Interpret confidence interval

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 \(15 \mathrm {~cm} ^ { 2 }\).
   (a) Calculate a \(95 \%\) confidence interval for the population mean.
   (b) 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.

4 A doctor wishes to investigate the mean fat content in low-fat burgers. He takes a random sample of 15 burgers and sends them to a laboratory where the mass, in grams, of fat in each burger is determined. The results are as follows. $$\begin{array} { l l l l l l l l l l l l l l l } 9 & 7 & 8 & 9 & 6 & 11 & 7 & 9 & 8 & 9 & 8 & 10 & 7 & 9 & 9 \end{array}$$ Assume that the mass, in grams, of fat in low-fat burgers is normally distributed with mean \(\mu\) and that the population standard deviation is 1.3 .

1. Calculate a \(99 \%\) confidence interval for \(\mu\).
2. Explain whether it was necessary to use the Central Limit theorem in the calculation in part (i).
3. The manufacturer claims that the mean mass of fat in burgers of this type is 8 g . Use your answer to part (i) to comment on this claim.

2 The lengths of a random sample of 50 roads in a certain region were measured.Using the results,a \(95 \%\) confidence interval for the mean length,in metres,of all roads in this region was found to be[245,263].

1. Find the mean length of the 50 roads in the sample.
2. Calculate an estimate of the standard deviation of the lengths of roads in this region.
3. It is now given that the lengths of roads in this region are normally distributed.
   State,with a reason,whether this fact would make any difference to your calculation in part(b).

2 Dipak carries out a test, at the \(10 \%\) significance level, using a normal distribution. The null hypothesis is \(\mu = 35\) and the alternative hypothesis is \(\mu \neq 35\).

1. Is this a one-tail or a two-tail test? State briefly how you can tell. Dipak finds that the value of the test statistic is \(z = - 1.750\).
2. Explain what conclusion he should draw.
3. This result is significant at the \(\alpha \%\) level. Find the smallest possible value of \(\alpha\), correct to the nearest whole number.

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.

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.

2 A six-sided die has faces marked \(1,2,3,4,5,6\). When the die is thrown 300 times it shows a six on 56 throws.

1. Calculate an approximate \(96 \%\) confidence interval for the probability that the die shows a six on one throw.
2. Maroulla claims that the die is biased. 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.

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.
   (a) Calculate a \(90 \%\) confidence interval for the population mean and explain what you understand by this confidence interval.
   (b) 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.

4 The volumes of juice in bottles of Apricola are normally distributed. In a random sample of 8 bottles, the volumes of juice, in millilitres, were found to be as follows. $$\begin{array} { l l l l l l l l } 332 & 334 & 330 & 328 & 331 & 332 & 329 & 333 \end{array}$$

1. Find unbiased estimates of the population mean and variance. A random sample of 50 bottles of Apricola gave unbiased estimates of 331 millilitres and 4.20 millilitres \({ } ^ { 2 }\) for the population mean and variance respectively.
2. Use this sample of size 50 to calculate a \(98 \%\) confidence interval for the population mean.
3. The manufacturer claims that the mean volume of juice in all bottles is 333 millilitres. State, with a reason, whether your answer to part (ii) supports this claim.

6 A variable \(X\) takes values \(1,2,3,4,5\), and these values are generated at random by a machine. Each value is supposed to be equally likely, but it is suspected that the machine is not working properly. A random sample of 100 values of \(X\), generated by the machine, gives the following results. $$n = 100 \quad \Sigma x = 340 \quad \Sigma x ^ { 2 } = 1356$$

1. Find a 95\% confidence interval for the population mean of the values generated by the machine.
2. Use your answer to part (i) to comment on whether the machine may be working properly.

6 An anthropologist was studying the inhabitants of two islands, Raloa and Tangi. Part of the study involved the incidence of blood group type A. The blood of 80 randomly chosen inhabitants of Raloa and 85 randomly chosen inhabitants of Tangi was tested. The number of inhabitants with type A blood was 28 for the Raloa sample and 46 for the Tangi sample. The anthropologist calculated \(90 \%\) confidence intervals for the population proportions of inhabitants with type A blood. They were \(( 0.262,0.438 )\) for Raloa and \(( 0.452,0.630 )\) for Tangi, where each figure is correct to 3 decimal places. It is known that \(43 \%\) of the world's population have type A blood.

1. State, giving your reasons, whether there is evidence for the following assertions about the proportions of people with type A blood.
   (a) The proportion in Raloa is different from the world proportion.
   (b) The proportion in Tangi is different from the world proportion.
2. Carry out a suitable test, at the \(2 \%\) significance level, of whether the proportions of people with type A blood differ on the two islands.

