Calculate CI from summary stats

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 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. 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.

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.

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 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.

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.

4

1. Give a reason why, in carrying out a statistical investigation, a sample rather than a complete population may be used.
2. Rose wishes to investigate whether men in her town have a different life-span from the national average of 71.2 years. She looks at government records for her town and takes a random sample of the ages of 110 men who have died recently. Their mean age in years was 69.3 and the unbiased estimate of the population variance was 65.61.
   1. Calculate a \(90 \%\) confidence interval for the population mean and explain what you understand by this confidence interval.
   2. State with a reason what conclusion about the life-span of men in her town Rose could draw from this confidence interval.

3 The masses of sweets produced by a machine are normally distributed with mean \(\mu\) grams and standard deviation 1.0 grams. A random sample of 65 sweets produced by the machine has a mean mass of 29.6 grams.

1. Find a \(99 \%\) confidence interval for \(\mu\). The manufacturer claims that the machine produces sweets with a mean mass of 30 grams.
2. Use the confidence interval found in part (i) to draw a conclusion about this claim.
3. Another random sample of 65 sweets produced by the machine is taken. This sample gives a \(99 \%\) confidence interval that leads to a different conclusion from that found in part (ii). Assuming that the value of \(\mu\) has not changed, explain how this can be possible.

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.

1 The times taken for students at a college to run 200 m have a normal distribution with mean \(\mu \mathrm { s }\). The times, \(x\) s, are recorded for a random sample of 10 students from the college. The results are summarised as follows, where \(\bar { x }\) is the sample mean. $$\bar { x } = 25.6 \quad \sum ( x - \bar { x } ) ^ { 2 } = 78.5$$

1. Find a 90\% confidence interval for \(\mu\).
   A test of the null hypothesis \(\mu = k\) is carried out on this sample, using a \(10 \%\) significance level. The test does not support the alternative hypothesis \(\mu < k\).
2. Find the greatest possible value of \(k\).

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.

3 Intensity of light is measured in lumens. The random variable \(X\) represents the intensity of the light from a standard 100 watt light bulb. \(X\) is Normally distributed with mean 1720 and standard deviation 90. You may assume that the intensities for different bulbs are independent.

1. Show that \(\mathrm { P } ( X < 1700 ) = 0.4121\).
2. These bulbs are sold in packs of 4 . Find the probability that the intensities of exactly 2 of the 4 bulbs in a randomly chosen pack are below 1700 lumens.
3. Use a suitable approximating distribution to find the probability that the intensities of at least 20 out of 40 randomly selected bulbs are below 1700 lumens. A manufacturer claims that the average intensity of its 25 watt low energy light bulbs is 1720 lumens. A consumer organisation suspects that the true figure may be lower than this. The intensities of a random sample of 20 of these bulbs are measured. A hypothesis test is then carried out to check the claim.
4. Write down a suitable null hypothesis and explain briefly why the alternative hypothesis should be \(\mathrm { H } _ { 1 } : \mu < 1720\). State the meaning of \(\mu\).
5. Given that the standard deviation of the intensity of such bulbs is 90 lumens and that the mean intensity of the sample of 20 bulbs is 1703 lumens, carry out the test at the \(5 \%\) significance level.

2. Krishi owns a farm on which he keeps chickens. He selects, at random, 10 of the eggs produced and weighs each of them.
You may assume that these weights are a random sample from a normal distribution with standard deviation 1.9 g The total weight of these 10 eggs is 537.2 g

1. Find a \(95 \%\) confidence interval for the mean weight of the eggs produced by Krishi's chickens. Krishi was hoping to obtain a \(99 \%\) confidence interval of width at most 1.5 g
2. Calculate the minimum sample size necessary to achieve this. \includegraphics[max width=\textwidth, alt={}, center]{fc43aabf-ad04-4852-8539-981cef608f31-04_2662_95_107_1962}

6 A garden centre sells bags of stones and large bags of gravel.
The weight, \(X\) kilograms, of stones in a bag can be modelled by a normal distribution with unknown mean \(\mu\) and known standard deviation 0.4 The stones in each of a random sample of 36 bags from a large batch is weighed. The total weight of stones in these 36 bags is found to be 806.4 kg

1. Find a 98\% confidence interval for the mean weight of stones in the batch.
2. Explain why the use of the Central Limit theorem is not required to answer part (a) The manufacturer of these bags of stones claims that bags in this batch have a mean weight of 22.5 kg
3. Using your answer to part (a), comment on the claim made by the manufacturer. The weight, \(Y\) kilograms, of gravel in a large bag can be modelled by a normal distribution with mean 850 kg and standard deviation 5 kg A builder purchases 10 large bags of gravel.
4. Find the probability that the mean weight of gravel in the 10 large bags is less than 848 kg

1. Assam produces bags of flour. The stated weight printed on the bags of flour is 3 kg . The weights of the bags of flour are normally distributed with standard deviation 0.015 kg .

