Two-sample z-test (known variances)

2 Two brands of car battery, ‘Invincible’ and ‘Excelsior’, have lifetimes which are normally distributed. Invincible batteries have a mean lifetime of 5 years with standard deviation 0.7 years. Excelsior batteries have a mean lifetime of 4.5 years with standard deviation 0.5 years. Random samples of 20 Invincible batteries and 25 Excelsior batteries are selected and the sample mean lifetimes are \(\bar { X } _ { I }\) years and \(\bar { X } _ { E }\) years respectively.

1. State the distributions of \(\bar { X } _ { I }\) and \(\bar { X } _ { E }\).
2. Calculate \(\mathrm { P } \left( \bar { X } _ { I } - \bar { X } _ { E } \geqslant 1 \right)\).

8 The tensile strength of rope is measured in kilograms. The standard deviation of the tensile strength of a particular design of 10 mm diameter rope is known to be 285 kilograms. A retail organisation, which buys such rope from two manufacturers, A and B , wishes to compare their ropes for mean tensile strength. The mean tensile strength, \(\bar { x }\), of a random sample of 80 lengths from manufacturer A was 3770 kilograms. The mean tensile strength, \(\bar { y }\), of a random sample of 120 lengths from manufacturer B was 3695 kilograms.

1. 1. Test, at the \(5 \%\) level of significance, the hypothesis that there is no difference between the mean tensile strength of rope from manufacturer A and that of rope from manufacturer B.
   2. Why was it not necessary to know the distributions of tensile strength in order for your test in part (a)(i) to be valid?
2. 1. Deduce that, for your test in part (a)(i), the critical values of \(( \bar { x } - \bar { y } )\) are \(\pm 80.63\), correct to two decimal places.
   2. In fact, the mean tensile strength of rope from manufacturer A exceeds that of rope from manufacturer B by 125 kilograms. Determine the probability of a Type II error for a test of the hypothesis in part (a)(i) at the \(5 \%\) level of significance, based upon a random sample of 80 lengths from manufacturer A and a random sample of 120 lengths from manufacturer B. (4 marks)

5

1. The random variable \(X\), which has distribution \(\mathrm { N } \left( \mu _ { X } , \sigma ^ { 2 } \right)\), is independent of the random variable \(Y\), which has distribution \(\mathrm { N } \left( \mu _ { Y } , \sigma ^ { 2 } \right)\). In order to test \(\mathrm { H } _ { 0 } : \mu _ { X } = 1.5 \mu _ { Y }\), samples of size \(n\) are taken on each of \(X\) and \(Y\) and the random variable \(\bar { D }\) is defined as $$\bar { D } = \bar { X } - 1.5 \bar { Y }$$ State the distribution of \(\bar { D }\) assuming that \(\mathrm { H } _ { 0 }\) is true.
2. A machine that fills bags with rice delivers weights that are normally distributed with a standard deviation of 4.5 grams. The machine fills two sizes of bags: large and extra-large.
   The mean weight of rice in a random sample of 50 large bags is 1509 grams.
   The mean weight of rice in an independent random sample of 50 extra-large bags is 2261 grams. Test, at the \(5 \%\) level of significance, the claim that, on average, the rice in an extra-large bag is \(1 \frac { 1 } { 2 }\) times as heavy as that in a large bag.
   [0pt] [6 marks]

4. Rugby players sometimes use protein powder to aid muscle increase. The monthly weight gains of rugby players taking protein powder may be modelled by a normal distribution having a standard deviation of 40 g and a mean which may depend on the type of protein powder they consume. A rugby team coach gives the same amount of protein powder over a trial month to 22 randomly selected players. Protein powder \(A\) was used by 12 players, randomly selected, and their mean weight gain was 900 g . Protein powder \(B\) was used by the other 10 players and their mean weight gain was 870 g . Let \(\mu _ { A }\) and \(\mu _ { B }\) be the mean monthly weight gains, in grams, of the populations of rugby players who use protein powder \(A\) and protein powder \(B\) respectively.

1. Calculate a 98\% confidence interval for \(\mu _ { A } - \mu _ { B }\).
2. In the given context, what can you conclude from your answer to part (a)? Give a reason for your answer.
3. Find the confidence level of the largest confidence interval that would lead the coach to favour protein powder \(A\) over protein powder \(B\).
4. State one non-statistical assumption you have made in order to reach these conclusions.