5.05d Confidence intervals: using normal distribution

1. A six-sided die has sides labelled \(1,2,3,4,5\) and 6

The random variable \(S\) represents the score when the die is rolled.
Alicia rolls the die 45 times and the mean score, \(\bar { S }\), is calculated.
Assuming the die is fair and using a suitable approximation,

1. find, to 3 significant figures, the value of \(k\) such that \(\mathrm { P } ( \bar { S } < k ) = 0.05\)
2. Explain the relevance of the Central Limit Theorem in part (a). Alicia considers the following hypotheses: \(\mathrm { H } _ { 0 }\) : The die is fair \(\mathrm { H } _ { 1 }\) : The die is not fair
   If \(\bar { S } < 3.1\) or \(\bar { S } > 3.9\), then \(\mathrm { H } _ { 0 }\) will be rejected.
   Given that the true distribution of \(S\) has mean 4 and variance 3
3. find the power of this test.
4. Describe what would happen to the power of this test if Alicia were to increase the number of rolls of the die.
   Give a reason for your answer.

A machine fills bags with flour. The weight of flour delivered by the machine into a bag, \(X\) grams, is normally distributed with mean \(\mu\) grams and standard deviation 30 grams. To check if there is any change to the mean weight of flour delivered by the machine into each bag, Olaf takes a random sample of 10 bags. The weight of flour, \(x\) grams, in each bag is recorded and \(\bar { x } = 1020\)

1. Test, at the \(5 \%\) level of significance, \(\mathrm { H } _ { 0 } : \mu = 1000\) against \(\mathrm { H } _ { 1 } : \mu \neq 1000\) Olaf decides to alter the test so that the hypotheses are \(\mathrm { H } _ { 0 } : \mu = 1000\) and \(\mathrm { H } _ { 1 } : \mu > 1000\) but keeps the level of significance at 5\% He takes a second sample of size \(n\) and finds the critical region, \(\bar { X } > c\)
2. Find an equation for \(c\) in terms of \(n\) When the true value of \(\mu\) is 1020 grams, the probability of making a Type II error is 0.0050 , to 2 significant figures.
3. Calculate the value of \(n\) and the value of \(c\)

1. A machine fills cartons with juice.

The amount of juice in a carton is normally distributed with mean \(\mu \mathrm { ml }\) and standard deviation 8 ml . A manager wants to test whether or not the amount of juice in the cartons, \(X \mathrm { ml }\), is less than 330 ml . The manager takes a random sample of 25 cartons of juice and calculates the mean amount of juice \(\bar { x } \mathrm { ml }\).

1. Using a \(5 \%\) level of significance, find the critical region of \(\bar { X }\) for this test. State your hypotheses clearly. The Director is concerned about the machine filling the cartons with more than 330 ml of juice as well as less than 330 ml of juice. The Director takes a sample of 55 cartons, records the mean amount of juice \(\bar { y } \mathrm { ml }\) and uses a test with a critical region of $$\{ \bar { Y } < 328 \} \cup \{ \bar { Y } > 332 \}$$
2. Find P (Type I error) for the Director's test. When \(\mu = 325 \mathrm { ml }\)
3. find P (Type II error) for the test in part (a)

1 A machine is set to fill pots with yoghurt such that the mean weight of yoghurt in a pot is 505 grams. To check that the machine is working properly, a random sample of 8 pots is selected. The weight of yoghurt, in grams, in each pot is as follows $$\begin{array} { l l l l l l l l } 508 & 510 & 500 & 500 & 498 & 503 & 508 & 505 \end{array}$$ Given that the weights of the yoghurt delivered by the machine follow a normal distribution with standard deviation 5.4 grams,

1. find a \(95 \%\) confidence interval for the mean weight, \(\mu\) grams, of yoghurt in a pot. Give your answers to 2 decimal places.
2. Comment on whether or not the machine is working properly, giving a reason for your answer.
3. State the probability that a \(95 \%\) confidence interval for \(\mu\) will not contain \(\mu\) grams.
4. Without carrying out any further calculations, explain the changes, if any, that would need to be made in calculating the confidence interval in part (a) if the standard deviation was unknown. Give a reason for your answer.
   You may assume that the weights of the yoghurt delivered by the machine still follow a normal distribution.