Assam weighs a random sample of 9 bags of flour and finds their mean weight is 2.977 kg .

1. Calculate the \(99 \%\) confidence interval for the mean weight of a bag of flour. Give your limits to 3 decimal places. Assam decides to increase the amount of flour put into the bags.
2. Explain why the confidence interval has led Assam to take this action. After the increase a random sample of \(n\) bags of flour is taken. The sample mean weight of these \(n\) bags is 2.995 kg . A \(95 \%\) confidence interval for \(\mu\) gave a lower limit of less than 2.991 kg .
3. Find the maximum value of \(n\).
   VILV SIHI NI IIII M I ON OC
   VIAV SIHI NI III IM I ON OO
   VIAV SIHI NI III HM ION OC

2. A random sample of 30 apples was taken from a batch. The mean weight of the sample was 124 g with standard deviation 20 g .

1. Find a \(99 \%\) confidence interval for the mean weight \(\mu\) grams of the population of apples. Write down any assumptions you made in your calculations. Given that the actual value of \(\mu\) is 140 ,
2. state, with a reason, what you can conclude about the sample of 30 apples.

1. The waiting times, in minutes, of patients at a doctor's surgery follows a normal distribution with unknown mean \(\mu\) and known standard deviation \(\sigma\)

A random sample of 120 patients was taken.

1. Find, in the form \(k \sigma\), the width of a \(99 \%\) confidence interval for \(\mu\) based on this sample. Give the value of \(k\) to 2 decimal places. A further random sample of 100 patients from the surgery gave a \(90 \%\) confidence interval for \(\mu\) of \(( 5.14,6.25 )\)
2. Use this confidence interval to determine whether or not it provides evidence that \(\mu = 6\) State the hypotheses being tested here and write down the significance level being used. You do not need to carry out any further calculations.
3. Find the value of \(\sigma\)

4 Rice that can be cooked in microwave ovens is sold in packets which the manufacturer claims contain a mean weight of more than 250 grams of rice. The weight of rice in a packet may be modelled by a normal distribution. A consumer organisation's researcher weighed the contents, \(x\) grams, of each of a random sample of 50 packets. Her summarised results are: $$\bar { x } = 251.1 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 184.5$$

1. Show that, correct to two decimal places, \(s = 1.94\), where \(s ^ { 2 }\) denotes the unbiased estimate of the population variance.
2. 1. Construct a \(96 \%\) confidence interval for the mean weight of rice in a packet, giving the limits to one decimal place.
   2. Hence comment on the manufacturer's claim.
3. The statement '250 grams' is printed on each packet. Explain, with reference to the values of \(\bar { x }\) and \(s\), why the consumer organisation may consider this statement to be dubious.

3. A random variable \(X\) is distributed normally with a standard deviation of 6.8 Sixty observations of \(X\) are made and found to have a mean of 31.4

1. Find a 90\% confidence interval for the mean of \(X\).
2. How many observations of \(X\) would be needed in order to obtain a \(90 \%\) confidence interval for the mean of \(X\) with a width of less than 1.5
   (5 marks)

7 A swimming coach believes that times recorded by people using stopwatches are on average 0.2 seconds faster than those recorded by an electronic timing system. In order to test this, the coach takes a random sample of 40 competitors' times recorded by both methods, and finds the differences between the times recorded by the two methods. The mean difference in the times (electronic time minus stopwatch time) is 0.1442 s and the standard deviation of the differences is 0.2580 s .

1. Find a 95\% confidence interval for the mean difference between electronic and stopwatch times.
2. Explain whether there is evidence to suggest that the coach's belief is correct.
3. Explain how you can calculate the confidence interval in part (a) even though you do not know the distribution of the parent population of differences.
4. If the coach wanted to produce a \(95 \%\) confidence interval of width no more than 0.12 s , what is the minimum sample size that would be needed, assuming that the standard deviation remains the same?