5.05d Confidence intervals: using normal distribution

1 Anita carried out a survey of 140 randomly selected students at her college. She found that 49 of these students watched a TV programme called Bunch.

1. Calculate an approximate \(98 \%\) confidence interval for the proportion, \(p\), of students at Anita's college who watch Bunch.
   Carlos says that the confidence interval found in (a) is not useful because it is too wide.
2. Without calculation, explain briefly how Carlos can use the results of Anita's survey to find a narrower confidence interval for \(p\).

1 The lengths, \(X\) centimetres, of a random sample of 7 leaves from a certain variety of tree are as follows.
3.9
4.8
4.8
4.4
5.2
5.5
6.1

1. Calculate unbiased estimates of the population mean and variance of \(X\).
   It is now given that the true value of the population variance of \(X\) is 0.55 , and that \(X\) has a normal distribution.
2. Find a 95\% confidence interval for the population mean of \(X\).

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?

2 The manager of a video hire shop wishes to estimate the proportion of videos damaged by customers. He takes a random sample of 120 videos and finds that 33 of them are damaged. Find a \(95 \%\) confidence interval for the true proportion of videos that are being damaged when hired from this shop.

3 A consumer group, interested in the mean fat content of a particular type of sausage, takes a random sample of 20 sausages and sends them away to be analysed. The percentage of fat in each sausage is as follows. $$\begin{array} { l l l l l l l l l l l l l l l l l l l l } 26 & 27 & 28 & 28 & 28 & 29 & 29 & 30 & 30 & 31 & 32 & 32 & 32 & 33 & 33 & 34 & 34 & 34 & 35 & 35 \end{array}$$ Assume that the percentage of fat is normally distributed with mean \(\mu\), and that the standard deviation is known to be 3 .

1. Calculate a 98\% confidence interval for the population mean percentage of fat.
2. The manufacturer claims that the mean percentage of fat in sausages of this type is 30 . Use your answer to part (i) to determine whether the consumer group should accept this claim.

5 Sunita has a six-sided die with faces marked \(1,2,3,4,5,6\). The probability that the die shows a six on any throw is \(p\). Sunita throws the die 500 times and finds that it shows a six 70 times.

1. Calculate an approximate \(99 \%\) confidence interval for \(p\).
2. Sunita believes that the die is fair. Use your answer to part (a) to comment on her belief.
3. Sunita uses the result of her 500 throws to calculate an \(\alpha \%\) confidence interval for \(p\). This interval has width 0.04 . Find the value of \(\alpha\).

6 The heights, \(h\) centimetres, of a random sample of 100 fully grown animals of a certain species were measured. The results are summarised below. $$n = 100 \quad \Sigma h = 7570 \quad \Sigma h ^ { 2 } = 588050$$

1. Find unbiased estimates of the population mean and variance.
2. Calculate a \(99 \%\) confidence interval for the mean height of animals of this species.
   Four random samples were taken and a \(99 \%\) confidence interval for the population mean, \(\mu\), was found from each sample.
3. Find the probability that all four of these confidence intervals contain the true value of \(\mu\).

4 The masses, \(m\) kilograms, of flour in a random sample of 90 sacks of flour are summarised as follows. $$n = 90 \quad \Sigma m = 4509 \quad \Sigma m ^ { 2 } = 225950$$

1. Find unbiased estimates of the population mean and variance.
2. Calculate a \(98 \%\) confidence interval for the population mean.
3. Explain why it was necessary to use the Central Limit theorem in answering part (b).
4. Find the probability that the confidence interval found in part (b) is wholly above the true value of the population mean.

1 The diameters, \(x\) millimetres, of a random sample of 200 discs made by a certain machine were recorded. The results are summarised below. $$n = 200 \quad \Sigma x = 2520 \quad \Sigma x ^ { 2 } = 31852$$

1. Calculate a 95\% confidence interval for the population mean diameter.
2. Jean chose 40 random samples and used each sample to calculate a 95\% confidence interval for the population mean diameter. How many of these 40 confidence intervals would be expected to include the true value of the population mean diameter?

1

1. A javelin thrower noted the lengths of a random sample of 50 of her throws. The sample mean was 72.3 m and an unbiased estimate of the population variance was \(64.3 \mathrm {~m} ^ { 2 }\). Find a \(92 \%\) confidence interval for the population mean length of throws by this athlete.
2. A discus thrower wishes to calculate a \(92 \%\) confidence interval for the population mean length of his throws. He bases his calculation on his first 50 throws in a week. Comment on this method.

3 Batteries of type \(A\) are known to have a mean life of 150 hours. It is required to test whether a new type of battery, type \(B\), has a shorter mean life than type \(A\) batteries.

1. Give a reason for using a sample rather than the whole population in carrying out this test.
   A random sample of 120 type \(B\) batteries are tested and it is found that their mean life is 147 hours, and an unbiased estimate of the population variance is 225 hours \(^ { 2 }\).
2. Test, at the \(2 \%\) significance level, whether type \(B\) batteries have a shorter mean life than type \(A\) batteries.
3. Calculate a \(94 \%\) confidence interval for the population mean life of type \(B\) batteries.

