4. Determine the degrees of freedom (df) using the formula:
df = n_train + n_coach - 2
5. Look up the critical t-value for the desired significance level (alpha) and degrees of freedom in the t-distribution table or using software.
6. Compare the t-statistic to the critical t-value.
- If t-statistic > critical t-value, reject the null hypothesis.
- If t-statistic <= critical t-value, fail to reject the null hypothesis.

Based on the evidence obtained from the statistical test, you can state your conclusion. If the null hypothesis is rejected, it means there is a significant difference in mean turnovers between train and coach companies. If the null hypothesis is not rejected, it means there is no sufficient evidence to conclude a difference in mean turnovers.

(b) To find the 99% confidence interval for the train companies' variation in turnover, you can use the following formula:
Confidence interval = mean ± (t * (sd / sqrt(n)))
where mean is the sample mean, sd is the sample standard deviation, n is the sample size, and t is the critical t-value for the desired confidence level and degrees of freedom.

(c) To perform an F test to determine whether the difference in variation between train and coach company turnover is significant, you can follow these steps:
1. Calculate the sample variances for both train and coach companies.
2. Determine the degrees of freedom for variances using the formulas:
df_train = n_train - 1
df_coach = n_coach - 1
3. Calculate the F-statistic using the formula:
F = variance_train / variance_coach
4. Look up the critical F-value for the desired significance level (alpha) and degrees of freedom in the F-distribution table or using software.
5. Compare the F-statistic to the critical F-value.
- If F-statistic > critical F-value, reject the null hypothesis.
- If F-statistic <= critical F-value, fail to reject the null hypothesis.

Remember to use the appropriate alpha level (significance level) for your test, as mentioned in the question.