Expected number of intervals containing parameter

1 The diameters, \(x\) millimetres, of a random sample of 200 discs made by a certain machine were recorded. The results are summarised below. $$n = 200 \quad \Sigma x = 2520 \quad \Sigma x ^ { 2 } = 31852$$

1. Calculate a 95\% confidence interval for the population mean diameter.
2. Jean chose 40 random samples and used each sample to calculate a 95\% confidence interval for the population mean diameter. How many of these 40 confidence intervals would be expected to include the true value of the population mean diameter?

4 The masses, in grams, of a certain type of plum are normally distributed with mean \(\mu\) and variance \(\sigma ^ { 2 }\). The masses, \(m\) grams, of a random sample of 150 plums of this type were found and the results are summarised by \(\Sigma m = 9750\) and \(\Sigma m ^ { 2 } = 647500\).

1. Calculate unbiased estimates of \(\mu\) and \(\sigma ^ { 2 }\).
2. Calculate a 98\% confidence interval for \(\mu\). Two more random samples of plums of this type are taken and a \(98 \%\) confidence interval for \(\mu\) is calculated from each sample.
3. Find the probability that neither of these two intervals contains \(\mu\).

5 The masses, \(m\) grams, of a random sample of 80 strawberries of a certain type were measured and summarised as follows. $$n = 80 \quad \Sigma m = 4200 \quad \Sigma m ^ { 2 } = 229000$$

1. Find unbiased estimates of the population mean and variance.
2. Calculate a 98\% confidence interval for the population mean. 50 random samples of size 80 were taken and a \(98 \%\) confidence interval for the population mean, \(\mu\), was found from each sample.
3. Find the number of these 50 confidence intervals that would be expected to include the true value of \(\mu\).

3 The times, in minutes, taken by competitors to complete a puzzle have mean \(\mu\) and standard deviation 3 . The times taken by a random sample of 10 competitors are noted and the results are given below. \(\begin{array} { l l l } 25.2 & 26.8 & 18.5 \end{array}\) 25.5
30.1 \(28.9 \quad 27.0\) \(26.1 \quad 26.0\) 24.9

1. Stating a necessary assumption, calculate a \(97 \%\) confidence interval for \(\mu\).
2. Two more random samples, each of 10 competitors, are taken. Their times are used to calculate two more \(97 \%\) confidence intervals for \(\mu\). Find the probability that neither of these intervals contains the true value of \(\mu\).

4 Diameters of golf balls are known to be normally distributed with mean \(\mu \mathrm { cm }\) and standard deviation \(\sigma \mathrm { cm }\). A random sample of 130 golf balls was taken and the diameters, \(x \mathrm {~cm}\), were measured. The results are summarised by \(\Sigma x = 555.1\) and \(\Sigma x ^ { 2 } = 2371.30\).

1. Calculate unbiased estimates of \(\mu\) and \(\sigma ^ { 2 }\).
2. Calculate a \(97 \%\) confidence interval for \(\mu\).
3. 300 random samples of 130 balls are taken and a \(97 \%\) confidence interval is calculated for each sample. How many of these intervals would you expect not to contain \(\mu\) ?

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.

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 \(( 29.74,31.86 )\)
4. Find, to 2 decimal places, the standard error of the mean.
5. Hence, or otherwise, find a \(90 \%\) confidence interval for \(\mu\) based on the same sample
   of sheets. (3)
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1 Vanya collected five samples of air and measured the carbon dioxide content of each sample, in parts per million by volume (ppmv). The results were as follows. $$\begin{array} { l l l l l } 387 & 375 & 382 & 379 & 381 \end{array}$$

1. Assuming that these data form a random sample from a normal distribution with mean \(\mu\) ppmv, construct a \(90 \%\) confidence interval for \(\mu\).
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2. Vanya repeated her sampling procedure on each of 30 days and, for each day's results, a \(90 \%\) confidence interval for \(\mu\) was constructed. On how many of these 30 days would you expect \(\mu\) to lie outside that day's confidence interval?
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