Anna cycles to her new school. She records the times taken for the first ten days with the following results (in minutes).

12.4 13.7 12.5 13.4 13.8 12.3 14.0 12.8 12.6 13.5

Assume that these times are a random sample from the \({\text{N}}(\mu ,{\text{ }}{\sigma ^2})\) distribution.

(a)     Determine unbiased estimates for \(\mu \) and \({\sigma ^2}\).

(b)     Calculate a 95 % confidence interval for \(\mu \).

(c)     Before Anna calculated the confidence interval she thought that the value of \(\mu \) would be 12.5. In order to check this, she sets up the null hypothesis \({{\text{H}}_0}:\mu  = 12.5\).

(i)     Use the above data to calculate the value of an appropriate test statistic. Find the corresponding p-value using a two-tailed test.

(ii)     Interpret your p-value at the 1 % level of significance, justifying your conclusion.

(a)     estimate of \(\mu = 13.1\)     A1

estimate of \({\sigma ^2} = 0.416\)     A1

[2 marks]

(b)     using a GDC (or otherwise), the 95% confidence interval is     (M1)

[12.6, 13.6]     A1A1

Note: Accept open or closed intervals.

[3 marks]

(c)     (i)     \(t = \frac{{13.1 - 12.5}}{{0.6446 \ldots /\sqrt {10} }} = 2.94\)     (M1)A1

\(v = 9\)     (A1)

p-value \( = 2 \times {\text{P}}(T > 2.9433 \ldots )\)     (M1)

\( = 0.0164\,\,\,\,\,\)(accept 0.0165)     A1

(ii)     we accept the null hypothesis (the mean travel time is 12.5 minutes)     A1

because 0.0164 (or 0.0165) > 0.01     R1

Note: Allow follow through on their p-value.

[7 marks]

Total [12 marks]

This was well answered by many candidates. In (a), some candidates chose the wrong standard deviation from their calculator and often failed to square their result to obtain the unbiased variance estimate. Candidates should realise that it is the smaller of the two values (ie the one obtained by dividing by (n – 1)) that is required. The most common error was to use the normal distribution instead of the t-distribution. The signpost towards the t-distribution is the fact that the variance had to be estimated in (a). Accuracy penalties were often given for failure to round the confidence limits, the t-statistic or the p-value to three significant figures.