5.05d Confidence intervals: using normal distribution

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'.

7 In the past the weekly profit at a store had mean \(\\) 34600\( and standard deviation \)\\( 4500\). Following a change of ownership, the mean weekly profit for 90 randomly chosen weeks was \(\\) 35400$.

1. Stating a necessary assumption, test at the \(5 \%\) significance level whether the mean weekly profit has increased.
2. State, with a reason, whether it was necessary to use the Central Limit theorem in part (i). The mean weekly profit for another random sample of 90 weeks is found and the same test is carried out at the 5\% significance level.
3. State the probability of a Type I error.
4. Given that the population mean weekly profit is now \(\\) 36500$, calculate the probability of a Type II error.

2 A die is biased. The mean and variance of a random sample of 70 scores on this die are found to be 3.61 and 2.70 respectively. Calculate a \(95 \%\) confidence interval for the population mean score.

3 A die is thrown 100 times and shows an odd number on 56 throws. Calculate an approximate \(97 \%\) confidence interval for the probability that the die shows an odd number on one throw.

4 In the past, the time taken by vehicles to drive along a particular stretch of road has had mean 12.4 minutes and standard deviation 2.1 minutes. Some new signs are installed and it is expected that the mean time will increase. In order to test whether this is the case, the mean time for a random sample of 50 vehicles is found. You may assume that the standard deviation is unchanged.

1. The mean time for the sample of 50 vehicles is found to be 12.9 minutes. Test at the \(2.5 \%\) significance level whether the population mean time has increased.
2. State what is meant by a Type II error in this context.
3. State what extra piece of information would be needed in order to find the probability of a Type II error.

5 The masses, \(m\) grams, of a random sample of 80 strawberries of a certain type were measured and summarised as follows. $$n = 80 \quad \Sigma m = 4200 \quad \Sigma m ^ { 2 } = 229000$$

1. Find unbiased estimates of the population mean and variance.
2. Calculate a 98\% confidence interval for the population mean. 50 random samples of size 80 were taken and a \(98 \%\) confidence interval for the population mean, \(\mu\), was found from each sample.
3. Find the number of these 50 confidence intervals that would be expected to include the true value of \(\mu\).

5 The volumes, \(v\) millilitres, of juice in a random sample of 50 bottles of Cooljoos are measured and summarised as follows. $$n = 50 \quad \Sigma v = 14800 \quad \Sigma v ^ { 2 } = 4390000$$

1. Find unbiased estimates of the population mean and variance.
2. An \(\alpha \%\) confidence interval for the population mean, based on this sample, is found to have a width of 5.45 millilitres. Find \(\alpha\). Four random samples of size 10 are taken and a \(96 \%\) confidence interval for the population mean is found from each sample.
3. Find the probability that these 4 confidence intervals all include the true value of the population mean.

2 A six-sided die is suspected of bias. The die is thrown 100 times and it is found that the score is 2 on 20 throws. It is given that the probability of obtaining a score of 2 on any throw is \(p\).

1. Find an approximate \(94 \%\) confidence interval for \(p\).
2. Use your answer to part (i) to comment on whether the die may be biased.

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.

6 Ramesh plans to carry out a survey in order to find out what adults in his town think about local sports facilities. He chooses a random sample from the adult members of a tennis club and gives each of them a questionnaire.

1. Give a reason why this will not result in Ramesh having a random sample of adults who live in the town.
2. Describe briefly a valid method that Ramesh could use to choose a random sample of adults in the town.
   Ramesh now uses a valid method to choose a random sample of 350 adults from the town. He finds that 47 adults think that the local sports facilities are good.
3. Calculate an approximate \(90 \%\) confidence interval for the proportion of all adults in the town who think that the local sports facilities are good.
4. Ramesh calculates a confidence interval whose width is 1.25 times the width of this \(90 \%\) confidence interval. Ramesh's new interval is an \(x \%\) confidence interval. Find the value of \(x\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

5 The manufacturer of a certain type of biscuit claims that \(10 \%\) of packets include a free offer printed on the packet. Jyothi suspects that the true proportion is less than \(10 \%\). He plans to test the claim by looking at 40 randomly selected packets and, if the number which include the offer is less than 2 , he will reject the manufacturer's claim.

1. State suitable hypotheses for the test.
2. Find the probability of a Type I error.
   On another occasion Jyothi looks at 80 randomly selected packets and finds that exactly 6 include the free offer.
3. Calculate an approximate \(90 \%\) confidence interval for the proportion of packets that include the offer.
4. Use your confidence interval to comment on the manufacturer's claim. \(6 X\) is a random variable with probability density function given by $$f ( x ) = \begin{cases} \frac { a } { x ^ { 2 } } & 1 \leqslant x \leqslant b \\ 0 & \text { otherwise } \end{cases}$$ where \(a\) and \(b\) are constants.

1 A coin is thrown 100 times and it shows heads 60 times. Calculate an approximate \(98 \%\) confidence interval for the probability, \(p\), that the coin shows heads on any throw.

