5.05d Confidence intervals: using normal distribution

4 A random sample of 160 observations of a random variable \(X\) is selected. The sample can be summarised as follows. \(n = 160 \quad \sum x = 2688 \quad \sum x ^ { 2 } = 48398\)

1. Calculate unbiased estimates of the following.
   1. \(\mathrm { E } ( X )\)
   2. \(\operatorname { Var } ( X )\)
2. Find a 99\% confidence interval for \(\mathrm { E } ( X )\), giving the end-points of the interval correct to 4 significant figures.
3. Explain whether it was necessary to use the Central Limit Theorem in answering
   1. part (a),
   2. part (b).

4. The scores in a national test of seven-year-old children are normally distributed with a standard deviation of 18
A random sample of 25 seven-year-old children from town \(A\) had a mean score of 52.4

1. Calculate a 98\% confidence interval for the mean score of the seven-year-old children from town \(A\).
   (4) An independent random sample of 30 seven-year-old children from town \(B\) had a mean score of 57.8
   A local newspaper claimed that the mean score of seven-year-old children from town \(B\) was greater than the mean score of seven-year-old children from town \(A\).
2. Stating your hypotheses clearly, use a \(5 \%\) significance level to test the newspaper's claim. You should show your working clearly. The mean score for the national test of seven-year-old children is \(\mu\). Considering the two samples of seven-year-old children separately, at the \(5 \%\) level of significance, there is insufficient evidence that the mean score for town \(A\) is less than \(\mu\), and insufficient evidence that the mean score for town \(B\) is less than \(\mu\).
3. Find the largest possible value for \(\mu\). \includegraphics[max width=\textwidth, alt={}, center]{ba3f3f9c-53d2-4e95-b2f3-3f617f1821ed-11_2255_50_314_34}
   

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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}

1. The weights, \(x \mathrm {~kg}\), of each of 10 watermelons selected at random from Priya's shop were recorded. The results are summarised as follows

$$\sum x = 114.2 \quad \sum x ^ { 2 } = 1310.464$$

1. Calculate unbiased estimates of the mean and the variance of the weights of the watermelons in Priya's shop. Priya researches the weight of watermelons, for the variety she has in her shop, and discovers that the weights of these watermelons are normally distributed with a standard deviation of 0.8 kg
2. Calculate a \(95 \%\) confidence interval for the mean weight of watermelons in Priya's shop. Give the limits of your confidence interval to 2 decimal places. Priya claims that the confidence interval in part (b) suggests that nearly all of the watermelons in her shop weigh more than 10.5 kg
3. Use your answer to part (b) to estimate the smallest proportion of watermelons in her shop that weigh less than 10.5 kg

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

6. A company produces a certain type of mug. The masses of these mugs are normally distributed with mean \(\mu\) and standard deviation 1.2 grams. A random sample of 5 mugs is taken and the mass, in grams, of each mug is measured. The results are given below. \section*{\(\begin{array} { l l l l l } 229.1 & 229.6 & 230.9 & 231.2 & 231.7 \end{array}\)}

1. Find a \(95 \%\) confidence interval for \(\mu\), giving your limits correct to 1 decimal place. Sonia plans to take 20 random samples, each of 5 mugs. A 95\% confidence interval for \(\mu\) is to be determined for each sample.
2. Find the probability that more than 3 of these intervals will not contain \(\mu\).

5. A factory produces steel sheets whose weights, \(X \mathrm {~kg}\), have a normal distribution with an unknown mean \(\mu \mathrm { kg }\) and known standard deviation \(\sigma \mathrm { kg }\). A random sample of 25 sheets gave both a

• \(95 \%\) confidence interval for \(\mu\) of \(( 30.612,31.788 )\)
• \(c \%\) confidence interval for \(\mu\) of \(( 30.66,31.74 )\)
  1. Find the value of \(\sigma\)
  2. Find the value of \(c\), giving your answer correct to 3 significant figures.