The time taken for a randomly chosen student at College \(P\) to complete a particular puzzle has a normal distribution with mean \(\mu\) minutes. The times, \(x\) minutes, are recorded for a random sample of 8 students chosen from the college. The results are summarised as follows. $$\Sigma x = 42.8 \quad \Sigma x ^ { 2 } = 236.0$$ Find a 95\% confidence interval for \(\mu\). A test is carried out on this sample data, at the \(10 \%\) significance level. The test supports the claim that \(\mu > k\). Find the greatest possible value of \(k\). A random sample, of size 12, is taken from the students at College \(Q\). Their times to complete the puzzle give a sample mean of 4.60 minutes and an unbiased variance estimate of 1.962 minutes \({ } ^ { 2 }\). Use a 2 -sample test at the \(10 \%\) significance level to test whether the mean time for students at College \(Q\) to complete the puzzle is less than the mean time for students at College \(P\) to complete the puzzle. You should state any assumptions necessary for the test to be valid.

7 The pulse rate of each member of a random sample of 25 adult UK males who exercise for a given period each week is measured in beats per minute. A \(98 \%\) confidence interval for the mean pulse rate, \(\mu\) beats per minute, for all such UK males was calculated as \(61.21 < \mu < 64.39\), based on a \(t\)-distribution.

1. Calculate the sample mean pulse rate and the standard deviation used in the calculation.
2. State an assumption necessary for the validity of the confidence interval.
3. The mean pulse rate for all UK males is 72 beats per minute. State, giving a reason, if it can be concluded that, on average, UK males who exercise have a reduced pulse rate.

12 A retailer sells bags of flour which are advertised as containing 1.5 kg of flour. A trading standards officer is investigating whether there is enough flour in each bag. He collects a random sample and uses software to carry out a hypothesis test at the \(5 \%\) level. The analysis is shown in the software printout below.

1. State the hypotheses the officer uses in the test, defining any parameters used.
2. State the distribution used in the analysis.
3. Carry out the hypothesis test, giving your conclusion in context.

1. The volume of water in a bottle has a normal distribution with unknown mean, \(\mu\) millilitres, and known standard deviation, \(\sigma\) millilitres.

A random sample of 150 of the bottles of water gave a 95\% confidence interval for \(\mu\) of
(327.84, 329.76)

1. Using the confidence interval given, test whether or not \(\mu = 328\) State your hypotheses clearly and write down the significance level you have used. A second random sample, of 200 of these bottles of water, had a mean volume of 328 millilitres.
2. Calculate a 98\% confidence interval for \(\mu\) based on this second sample. You must show all steps in your working.
   (Solutions relying entirely on calculator technology are not acceptable.) Using five different random samples of 200 of these bottles of water, five \(98 \%\) confidence intervals for \(\mu\) are to be found.
3. Calculate the probability that more than 3 of these intervals will contain \(\mu\)

6

1. A district council claimed that more than 80 per cent of the complaints that it received about the delivery of its services were answered to the satisfaction of complainants before reaching formal status. An analysis of a random sample of 175 complaints revealed that 28 reached formal status.
   1. Construct an approximate \(95 \%\) confidence interval for the proportion of complaints that reach formal status.
   2. Hence comment on the council's claim.
2. The district council also claimed that less than 40 per cent of all formal complaints were due to a failing in the delivery of its services. An analysis of the 50 formal complaints received during 2007/08 showed that 16 were due to a failing in the delivery of its services.
   1. Using an exact test, investigate the council's claim at the \(10 \%\) level of significance. The 50 formal complaints received during 2007/08 may be assumed to be a random sample.
   2. Determine the critical value for your test in part (b)(i).
   3. In fact, only 25 per cent of all formal complaints were due to a failing in the delivery of the council's services. Determine the probability of a Type II error for a test of the council's claim at the \(10 \%\) level of significance and based on the analysis of a random sample of 50 formal complaints.
      (4 marks)
      \includegraphics[max width=\textwidth, alt={}]{b855b5b3-097e-4894-aaec-d77f515949b0-15_2484_1709_223_153}
      \includegraphics[max width=\textwidth, alt={}]{b855b5b3-097e-4894-aaec-d77f515949b0-16_2484_1712_223_153}
      \includegraphics[max width=\textwidth, alt={}]{b855b5b3-097e-4894-aaec-d77f515949b0-17_2484_1709_223_153}

4. A town council is concerned that the mean price of renting two bedroom flats in the town has exceeded \(\pounds 650\) per month. A random sample of eight two bedroom flats gave the following results, \(\pounds x\), per month. $$705 , \quad 640 , \quad 560 , \quad 680 , \quad 800 , \quad 620 , \quad 580 , \quad 760$$ [You may assume \(\sum x = 5345 \quad \sum x ^ { 2 } = 3621025\) ]

1. Find a 90\% confidence interval for the mean price of renting a two bedroom flat.
2. State an assumption that is required for the validity of your interval in part (a).
3. Comment on whether or not the town council is justified in being concerned. Give a reason for your answer.