5 The 150 oranges in a random sample from a certain supplier were weighed and the masses, \(X\) grams, were recorded. The results are summarised below. $$n = 150 \quad \Sigma x = 14910 \quad \Sigma x ^ { 2 } = 1525000$$

1. Calculate a \(99 \%\) confidence interval for the population mean of \(X\).
2. The supplier claims that the mean mass of his oranges is 100 grams. Use your answer to part (i) to explain whether this claim should be accepted.
3. State briefly why the sample should be random.

1 In a survey, 36 out of 120 randomly selected voters in Hungton said that if there were an election next week they would vote for the Alpha party. Calculate an approximate \(90 \%\) confidence interval for the proportion of voters in Hungton who would vote for the Alpha party.

2 A random sample of 250 people living in Barapet was chosen. It was found that 78 of these people owned a BETEC phone.

1. Calculate an approximate \(98 \%\) confidence interval for the proportion of people living in Barapet who own a BETEC phone.
2. Manjit claims that more than \(40 \%\) of the people living in Barapet own a BETEC phone. Use your answer to part (a) to comment on this claim.

4 The lengths, in millimetres, of rods produced by a machine are normally distributed with mean \(\mu\) and standard deviation 0.9. A random sample of 75 rods produced by the machine has mean length 300.1 mm .

1. Find a \(99 \%\) confidence interval for \(\mu\), giving your answer correct to 2 decimal places.
   The manufacturer claims that the machine produces rods with mean length 300 mm .
2. Use the confidence interval found in part (i) to comment on this claim.

2 The number of words in History essays by students at a certain college has mean \(\mu\) and standard deviation 1420.

1. The mean number of words in a random sample of 125 History essays was found to be 4820 . Calculate a \(98 \%\) confidence interval for \(\mu\).
2. Another random sample of \(n\) History essays was taken. Using this sample, a \(95 \%\) confidence interval for \(\mu\) was found to be 4700 to 4980 , both correct to the nearest integer. Find the value of \(n\).

1 The standard deviation of the heights of adult males is 7.2 cm . The mean height of a sample of 200 adult males is found to be 176 cm .

1. Calculate a \(97.5 \%\) confidence interval for the mean height of adult males.
2. State a necessary condition for the calculation in part (i) to be valid.

2 The standard deviation of the volume of drink in cans of Koola is 4.8 centilitres. A random sample of 180 cans is taken and the mean volume of drink in these 180 cans is found to be 330.1 centilitres.

1. Calculate a \(95 \%\) confidence interval for the mean volume of drink in all cans of Koola. Give the end-points of your interval correct to 1 decimal place.
2. Explain whether it was necessary to use the Central Limit theorem in your answer to part (i).

6 In the past, Angus found that his train was late on \(15 \%\) of his daily journeys to work. Following a timetable change, Angus found that out of 60 randomly chosen days, his train was late on 6 days.

1. Test at the \(10 \%\) significance level whether Angus' train is late less often than it was before the timetable change.
   Angus used his random sample to find an \(\alpha \%\) confidence interval for the proportion of days on which his train is late. The upper limit of his interval was 0.150 , correct to 3 significant figures.
2. Calculate the value of \(\alpha\) correct to the nearest integer.

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.

3 The times, in minutes, taken by competitors to complete a puzzle have mean \(\mu\) and standard deviation 3 . The times taken by a random sample of 10 competitors are noted and the results are given below. \(\begin{array} { l l l } 25.2 & 26.8 & 18.5 \end{array}\) 25.5
30.1 \(28.9 \quad 27.0\) \(26.1 \quad 26.0\) 24.9

1. Stating a necessary assumption, calculate a \(97 \%\) confidence interval for \(\mu\).
2. Two more random samples, each of 10 competitors, are taken. Their times are used to calculate two more \(97 \%\) confidence intervals for \(\mu\). Find the probability that neither of these intervals contains the true value of \(\mu\).

7 Bob is a self-employed builder. In the past his weekly income had mean \(\\) 546\( and standard deviation \)\\( 120\). Following a change in Bob's working pattern, his mean weekly income for 40 randomly chosen weeks was \(\\) 581\(. You should assume that the standard deviation remains unchanged at \)\\( 120\).

1. Test at the \(2.5 \%\) significance level whether Bob's mean weekly income has increased.
   Bob finds his mean weekly income for another random sample of 40 weeks and carries out a similar test at the \(2.5 \%\) significance level.
2. Given that Bob's mean weekly income is now in fact \(\\) 595$, find the probability of a Type II error.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

2 The mean and standard deviation of the time spent by people in a certain library are 29 minutes and 6 minutes respectively.

1. Find the probability that the mean time spent in the library by a random sample of 120 people is more than 30 minutes.
2. Explain whether it was necessary to assume that the time spent by people in the library is normally distributed in the solution to part (i).

3 Jagdeesh measured the lengths, \(x\) minutes, of 60 randomly chosen lectures. His results are summarised below.

1. Calculate unbiased estimates of the population mean and variance.
2. Calculate a \(98 \%\) confidence interval for the population mean.