4 A certain train journey takes place every day throughout the year. The time taken, in minutes, for the journey is normally distributed with variance 11.2.

1. The mean time for a random sample of \(n\) of these journeys was found. A \(94 \%\) confidence interval for the population mean time was calculated and was found to have a width of 1.4076 minutes, correct to 4 decimal places. Find the value of \(n\).
2. A passenger noted the times for 50 randomly chosen journeys in January, February and March. Give a reason why this sample is unsuitable for use in finding a confidence interval for the population mean time.
3. A researcher took 4 random samples and a \(94 \%\) confidence interval for the population mean was found from each sample. Find the probability that exactly 3 of these confidence intervals contain the true value of the population mean.

1 In a survey of 200 randomly chosen students from a certain college, 23\% of the students said that they owned a car. Calculate an approximate \(93 \%\) confidence interval for the proportion of students from the college who own a car.

3 In a random sample of 100 students at Luciana's college, \(x\) students said that they liked exams. Luciana used this result to find an approximate \(90 \%\) confidence interval for the proportion, \(p\), of all students at her college who liked exams. Her confidence interval had width 0.15792 .

1. Find the two possible values of \(x\).
   Suzma independently took another random sample and found another approximate \(90 \%\) confidence interval for \(p\).
2. Find the probability that neither of the two confidence intervals contains the true value of \(p\). [1]

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 A student wishes to estimate the proportion, \(p\), of students at her college who have exactly one brother. She surveys a random sample of 50 students at her college and finds that 18 of them have exactly one brother. She calculates an approximate \(\alpha \%\) confidence interval for \(p\) and finds that the lower limit of the confidence interval is 0.244 correct to 3 significant figures. Find \(\alpha\) correct to the nearest integer.

2 The widths, \(w \mathrm {~cm}\), of a random sample of 150 leaves of a certain kind were measured. The sample mean of \(w\) was found to be 3.12 cm . Using this sample, an approximate \(95 \%\) confidence interval for the population mean of the widths in centimetres was found to be [3.01, 3.23].

1. Calculate an estimate of the population standard deviation.
2. Explain whether it was necessary to use the Central Limit theorem in your answer to part (a). [1]

2 Lengths of a certain species of lizard are known to be normally distributed with standard deviation 3.2 cm . A naturalist measures the lengths of a random sample of 100 lizards of this species and obtains an \(\alpha \%\) confidence interval for the population mean. He finds that the total width of this interval is 1.25 cm . Find \(\alpha\).

1 A construction company notes the time, \(t\) days, that it takes to build each house of a certain design. The results for a random sample of 60 such houses are summarised as follows. $$\Sigma t = 4820 \quad \Sigma t ^ { 2 } = 392050$$

1. Calculate a 98\% confidence interval for the population mean time.
2. Explain why it was necessary to use the Central Limit theorem in part (a).

3 A random sample of 500 households in a certain town was chosen. Using this sample, a confidence interval for the proportion, \(p\), of all households in that town that owned two or more cars was found to be \(0.355 < p < 0.445\). Find the confidence level of this confidence interval. Give your answer correct to the nearest integer.

5 Mahmoud throws a coin 400 times and finds that it shows heads 184 times. The probability that the coin shows heads on any throw is denoted by \(p\).

1. Calculate an approximate \(95 \%\) confidence interval for \(p\).
2. Mahmoud claims that the coin is not fair. Use your answer to part (i) to comment on this claim.
3. Mahmoud's result of 184 heads in 400 throws gives an \(\alpha \%\) confidence interval for \(p\) with width 0.1 . Calculate the value of \(\alpha\).

3 A die is biased so that the probability that it shows a six on any throw is \(p\).

1. In an experiment, the die shows a six on 22 out of 100 throws. Find an approximate \(97 \%\) confidence interval for \(p\).
2. The experiment is repeated and another \(97 \%\) confidence interval is found. Find the probability that exactly one of the two confidence intervals includes the true value of \(p\).

3 Based on a random sample of 700 people living in a certain area, a confidence interval for the proportion, \(p\), of all people living in that area who had travelled abroad was found to be \(0.5672 < p < 0.6528\).

1. Find the proportion of people in the sample who had travelled abroad.
2. Find the confidence level of this confidence interval. Give your answer correct to the nearest integer.

3

1. Give a reason for using a sample rather than the whole population in carrying out a statistical investigation.
2. Tennis balls of a certain brand are known to have a mean height of bounce of 64.7 cm , when dropped from a height of 100 cm . A change is made in the manufacturing process and it is required to test whether this change has affected the mean height of bounce. 100 new tennis balls are tested and it is found that their mean height of bounce when dropped from a height of 100 cm is 65.7 cm and the unbiased estimate of the population variance is \(15 \mathrm {~cm} ^ { 2 }\).
   1. Calculate a \(95 \%\) confidence interval for the population mean.
   2. Use your answer to part (ii) (a) to explain what conclusion can be drawn about whether the change has affected the mean height of bounce.

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.