Confidence interval interpretation

1. Components are manufactured such that their length in mm is normally distributed with mean \(\mu\) and variance \(\sigma ^ { 2 }\). Below is a 95\% confidence interval for \(\mu\) calculated from a random sample of components.
   (11.52, 13.75)

Using the same random sample,

1. find a \(90 \%\) confidence interval for \(\mu\). Four 90\% confidence intervals are found from independent random samples.
2. Calculate the probability that only 3 of these 4 intervals will contain \(\mu\).

6 The label on a pack of strawberries in a large batch states that it holds 250 g of strawberries. A random sample of 40 packs from the batch is selected and software is used to produce a \(95 \%\) confidence interval for the mean weight of strawberries per pack. An extract from the software output is shown in Fig. 6. \begin{table}[h] \end{table}

1. Explain whether the confidence interval suggests that the mean weight of strawberries per pack in the batch is different from 250 g .
2. A manager looking at the data says that the conclusion would have been different if a \(90 \%\) confidence interval had been used.
   Determine whether the manager is correct.
3. Explain briefly whether or not it is appropriate for the manager to vary the confidence level before coming to any conclusions.
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4. On another occasion, using the same sample size, a 95\% confidence interval for the mean weight of strawberries per pack is [248.05, 249.95].
   Find the sample variance in this case.
5. Explain the meaning of a 95\% confidence interval.

5 Rebekah is investigating the distances, \(X\) light years, between the Earth and visible stars in the night sky. She determines the distance between the Earth and a star for a random sample of 100 visible stars. The summarised results are as follows: $$\sum x = 35522 \quad \text { and } \quad \sum x ^ { 2 } = 32902257$$ 5

1. Calculate a 97\% confidence interval for the population mean of \(X\), giving your values to the nearest light year.
   5
2. Mike claims that the population mean is 267 light years. Rebekah says that the confidence interval supports Mike's claim. State, with a reason, whether Rebekah is correct.

10 The label on a particular size of milk carton states that it contains 1.5 litres of milk. In an investigation at the packaging plant the contents, \(x\) litres, of each of 60 randomly selected cartons are measured. The data are summarised as follows. $$\Sigma x = 89.758 \quad \Sigma x ^ { 2 } = 134.280$$

1. Estimate the variance of the underlying population.
2. Find a 95\% confidence interval for the mean of the underlying population.
3. What does the confidence interval which you have calculated suggest about the statement on the carton? Each day for 300 days a random sample of 60 cartons is selected and for each sample a \(95 \%\) confidence interval is constructed.
4. Explain why the confidence intervals will not be identical.
5. What is the expected number of confidence intervals to contain the population mean?