Confidence interval interpretation or related probability

1 The result of a fitness trial is a random variable \(X\) which is normally distributed with mean \(\mu\) and standard deviation 2.4. A researcher uses the results from a random sample of 90 trials to calculate a \(98 \%\) confidence interval for \(\mu\). What is the width of this interval?

3 The time taken in minutes for a certain daily train journey has a normal distribution with standard deviation 5.8. For a random sample of 20 days the journey times were noted and the mean journey time was found to be 81.5 minutes.

1. Calculate a \(98 \%\) confidence interval for the population mean journey time.
   A student was asked for the meaning of this confidence interval. The student replied as follows.
   'The times for \(98 \%\) of these journeys are likely to be within the confidence interval.'
2. Explain briefly whether this statement is true or not.
   Two independent 98\% confidence intervals are found.
3. Given that at least one of these intervals contains the population mean, find the probability that both intervals contain the population mean.

3

1. The waiting time at a certain bus stop has variance 2.6 minutes \({ } ^ { 2 }\). For a random sample of 75 people, the mean waiting time was 7.1 minutes. Calculate a \(92 \%\) confidence interval for the population mean waiting time.
2. A researcher used 3 random samples to calculate 3 independent \(92 \%\) confidence intervals. Find the probability that all 3 of these confidence intervals contain only values that are greater than the actual population mean.
3. Another researcher surveyed the first 75 people who waited at a bus stop on a Monday morning. Give a reason why this sample is unsuitable for use in finding a confidence interval for the mean waiting time.

2

1. The time taken by a worker to complete a task was recorded for a random sample of 50 workers. The sample mean was 41.2 minutes and an unbiased estimate of the population variance was 32.6 minutes \({ } ^ { 2 }\). Find a \(95 \%\) confidence interval for the mean time taken to complete the task.
2. The probability that an \(\alpha \%\) confidence interval includes only values that are lower than the population mean is \(\frac { 1 } { 16 }\). Find the value of \(\alpha\).

5 A builders' merchant sells stones of different sizes.

1. The masses of size \(A\) stones have standard deviation 6 grams. The mean mass of a random sample of 200 size \(A\) stones is 45 grams. Find a 95\% confidence interval for the population mean mass of size \(A\) stones.
2. The masses of size \(B\) stones have standard deviation 11 grams. Using a random sample of size 200, an \(\alpha \%\) confidence interval for the population mean mass is found to have width 4 grams. Find \(\alpha\).

2 The length, in minutes, of mathematics lectures at a certain college has mean \(\mu\) and standard deviation 8.3.

1. The total length of a random sample of 85 lectures was 4590 minutes. Calculate a 95\% confidence interval for \(\mu\).
   The length, in minutes, of history lectures at the college has mean \(m\) and standard deviation \(s\).
2. Using a random sample of 100 history lectures, a 95\% confidence interval for \(m\) was found to have width 2.8 minutes. Find the value of \(s\).

1 Leaves from a certain type of tree have lengths that are distributed with standard deviation 3.2 cm . A random sample of 250 of these leaves is taken and the mean length of this sample is found to be 12.5 cm .

1. Calculate a \(99 \%\) confidence interval for the population mean length.
2. Write down the probability that the whole of a \(99 \%\) confidence interval will lie below the population mean.

3 Each of a random sample of 15 students was asked how long they spent revising for an exam. The results, in minutes, were as follows. $$\begin{array} { l l l l l l l l l l l l l l l } 50 & 70 & 80 & 60 & 65 & 110 & 10 & 70 & 75 & 60 & 65 & 45 & 50 & 70 & 50 \end{array}$$ Assume that the times for all students are normally distributed with mean \(\mu\) minutes and standard deviation 12 minutes.

1. Calculate a \(92 \%\) confidence interval for \(\mu\).
2. Explain what is meant by a \(92 \%\) confidence interval for \(\mu\).
3. Explain what is meant by saying that a sample is 'random'.

3 The management of a factory wished to find a range within which the time taken to complete a particular task generally lies. It is given that the times, in minutes, have a normal distribution with mean \(\mu\) and standard deviation 6.5. A random sample of 15 employees was chosen and the mean time taken by these employees was found to be 52 minutes.

1. Calculate a \(95 \%\) confidence interval for \(\mu\).
   Later another \(95 \%\) confidence interval for \(\mu\) was found, based on a random sample of 30 employees.
2. State, with a reason, whether the width of this confidence interval was less than, equal to or greater than the width of the previous interval.

3 The masses, in grams, of bags of flour are normally distributed with mean \(\mu\). The masses, \(m\) grams, of a random sample of 50 bags are summarised by \(\Sigma m = 25110\) and \(\Sigma m ^ { 2 } = 12610300\).

1. Calculate a \(96 \%\) confidence interval for \(\mu\), giving the end-points correct to 1 decimal place.
   Another random sample of 50 bags of flour is taken and a \(99 \%\) confidence interval for \(\mu\) is calculated.
2. Without calculation, state whether this confidence interval will be wider or narrower than the confidence interval found in part (i). Give a reason for your answer.

5

1. The masses, in grams, of certain tomatoes are normally distributed with standard deviation 9 grams. A random sample of 100 tomatoes has a sample mean of 63 grams. Find a \(90 \%\) confidence interval for the population mean mass of these tomatoes.
2. The masses, in grams, of certain potatoes are normally distributed with known population standard deviation but unknown population mean. A random sample of potatoes is taken in order to find a confidence interval for the population mean. Using a sample of size 50 , a \(95 \%\) confidence interval is found to have width 8 grams.
   1. Using another sample of size 50 , an \(\alpha \%\) confidence interval has width 4 grams. Find \(\alpha\).
   2. Find the sample size \(n\), such that a \(95 \%\) confidence interval has width 4 grams.

2 The times taken for customers' phone complaints to be handled were monitored regularly by a company. During a particular week a researcher checked a random sample of 20 complaints and the times, \(x\) minutes, taken to handle the complaints are summarised by \(\Sigma x = 337.5\). Handling times may be assumed to have a normal distribution with mean \(\mu\) minutes and standard deviation 3.8 minutes.

1. Calculate a \(98 \%\) confidence interval for \(\mu\). During the same week two other researchers each calculated a \(98 \%\) confidence interval for \(\mu\) based on independent samples.
2. Calculate the probability that at least one of the three intervals does not contain \(\mu\).
3. State two ways in which the calculation in part (i) would differ if the standard deviation were unknown.