One-sample z-test known variance

13 A large supermarket chain advertises that the mean mass of apples of a certain variety on sale in their stores is 0.14 kg . Following a poor growing season, the head of quality control believes that the mean mass of these apples is less than 0.14 kg and she decides to carry out a hypothesis test at the \(5 \%\) level of significance. She collects a random sample of this variety of apple from the supermarket chain and records the mass, in kg, of each apple. She uses software to analyse the data. The results are summarised in the output below.

1. State the null hypothesis and the alternative hypothesis for the test, defining the parameter used.
2. Write down the distribution of the sample mean for this hypothesis test.
3. Determine the critical region for the test.
4. Carry out the test, giving your conclusion in context.

3. Star Farm produces duck eggs. Xander takes a random sample of 20 duck eggs from Star Farm and their widths, \(x \mathrm {~cm}\), are recorded. Xander's results are summarised as follows. $$\sum x = 92.0 \quad \sum x ^ { 2 } = 433.4974$$

1. Calculate unbiased estimates of the mean and the variance of the width of duck eggs produced by Star Farm. Yinka takes an independent random sample of 30 duck eggs from Star Farm and their widths, \(y \mathrm {~cm}\), are recorded. Yinka's results are summarised as follows. $$\sum y = 142.5 \quad \sum y ^ { 2 } = 689.5078$$
2. Treating the combined sample of 50 duck eggs as a single sample, estimate the standard error of the mean.
   (5) Research shows that the population of duck egg widths is normally distributed with standard deviation 0.71 cm . The farmer claims that the mean width of duck eggs produced by Star Farm is greater than 4.5 cm .
3. Using your combined mean, test, at the \(5 \%\) level of significance, the farmer's claim. State your hypotheses clearly.

6 A supermarket buys pears from a local supplier. The supermarket requires the mean weight of the pears to be at least 175 grams. William, the fresh-produce manager at the supermarket, suspects that the latest batch of pears delivered does not meet this requirement.

1. William weighs a random sample of 6 pears, obtaining the following weights, in grams. $$\begin{array} { l l l l l l } 160.6 & 155.4 & 181.3 & 176.2 & 162.3 & 172.8 \end{array}$$ Previous batches of pears have had weights that could be modelled by a normal distribution with standard deviation 9.4 grams. Assuming that this still applies, show that a hypothesis test at the \(5 \%\) level of significance supports William's suspicion.
   (7 marks)
2. William then weighs a random sample of 20 pears. The mean of this sample is 169.4 grams and \(s = 11.2\) grams, where \(s ^ { 2 }\) is an unbiased estimate of the population variance. Assuming that the population from which this sample is taken has a normal distribution but with unknown standard deviation, test William's suspicion at the \(\mathbf { 1 \% }\) level of significance.
3. Give a reason why the probability of a Type I error occurring was smaller when conducting the test in part (b) than when conducting the test in part (a).

Roastie's Coffee is sold in packets with a stated weight of 250 g. A supermarket manager claims that the mean weight of the packets is less than the stated weight. She weighs a random sample of 90 packets from their stock and finds that their weights have a mean of 248 g and a standard deviation of 5.4 g.

1. Using a 5\% level of significance, test whether or not the manager's claim is justified. State your hypotheses clearly. [5]
2. Find the 98\% confidence interval for the mean weight of a packet of coffee in the supermarket's stock. [4]
3. State, with a reason, the action you would recommend the manager to take over the weight of a packet of Roastie's Coffee. [2]

Roastie's Coffee company increase the mean weight of their packets to \(\mu\) g and reduce the standard deviation to 3 g. The manager takes a sample of size \(n\) from these new packets. She uses the sample mean \(\bar{X}\) as an estimator of \(\mu\).

1. Find the minimum value of \(n\) such that P\((|\bar{X} - \mu| < 1) \geq 0.98\) [5]

Gerald is a scientist who studies sand lizards. He believes that sand lizards on islands are, on average, shorter than those on the mainland. The population of sand lizards on the mainland has a mean length of 18.2 cm and a standard deviation of 1.8 cm. Gerald visited three islands, A, B and C, and measured the length, \(X\) centimetres, of each of a sample of \(n\) sand lizards on each island. The samples may be regarded as random. The data are shown in the table.

Island	\(\sum x\)	\(n\)
A	1384.5	78
B	116.9	7
C	394.6	20

1. Carry out a hypothesis test to investigate whether the data from Island A provide support for Gerald's belief at the 2% significance level. Assume that the standard deviation of the lengths of sand lizards on Island A is 1.8 cm. [7 marks]
2. For Island B, it is also given that $$\sum(x - \bar{x})^2 = 22.64$$
   1. Construct a 95% confidence interval for \(\mu_B\), where \(\mu_B\) centimetres is the mean length of sand lizards on Island B. Assume that the lengths of sand lizards on Island B are normally distributed with unknown standard deviation.
   2. Comment on whether your confidence interval provides support for Gerald's belief.
   [7 marks]
3. Comment on whether the data from Island C provide support for Gerald's belief. [2 marks]

Sweet pea plants grown using a standard plant food have a mean height of 1.6 m. A new plant food is used for a random sample of 49 randomly chosen plants and the heights, \(x\) metres, of this sample can be summarised by the following. $$n = 49$$ $$\sum x = 74.48$$ $$\sum x^2 = 120.8896$$ Test, at the 5\% significance level, whether, when the new plant food is used, the mean height of sweet pea plants is less than 1.6 m. [9]