2 Jemima makes jam to sell in a local shop. The jam is sold in jars and the weight of jam in a jar is normally distributed. Jemima takes a random sample of 8 of her jars of jam and weighs the contents of each jar, \(x\) grams. Her results are summarised as follows $$\sum x = 3552 \quad \sum x ^ { 2 } = 1577314$$

1. Calculate a 95\% confidence interval for the mean weight of jam in a jar. The labels on the jars state that the average contents weigh 440 grams.
2. State, giving a reason, whether or not Jemima should be concerned about the labels on her jars of jam.

6 A new employee, Kim, joins an existing employee, Jiang, to work in the quality control department of a company producing steel rods.
Each day a random sample of rods is taken, their lengths measured and a \(95 \%\) confidence interval for the mean length of the rods, in metres, is calculated. It is assumed that the lengths of the rods produced are normally distributed. Kim took a random sample of 25 rods and used the \(t\) distribution to obtain a \(95 \%\) confidence interval of \(( 1.193,1.367 )\) for the mean length of the rods. Jiang commented that this interval was a little wider than usual and explained that they usually assume that the standard deviation does not change and can be taken as 0.175 metres.

1. Test, at the \(10 \%\) level of significance, whether or not Kim's sample suggests that the standard deviation is different from 0.175 metres. State your hypotheses clearly. Using Kim's sample and the normal distribution with a standard deviation of 0.175 metres, (b) find a 95\% confidence interval for the mean length of the rods.

1. Elsa is collecting information on the wingspan of two different species of butterfly, Ringlet and Meadow Brown. She takes a random sample of each type of butterfly. The wingspans, \(w \mathrm {~cm}\), are summarised in the table below. The wingspans of Ringlet and Meadow Brown butterflies each follow normal distributions.

1. Test, at the \(2 \%\) level of significance, whether or not there is evidence that the variance of the wingspans of Ringlet butterflies is different from the variance of the wingspans of Meadow Brown butterflies. You should state your hypotheses clearly. The \(k \%\) confidence interval for the variance of the wingspans of Meadow Brown butterflies is (1.194, 48.54)
2. Find the value of \(k\)
3. Calculate a \(95 \%\) confidence interval for the difference between the mean wingspan of the Ringlet butterfly and the mean wingspan of the Meadow Brown butterfly.

1. A factory produces yellow tennis balls and white tennis balls. Independent samples, one of yellow tennis balls and one of white tennis balls, are taken. The table shows information about the weights of the yellow tennis balls, \(Y\) grams, and the weights of the white tennis balls, \(W\) grams.

1. Find a 95\% confidence interval for the mean weight of yellow tennis balls. Jamie claims that the mean weight of the population of yellow tennis balls is greater than the mean weight of the population of white tennis balls. A test of Jamie's claim is carried out.
2. 1. Specify the approximate distribution of \(\bar { Y } - \bar { W }\) under the null hypothesis of the test.
   2. Explain the relevance of the large sample sizes to your answer to part (i).
3. Complete the hypothesis test using a \(5 \%\) level of significance. You should state your hypotheses and the value of your test statistic clearly.

1. A doctor believes that a four-week exercise programme can reduce the resting heart rate of her patients. She takes a random sample of 7 patients and records their resting heart rate before the exercise programme and again after the exercise programme.

1. Using a \(5 \%\) level of significance, carry out an appropriate test of the doctor's belief. You should state your hypotheses, test statistic and critical value.
2. State the assumption made about the resting heart rates that was required to carry out the test.

