State the distribution of \(\bar X\) , giving its mean and standard deviation.

The sample values are summarized as \(\sum {x = 3782} \) and \(\sum {{x^2} = 155341} \) where \(x\) kg is the weight of a sheep.

(i)     Find unbiased estimates for \(\mu \) and \({\sigma ^2}\).

(ii)     Find a \(95\%\) confidence interval for \(\mu \).

Test, at the \(1\%\) level of significance, the null hypothesis \(\mu  = 35\) against the alternative hypothesis that \(\mu  > 35\).

\(\bar X \sim N\left( {\mu ,{\text{ }}\frac{{{\sigma ^2}}}{{100}}} \right)\)     A1A1

Note: Award A1 for \(N\), A1 for the parameters.

(i)     \(\bar x = \frac{{\sum x }}{n} = \frac{{3782}}{{100}} = 37.8\)     A1

\(s_{n - 1}^2 = \frac{{155341}}{{99}} - \frac{{{{3782}^2}}}{{9900}} = 124\)     M1A1

(ii)     \(95\% CI = 37.82 \pm 1.98\sqrt {\frac{{124.3006}}{{100}}} \)     (M1)(A1)

\( = (35.6,{\text{ }}40.0)\)     A1

METHOD 1

one tailed t-test     A1

testing \(37.82\)     A1

\(99\) degrees of freedom

reject if \(t > 2.36\)     A1

t-value being tested is \(2.5294\)     A1

since \(2.5294 > 2.36\) we reject the null hypothesis and accept the alternative hypothesis     R1

METHOD 2

one tailed t-test     (A1)

\(p = 0.00650\)     A3

since \(p{\text{ - value}} < 0.01\) we reject the null hypothesis and accept the alternative hypothesis     R1

Almost all candidates recognised the sample distribution as normal but were not always successful in stating the mean and the standard deviation. Similarly almost all candidates knew how to find an unbiased estimator for \(\mu \), but a number failed to find the correct answer for the unbiased estimator for \({\sigma ^2}\). Most candidates were successful in finding the 95% confidence interval for \(\mu \). In part c) many fully correct answers were seen but a significant number of candidates did not recognise they were working with a t-distribution.

Almost all candidates recognised the sample distribution as normal but were not always successful in stating the mean and the standard deviation. Similarly almost all candidates knew how to find an unbiased estimator for \(\mu \), but a number failed to find the correct answer for the unbiased estimator for \({\sigma ^2}\). Most candidates were successful in finding the 95% confidence interval for \(\mu \). In part c) many fully correct answers were seen but a significant number of candidates did not recognise they were working with a t-distribution.

Almost all candidates recognised the sample distribution as normal but were not always successful in stating the mean and the standard deviation. Similarly almost all candidates knew how to find an unbiased estimator for \(\mu \), but a number failed to find the correct answer for the unbiased estimator for \({\sigma ^2}\). Most candidates were successful in finding the 95% confidence interval for \(\mu \). In part c) many fully correct answers were seen but a significant number of candidates did not recognise they were working with a t-distribution.