Known variance confidence interval

1 The standard deviation of the heights of adult males is 7.2 cm . The mean height of a sample of 200 adult males is found to be 176 cm .

1. Calculate a \(97.5 \%\) confidence interval for the mean height of adult males.
2. State a necessary condition for the calculation in part (i) to be valid.

6 The Body Mass Index (BMI) of each of a random sample of 100 army recruits from a large intake in 2008 was measured. The results are summarised by $$\Sigma x = 2605.0 , \quad \Sigma x ^ { 2 } = 68636.41 .$$ It may be assumed that BMI has a normal distribution.

1. Find a 98\% confidence interval for the mean BMI of all recruits in 2008.
2. Estimate the percentage of the intake with a BMI greater than 30.0.
3. The BMIs of two randomly chosen recruits are denoted by \(\boldsymbol { B } _ { 1 }\) and \(\boldsymbol { B } _ { 2 }\). Estimate \(\mathrm { P } \left( \boldsymbol { B } _ { 1 } - \boldsymbol { B } _ { 2 } < 5 \right)\).
4. State, giving a reason, for which of the above calculations the normality assumption is unnecessary.

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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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 A factory produces bottles of brown sauce and bottles of tomato sauce.

1. The content, \(Y\) grams, of a bottle of brown sauce is normally distributed with mean \(\mu _ { Y }\) and variance 4. A quality control inspection found that the mean content, \(\bar { y }\) grams, of a random sample of 16 bottles of brown sauce was 450 . Construct a \(95 \%\) confidence interval for \(\mu _ { Y }\).
2. The content, \(X\) grams, of a bottle of tomato sauce is normally distributed with mean \(\mu _ { X }\) and variance \(\sigma ^ { 2 }\). A quality control inspection found that the content, \(x\) grams, of a random sample of 9 bottles of tomato sauce was summarised by $$\sum x = 4950 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 334$$
   1. Construct a 90\% confidence interval for \(\mu _ { X }\).
   2. Holly, the supervisor at the factory, claims that the mean content of a bottle of tomato sauce is 545 grams. Comment, with a justification, on Holly's claim. State the level of significance on which your conclusion is based.
      (3 marks)

1. The continuous random variable \(X\) has the distribution \(\mathrm { N } ( \mu , 30 )\). The mean of a random sample of 8 observations of \(X\) is 53.1. Determine a \(95 \%\) confidence interval for \(\mu\). You should give the end points of the interval correct to 4 significant figures.

3 The masses of tins of a particular brand of spaghetti are normally distributed with mean \(\mu\) grams and standard deviation 4.1 grams. A random sample of 11 tins of spaghetti has a mean mass of 401.8 grams.
Construct a \(98 \%\) confidence interval for \(\mu\), giving your values to one decimal place.

1. Rachel records the times taken, in minutes, to cycle into town from her house on a random sample of 10 days. Her results are shown below.

$$\begin{array} { l l l l l l l l l l } 15 \cdot 5 & 14 \cdot 9 & 16 \cdot 2 & 17 \cdot 3 & 14 \cdot 8 & 14 \cdot 2 & 16 \cdot 0 & 14 \cdot 2 & 15 \cdot 5 & 15 \cdot 1 \end{array}$$ Assuming that these data come from a normal distribution with mean \(\mu\) and variance \(0 \cdot 9\), calculate a \(90 \%\) confidence interval for \(\mu\).