1. Camilo grows two types of apple, green apples and red apples.

The standard deviation of the weights of green apples is known to be 3.5 grams.
A random sample of 80 green apples has a mean weight of 128 grams.

1. Find a 98\% confidence interval for the mean weight of the population of green apples. Show your working clearly and give the confidence interval limits to 2 decimal places. Camilo believes that the mean weight of the population of green apples is more than 10 grams greater than the mean weight of the population of red apples. A random sample of \(n\) red apples has a mean weight of 117 grams.
   The standard deviation of the weights of the red apples is known to be 4 grams.
   A test of Camilo's belief is carried out at the 5\% level of significance.
2. State the null and alternative hypotheses for this test.
3. Find the smallest value of \(n\) for which the null hypothesis will be rejected.
4. Explain the relevance of the Central Limit Theorem in parts (a) and (c).
5. Given that \(n = 85\), state the conclusion of the hypothesis test.

1. A factory produces bolts. The lengths of the bolts are normally distributed with mean \(\mu \mathrm { mm }\) and standard deviation 0.868 mm

A random sample of 15 of these bolts is taken and the mean length is 30.03 mm

1. Calculate a 90\% confidence interval for \(\mu\) A suitable test, at the \(10 \%\) level of significance, is carried out using these 15 bolts, to see whether or not there is evidence that the variance of the length of the bolts has increased.
2. Calculate the critical region for \(S ^ { 2 }\) The manager of the factory decides that, in future, he will check each month whether the machine making the bolts is working properly. He uses a \(10 \%\) level of significance to test whether or not there is evidence that
   The next month a random sample of 15 bolts is taken.
   The mean length of these bolts is 30.06 mm and the standard deviation is 1.02 mm
3. With reference to your answers to part (a) and part (b), state whether or not there is any evidence that the machine is not working properly.
   Give reasons for your answer.

1. A researcher set up a trial to assess the effect that a food supplement has on the increase in weight of Herdwick lambs. The researcher randomly selected 8 sets of twin lambs. One of each set of twins was given the food supplement and the other had no food supplement. The gain in weight, in kg, of each lamb over the period of the trial was recorded.

1. State why a two sample \(t\)-test is not suitable for use with these data.
2. Suggest 2 other factors about the lambs that the researcher may need to control when selecting the sample.
3. State one assumption, in context, that needs to be made for a paired \(t\)-test to be valid. For a pair of twin lambs, the random variable \(W\) represents the weight gain of the lamb given the food supplement minus the weight gain of the lamb not given the food supplement.
4. Using the data in the table, calculate a \(98 \%\) confidence interval for the mean of \(W\) Show your working clearly. The researcher believes that the mean of \(W\) is greater than 200 g
5. Stating your hypotheses clearly, use your confidence interval to explain whether or not there is evidence to support the researcher's belief.

1. A nutritionist studied the levels of cholesterol, \(X \mathrm { mg } / \mathrm { cm } ^ { 3 }\), of male students at a large college. She assumed that \(X\) was distributed \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\) and examined a random sample of 25 male students. Using this sample she obtained unbiased estimates of \(\mu\) and \(\sigma ^ { 2 }\) as \(\hat { \mu }\) and \(\hat { \sigma } ^ { 2 }\)

A \(95 \%\) confidence interval for \(\mu\) was found to be \(( 1.128,2.232 )\)

1. Show that \(\hat { \sigma } ^ { 2 } = 1.79\) (correct to 3 significant figures)
2. Obtain a \(95 \%\) confidence interval for \(\sigma ^ { 2 }\)

9 The values of a set of bivariate data \(\left( x _ { i } , y _ { i } \right)\) can be summarised by $$n = 50 , \sum x = 1270 , \sum y = 5173 , \sum x ^ { 2 } = 42767 , \sum y ^ { 2 } = 701301 , \sum x y = 173161 .$$ Ten independent observations of \(Y\) are obtained, all corresponding to \(x = 20\). It may be assumed that the variance of \(Y\) is 1.9 , independently of the value of \(x\). Find a \(95 \%\) confidence interval for the mean \(\bar { Y }\) of the 10 observations of \(Y\). \section*{END OF QUESTION PAPER}

2 The volume, in millilitres, of lemonade in mini-cans may be assumed to be normally distributed with a standard deviation of 3.5. The volumes, in millilitres, of lemonade in a random sample of 12 mini-cans were as follows.