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. Components are manufactured such that their length in mm is normally distributed with mean \(\mu\) and variance \(\sigma ^ { 2 }\). Below is a 95\% confidence interval for \(\mu\) calculated from a random sample of components.
   (11.52, 13.75)

Using the same random sample,

1. find a \(90 \%\) confidence interval for \(\mu\). Four 90\% confidence intervals are found from independent random samples.
2. Calculate the probability that only 3 of these 4 intervals will contain \(\mu\).

1. The random variable \(X\) is normally distributed with unknown mean \(\mu\) and known variance \(\sigma ^ { 2 }\)

A random sample of 25 observations of \(X\) produced a \(95 \%\) confidence interval for \(\mu\) of (26.624, 28.976)

1. Find the mean of the sample.
2. Show that the standard deviation is 3 The \(a\) \% confidence interval using the 25 observations has a width of 2.1
3. Calculate the value of \(a\)
4. Find the smallest sample size, of observations from \(X\), that would be required to obtain a 95\% confidence interval of width at most 1.5

1. The continuous random variable \(X\) is normally distributed with

$$X \sim \mathrm {~N} \left( \mu , 5 ^ { 2 } \right)$$ A random sample of 10 observations of \(X\) is taken and \(\bar { X }\) denotes the sample mean.

1. Show that a \(90 \%\) confidence interval for \(\mu\), in terms of \(\bar { x }\), is given by $$( \bar { x } - 2.60 , \bar { x } + 2.60 )$$ The continuous random variable \(Y\) is normally distributed with $$Y \sim \mathrm {~N} \left( \mu , 3 ^ { 2 } \right)$$ A random sample of 20 observations of \(Y\) are taken and \(\bar { Y }\) denotes the sample mean.
2. Find a 95\% confidence interval for \(\mu\), in terms of \(\bar { y }\)
3. Given that \(X\) and \(Y\) are independent,
   1. find the distribution of \(\bar { X } - \bar { Y }\)
   2. calculate the probability that the two confidence intervals from part (a) and part (b) do not overlap.

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. The number of toasters sold by a shop each week may be modelled by a Poisson distribution with mean 4 A random sample of 35 weeks is taken and the mean number of toasters sold per week is found.

1. Write down the approximate distribution for the mean number of toasters sold per week from a random sample of 35 weeks. The number of kettles sold by the shop each week may be modelled by a Poisson distribution with mean \(\lambda\) A random sample of 40 weeks is taken and the mean number of kettles sold per week is found. The width of the \(99 \%\) confidence interval for \(\lambda\) is 2.6
2. Find an estimate for \(\lambda\) A second, independent random sample of 40 weeks is taken and a second \(99 \%\) confidence interval for \(\lambda\) is found.
3. Find the probability that only one of these two confidence intervals contains \(\lambda\)

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 \(n\).
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8. A factory produces steel sheets whose weights \(X \mathrm {~kg}\), are such that \(X \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\) A random sample of these sheets is taken and a \(95 \%\) confidence interval for \(\mu\) is found to be (29.74, 31.86)

1. Find, to 2 decimal places, the standard error of the mean.
2. Hence, or otherwise, find a \(90 \%\) confidence interval for \(\mu\) based on the same sample of sheets. Using four different random samples, four \(90 \%\) confidence intervals for \(\mu\) are to be found.
3. Calculate the probability that at least 3 of these intervals will contain \(\mu\). \section*{8. A factory produces steel sheets whose weights \(X \mathrm { gg }\), are such \(X \sim N ( \mu , \sigma ) ^ { 2 }\)} A. A. A random sample of these sheets is taken and a \(95 \%\) confidence interval for \(\mu\) is found to
   be \(( 29.74,31.86 )\)
   1. Find, to 2 decimal places, the standard error of the mean.
   2. Hence, or otherwise, find a \(90 \%\) confidence interval for \(\mu\) based on the same sample
      of sheets. (3)
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3. A woodwork teacher measures the width, \(w \mathrm {~mm}\), of a board. The measured width, \(X \mathrm {~mm}\), is normally distributed with mean \(w \mathrm {~mm}\) and standard deviation 0.5 mm .