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?

5 Amari is investigating how accurately people can estimate a short time period. He asks each of a random sample of 40 people to estimate a period of 20 seconds. For each person, he starts a stopwatch and then stops it when they tell him that they think that 20 s has elapsed. The times which he records are denoted by \(x \mathrm {~s}\). You are given that \(\sum x = 765 , \quad \sum x ^ { 2 } = 15065\).

1. Determine a 95\% confidence interval for the mean estimated time.
2. Amari says that the confidence interval supports the suggestion that people can estimate 20 s accurately. Make two comments about Amari's statement.
3. Discuss whether you could have constructed the confidence interval if there had only been 10 people involved in the experiment. Amari thinks that people would be able to estimate more accurately if he gave them a second attempt. He repeats the experiment with each person and again records the times. Software is used to produce a \(95 \%\) confidence interval for the mean estimated time. The output from the software is shown below. Z Estimate of a Mean Confidence level 0.95 Sample

   Mean	19.68
   s	1.38
   N	40

   Result
   Z Estimate of a Mean

   Mean	19.68
   s	1.38
   SE	0.2182
   N	40
   Interval	\(19.68 \pm 0.4277\)

4. State the confidence interval in the form \(\mathrm { a } < \mu < \mathrm { b }\).
5. Make two comments based on this confidence interval about Amari's opinion that second attempts result in more accurate estimates.

7 An analyst routinely examines bottles of hair shampoo in order to check that the average percentage of a particular chemical which the shampoo contains does not exceed the value of \(1.0 \%\) specified by the manufacturer. The percentages of the chemical in a random sample of 12 bottles of the shampoo are as follows. \(\begin{array} { l l l l l l l l l l l } 1.087 & 1.171 & 1.047 & 0.846 & 0.909 & 1.052 & 1.042 & 0.893 & 1.021 & 1.085 & 1.096 \end{array} 0.931\) The analyst uses software to draw a Normal probability plot for these data, and to carry out a Normality test as shown below. \includegraphics[max width=\textwidth, alt={}, center]{c692fb20-436f-4bc1-89bd-10fdba41ceba-08_524_1539_694_264}

1. The analyst is going to carry out a hypothesis test to check whether the average percentage exceeds 1.0\%. Explain which test the analyst should use, referring to each of the following.
   • The Normal probability plot
   • The \(p\)-value of the Kolmogorov-Smirnov test
   • In this question you must show detailed reasoning.
   Carry out the test at the 5\% significance level.

16

1. The graph below shows the amount of salt, in grams, purchased per person per week in England between 2001-02 and 2014, based upon the Large Data Set. \includegraphics[max width=\textwidth, alt={}, center]{c8a41c47-bbda-4e91-a7a2-d0bcf6a46f25-22_821_1349_406_347} Meera and Gemma are arguing about what this graph shows.
   Meera believes that the amount of salt consumed by people decreased greatly during this period. Gemma says that this is not the case.
   Using your knowledge of the Large Data Set, give two reasons why Gemma may be correct.
   [0pt] [2 marks]
   16
2. It is known that the mean amount of sugar purchased per person in England in 2014 was 78.9 grams, with a standard deviation of 25.0 grams. In 2018, a sample of 918 people had a mean of 80.4 grams of sugar purchased per person. Investigate, at the \(5 \%\) level of significance, whether the mean amount of sugar purchased per person in England has changed between 2014 and 2018. Assume that the survey data is a random sample taken from a normal distribution and that the standard deviation has remained the same.
   16
3. Another test is performed to determine whether the mean amount of fat purchased per person has changed between 2014 and 2018. At the \(10 \%\) significance level, the null hypothesis is rejected.
   With reference to the \(10 \%\) significance level, explain why it is not necessarily true that there has been a change.
   [0pt] [2 marks] \includegraphics[max width=\textwidth, alt={}, center]{c8a41c47-bbda-4e91-a7a2-d0bcf6a46f25-25_2488_1716_219_153}