1. Construct a \(98 \%\) confidence interval for the mean volume of lemonade in a mini-can, giving the limits to one decimal place.
2. On each mini-can is printed " 150 ml ". Comment on this, using the given sample and your confidence interval in part (a).
3. State why, in part (a), use of the Central Limit Theorem was not necessary.

4 A very popular play has been performed at a London theatre on each of 6 evenings per week for about a year. Over the past 13 weeks ( 78 performances), records have been kept of the proceeds from the sales of programmes at each performance. An analysis of these records has found that the mean was \(\pounds 184\) and the standard deviation was \(\pounds 32\).

1. Assuming that the 78 performances may be considered to be a random sample, construct a \(90 \%\) confidence interval for the mean proceeds from the sales of programmes at an evening performance of this play.
2. Comment on the likely validity of the assumption in part (a) when constructing a confidence interval for the mean proceeds from the sales of programmes at an evening performance of:
   1. this particular play;
   2. any play.

5

1. Wooden lawn edging is supplied in 1.8 m length rolls. The actual length, \(X\) metres, of a roll may be modelled by a normal distribution with mean 1.81 and standard deviation 0.08 . Determine the probability that a randomly selected roll has length:
   1. less than 1.90 m ;
   2. greater than 1.85 m ;
   3. between 1.81 m and 1.85 m .
2. Plastic lawn edging is supplied in 9 m length rolls. The actual length, \(Y\) metres, of a roll may be modelled by a normal distribution with mean \(\mu\) and standard deviation \(\sigma\). An analysis of a batch of rolls, selected at random, showed that $$\mathrm { P } ( Y < 9.25 ) = 0.88$$
   1. Use this probability to find the value of \(z\) such that $$9.25 - \mu = z \times \sigma$$ where \(z\) is a value of \(Z \sim \mathrm {~N} ( 0,1 )\).
   2. Given also that $$\mathrm { P } ( Y > 8.75 ) = 0.975$$ find values for \(\mu\) and \(\sigma\).

7

1. The weight of a sack of mixed dog biscuits can be modelled by a normal distribution with a mean of 10.15 kg and a standard deviation of 0.3 kg . A pet shop purchases 12 such sacks that can be considered to be a random sample.
   Calculate the probability that the mean weight of the 12 sacks is less than 10 kg .
2. The weight of dry cat food in a pouch can also be modelled by a normal distribution. The contents, \(x\) grams, of each of a random sample of 40 pouches were weighed. Subsequent analysis of these weights gave $$\bar { x } = 304.6 \quad \text { and } \quad s = 5.37$$
   1. Construct a \(99 \%\) confidence interval for the mean weight of dry cat food in a pouch. Give the limits to one decimal place.
   2. Comment, with justification, on each of the following two claims. Claim 1: The mean weight of dry cat food in a pouch is more than 300 grams.
      Claim 2: All pouches contain more than 300 grams of dry cat food.
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5 Jane, who supplies fruit to a jam manufacturer, knows that the weight of fruit in boxes that she sends to the manufacturer can be modelled by a normal distribution with unknown mean, \(\mu\) grams, and unknown standard deviation, \(\sigma\) grams. Jane selects a random sample of 16 boxes and, using the \(t\)-distribution, calculates correctly that a \(98 \%\) confidence interval for \(\mu\) is \(( 70.65,80.35 )\).

