Find unbiased estimates of the mean of $T$.

Find unbiased estimates of the variance of $T$.

Assuming that $T$ is normally distributed, find

(i)  the 90% confidence interval for the mean time taken to travel to work by the workers of this company,

(ii)  the 95% confidence interval for the mean time taken to travel to work by the workers of this company.

Before seeing these results the managing director believed that the mean time was 26 minutes.

Explain whether your answers to part (b) support her belief.

$\overline{t}=\frac{\sum _{i=1}^{75}{t}_i}{75}=28.886\dots =28.9$       A1

[1 mark]

${s_{n-1}}^2=\frac{75}{74}\left(\frac{\sum _{i=1}^{75}{t}_i^2}{75}-{\overline{t}}^2\right)=188.9009\dots =189$      (M1)A1

Note: Accept all answers that round to 28.9 and 189.

Note: Award M0 if division by 75.

[2 marks]

attempting to find a confidence interval.     (M1)

(i)   90% interval: (26.2, 31.5)       A1

(ii)  95% interval: (25.7, 32.0)       A1

Note: Accept any values which round to within 0.1 of the correct value.

Note: Award M1A1A0 if only confidence limits are given in the form 28.9 ± 2.6.

[3 marks]

26 lies within the 95% interval but not within the 90% interval        R1

Note: Award R1 for considering whether or not one or two of the intervals contain 26.

the belief is supported at the 5% level (accept 95%)        A1

the belief is not supported at the 10% level (accept 90%)        A1

Note: FT their intervals but award R1A1A0 if both intervals give the same conclusion.

[3 marks]