Unknown variance confidence intervals

6. The continuous random variable \(Y\) is uniformly distributed over the interval $$[ a - 3 , a + 6 ]$$ where \(a\) is a constant. A random sample of 60 observations of \(Y\) is taken.
Given that \(\bar { Y } = \frac { \sum _ { i = 1 } ^ { 60 } Y _ { i } } { 60 }\)

1. use the Central Limit Theorem to find an approximate distribution for \(\bar { Y }\) Given that the 60 observations of \(Y\) have a sample mean of 13.4
2. find a \(98 \%\) confidence interval for the maximum value that \(Y\) can take.

1. The continuous random variable \(D\) is uniformly distributed over the interval \([ x - 1 , x + 5 ]\) where \(x\) is a constant.

A random sample of \(n\) observations of \(D\) is taken, where \(n\) is large.

1. Use the Central Limit Theorem to find an approximate distribution for \(\bar { D }\) Give your answer in terms of \(n\) and \(x\) where appropriate. The \(n\) observations of \(D\) have a sample mean of 24.6
   Given that the lower bound of the \(99 \%\) confidence interval for \(x\) is 22.101 to 3 decimal places,
2. find the value of \(n\) Show your working clearly.

4. Kylie regularly travels from home to visit a friend. On 10 randomly selected occasions the journey time \(x\) minutes was recorded. The results are summarised as follows. $$\Sigma x = 753 , \quad \Sigma x ^ { 2 } = 57455 .$$

1. Calculate unbiased estimates of the mean and the variance of the population of journey times. After many journeys, a random sample of 100 journeys gave a mean of 74.8 minutes and a variance of 84.6 minutes \({ } ^ { 2 }\).
2. Calculate a 95\% confidence interval for the mean of the population of journey times.
3. Write down two assumptions you made in part (b).

1. A sociologist is studying how much junk food teenagers eat. A random sample of 100 female teenagers and an independent random sample of 200 male teenagers were asked to estimate what their weekly expenditure on junk food was. The results are summarised below.

1. Using a 5\% significance level, test whether or not there is a difference in the mean amounts spent on junk food by male teenagers and female teenagers. State your hypotheses clearly.
2. Explain briefly the importance of the central limit theorem in this problem.

6. The continuous random variable \(X\) is uniformly distributed over the interval $$[ a - 1 , a + 5 ]$$ where \(a\) is a constant.
Fifty observations of \(X\) are taken, giving a sample mean of 17.2

1. Use the Central Limit Theorem to find an approximate distribution for \(\bar { X }\).
2. Hence find a 95\% confidence interval for \(a\).

4 The time, \(x\) seconds, spent by each of a random sample of 100 customers at an automatic teller machine (ATM) is recorded. The times are summarised in the table.

1. Calculate estimates for the mean and standard deviation of the time spent at the ATM by a customer.
2. The mean time spent at the ATM by a random sample of \(\mathbf { 3 6 }\) customers is denoted by \(\bar { Y }\).
   1. State why the distribution of \(\bar { Y }\) is approximately normal.
   2. Write down estimated values for the mean and standard error of \(\bar { Y }\).
   3. Hence estimate the probability that \(\bar { Y }\) is less than \(1 \frac { 1 } { 2 }\) minutes.

6

1. The length of one-metre galvanised-steel straps used in house building may be modelled by a normal distribution with a mean of 1005 mm and a standard deviation of 15 mm . The straps are supplied to house builders in packs of 12, and the straps in a pack may be assumed to be a random sample. Determine the probability that the mean length of straps in a pack is less than one metre.
2. Tania, a purchasing officer for a nationwide house builder, measures the thickness, \(x\) millimetres, of each of a random sample of 24 galvanised-steel straps supplied by a manufacturer. She then calculates correctly that the value of \(\bar { x }\) is 4.65 mm .
   1. Assuming that the thickness, \(X \mathrm {~mm}\), of such a strap may be modelled by the distribution \(\mathrm { N } \left( \mu , 0.15 ^ { 2 } \right)\), construct a \(99 \%\) confidence interval for \(\mu\).
   2. Hence comment on the manufacturer's specification that the mean thickness of such straps is greater than 4.5 mm .

3

1. A sample of 50 washed baking potatoes was selected at random from a large batch.
   The weights of the 50 potatoes were found to have a mean of 234 grams and a standard deviation of 25.1 grams. Construct a \(95 \%\) confidence interval for the mean weight of potatoes in the batch.
   (4 marks)
2. The batch of potatoes is purchased by a market stallholder. He sells them to his customers by allowing them to choose any 5 potatoes for \(\pounds 1\). Give a reason why such chosen potatoes are unlikely to represent a random sample from the batch.

4 David, a zoologist, is investigating a particular species of monitor lizard. He measures the lengths, in centimetres, of a random sample of this particular species of lizard. His measured lengths are $$\begin{array} { l l l l l l l l l l } 53.2 & 57.8 & 55.3 & 58.9 & 59.0 & 60.2 & 61.8 & 62.3 & 65.4 & 66.5 \end{array}$$ The lengths may be assumed to be normally distributed.
David correctly constructed a 90\% confidence interval for the mean length of lizard using the measured lengths given and the formula \(\bar { x } \pm \left( b \times \frac { s } { \sqrt { n } } \right)\) This interval had limits of 57.63 and 62.45, correct to two decimal places.
4

1. State the value for \(b\) used in David's formula. 4
2. David interprets his interval and states,
   "My confidence interval indicates that exactly 90\% of the population of lizard lengths for this particular species lies between 57.63 cm and \(62.45 \mathrm {~cm} ^ { \prime \prime }\). Do you think David's statement is true? Explain your reasoning. 4
3. David's assistant, Amina, correctly constructs a \(\beta \%\) confidence interval from David's random sample of measured lengths. Amina informs David that the width of her confidence interval is 8.54 .
   Find the value of \(\beta\).
   [0pt] [3 marks]
   Turn over for the next question

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.

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.

6 On arrival at a business centre, all visitors are required to register at the reception desk. An analysis of the register, for a random sample of 100 days, results in the following information on the number, \(X\), of visitors per day.

1. Calculate an estimate of:
   1. \(\mu\), the mean number of visitors per day;
   2. \(\sigma\), the standard deviation of the number of visitors per day.
2. Give a reason, based upon the data provided, why \(X\) is unlikely to be normally distributed.
3. 1. Give a reason why \(\bar { X }\), the mean of a random sample of 100 observations on \(X\), may be assumed to be normally distributed.
   2. State, in terms of \(\mu\) and \(\sigma\), the mean and variance of \(\bar { X }\).
4. Hence construct a \(99 \%\) confidence interval for \(\mu\).
5. The receptionist claims that she registers on average more than 30 visitors per day, and frequently registers more than 50 visitors on any one day. Comment on each of these two claims.

3 Fiona is studying the heights of corn plants on a farm. She measures the height, \(x \mathrm {~cm}\), of a random sample of 200 corn plants on the farm.
The summarised results are as follows: $$\sum x = 60255 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 995$$ Calculate a \(96 \%\) confidence interval for the population mean of heights of corn plants on the farm, giving your values to one decimal place.