1. 1. Show that the sample mean is 75.5 grams.
   2. Find the width of the confidence interval.
   3. Calculate an estimate of the standard error of the mean.
   4. Hence, or otherwise, show that an unbiased estimate of \(\sigma ^ { 2 }\) is 55.6 , correct to three significant figures.
2. Jane decides that the width of the \(98 \%\) confidence interval is too large. Construct a \(95 \%\) confidence interval for \(\mu\), based on her sample of 16 boxes.
3. Jane is informed that the manufacturer would prefer the confidence interval to have a width of at most 5 grams.
   1. Write down a confidence interval for \(\mu\), again based on Jane's sample of 16 boxes, which has a width of 5 grams.
   2. Determine the percentage confidence level for your interval in part (c)(i).

5 Members of a residents' association are concerned about the speeds of cars travelling through their village. They decide to record the speed, in mph , of each of a random sample of 10 cars travelling through their village, with the following results: $$\begin{array} { l l l l l l l l l l } 33 & 27 & 34 & 30 & 48 & 35 & 34 & 33 & 43 & 39 \end{array}$$

1. Construct a \(99 \%\) confidence interval for \(\mu\), the mean speed of cars travelling through the village, stating any assumption that you make.
2. Comment on the claim that a 30 mph speed limit is being adhered to by most motorists.
   (3 marks)

6 Bishen believes that the mean weight of boxes of black peppercorns is 45 grams. Abi, thinking that this is not the case, weighs, in grams, a random sample of 8 boxes of black peppercorns, with the following results. $$\begin{array} { l l l l l l l l } 44 & 44 & 43 & 46 & 42 & 40 & 43 & 46 \end{array}$$

1. 1. Construct a \(95 \%\) confidence interval for the mean weight of boxes of black peppercorns, stating any assumption that you make.
   2. Comment on Bishen's belief.
2. 1. Abi claims that the mean weight of boxes of black peppercorns is less than 45 grams. Test this claim at the \(5 \%\) level of significance.
   2. If Bishen's belief is true, state, with a reason, what type of error, if any, may have occurred when conclusions to the test in part (b)(i) were drawn.
      (2 marks)

1 A council claims that 80 per cent of households are generally satisfied with the services it provides. A random sample of 250 households shows that 209 are generally satisfied with the council's provision of services.

1. Construct an approximate \(95 \%\) confidence interval for the proportion of households that are generally satisfied with the council's provision of services.
2. Hence comment on the council's claim.

7 A shop sells cooked chickens in two sizes: medium and large.
The weights, \(X\) grams, of medium chickens may be assumed to be normally distributed with mean \(\mu _ { X }\) and standard deviation 45. The weights, \(Y\) grams, of large chickens may be assumed to be normally distributed with mean \(\mu _ { Y }\) and standard deviation 65. A random sample of 20 medium chickens had a mean weight, \(\bar { x }\) grams, of 936 .
A random sample of 10 large chickens had the following weights in grams: $$\begin{array} { l l l l l l l l l l } 1165 & 1202 & 1077 & 1144 & 1195 & 1275 & 1136 & 1215 & 1233 & 1288 \end{array}$$

1. Calculate the mean weight, \(\bar { y }\) grams, of this sample of large chickens.
2. Hence investigate, at the \(1 \%\) level of significance, the claim that the mean weight of large chickens exceeds that of medium chickens by more than 200 grams.
3. 1. Deduce that, for your test in part (b), the critical value of \(( \bar { y } - \bar { x } )\) is 253.24, correct to two decimal places.
   2. Hence determine the power of your test in part (b), given that \(\mu _ { Y } - \mu _ { X } = 275\).
   3. Interpret, in the context of this question, the value that you obtained in part (c)(ii).
      (3 marks)

1 As part of an investigation into the starting salaries of graduates in a European country, the following information was collected.

1. Stating a necessary assumption about the samples, construct a \(98 \%\) confidence interval for the difference between the mean starting salary of science graduates and that of arts graduates.
2. What can be concluded from your confidence interval?