1. Find the probability that \(X\) is within 0.6 mm of \(w\). The same board is measured 16 times and the results are recorded.
2. Find the probability that the mean of these results is within 0.3 mm of \(w\). Given that the mean of these 16 measurements is 35.6 mm ,
3. find a 98\% confidence interval for \(w\).

3. The drying times of paint can be assumed to be normally distributed. A paint manufacturer paints 10 test areas with a new paint. The following drying times, to the nearest minute, were recorded. $$82 , \quad 98 , \quad 140 , \quad 110 , \quad 90 , \quad 125 , \quad 150 , \quad 130 , \quad 70 , \quad 110 .$$

1. Calculate unbiased estimates for the mean and the variance of the population of drying times of this paint. Given that the population standard deviation is 25 ,
2. find a 95\% confidence interval for the mean drying time of this paint. Fifteen similar sets of tests are done and the \(95 \%\) confidence interval is determined for each set.
3. Estimate the expected number of these 15 intervals that will enclose the true value of the population mean \(\mu\).

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.

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 random sample of the daily sales (in £s) of a small company is taken and, using tables of the normal distribution, a 99\% confidence interval for the mean daily sales is found to be
   (123.5, 154.7)

Find a \(95 \%\) confidence interval for the mean daily sales of the company.
(6)

1. Some biologists were studying a large group of wading birds. A random sample of 36 were measured and the wing length, \(x \mathrm {~mm}\) of each wading bird was recorded. The results are summarised as follows

$$\sum x = 6046 \quad \sum x ^ { 2 } = 1016338$$

1. Calculate unbiased estimates of the mean and the variance of the wing lengths of these birds. Given that the standard deviation of the wing lengths of this particular type of bird is actually 5.1 mm ,
2. find a \(99 \%\) confidence interval for the mean wing length of the birds from this group.
   

3. A woodwork teacher measures the width, \(w \mathrm {~mm}\), of a board. The measured width, \(X \mathrm {~mm}\), is normally distributed with mean \(w \mathrm {~mm}\) and standard deviation 0.5 mm .

1. Find the probability that \(X\) is within 0.6 mm of \(w\). The same board is measured 16 times and the results are recorded.
2. Find the probability that the mean of these results is within 0.3 mm of \(w\). Given that the mean of these 16 measurements is 35.6 mm ,
3. find a \(98 \%\) confidence interval for \(w\).

3.

1. Explain what you understand by the Central Limit Theorem. A garage services hire cars on behalf of a hire company. The garage knows that the lifetime of the brake pads has a standard deviation of 5000 miles. The garage records the lifetimes, \(x\) miles, of the brake pads it has replaced. The garage takes a random sample of 100 brake pads and finds that \(\sum x = 1740000\)
2. Find a 95\% confidence interval for the mean lifetime of a brake pad.
3. Explain the relevance of the Central Limit Theorem in part (b). Brake pads are made to be changed every 20000 miles on average.
   The hire car company complain that the garage is changing the brake pads too soon.
4. Comment on the hire company's complaint. Give a reason for your answer.
   

A manufacturer produces circular discs with diameter \(D \mathrm {~mm}\), such that \(D \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\). A random sample of discs is taken and, using tables of the normal distribution, a \(90 \%\) confidence interval for \(\mu\) is found to be
(118.8, 121.2)

1. Find a 98\% confidence interval for \(\mu\).
2. Hence write down a 98\% confidence interval for the circumference of the discs. Using three different random samples, three \(98 \%\) confidence intervals for \(\mu\) are to be found.
3. Calculate the probability that all the intervals will contain \(\mu\).
   

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 \(a\).