Estimated variance confidence interval

Questions where the variance must be estimated from the sample data using s² or unbiased estimates, typically requiring t-distribution (though some may use normal approximation for large samples).

5 questions

CAIE S2 2014 November Q3
3 The times, in minutes, taken by people to complete a walk are normally distributed with mean \(\mu\). The times, \(t\) minutes, for a random sample of 80 people were summarised as follows. $$\Sigma t = 7220 \quad \Sigma t ^ { 2 } = 656060$$
  1. Calculate a \(97 \%\) confidence interval for \(\mu\).
  2. Explain whether it was necessary to use the Central Limit theorem in part (i).
CAIE S2 2015 November Q7
7 The diameter, in cm, of pistons made in a certain factory is denoted by \(X\), where \(X\) is normally distributed with mean \(\mu\) and variance \(\sigma ^ { 2 }\). The diameters of a random sample of 100 pistons were measured, with the following results. $$n = 100 \quad \Sigma x = 208.7 \quad \Sigma x ^ { 2 } = 435.57$$
  1. Calculate unbiased estimates of \(\mu\) and \(\sigma ^ { 2 }\). The pistons are designed to fit into cylinders. The internal diameter, in cm , of the cylinders is denoted by \(Y\), where \(Y\) has an independent normal distribution with mean 2.12 and variance 0.000144 . A piston will not fit into a cylinder if \(Y - X < 0.01\).
  2. Using your answers to part (i), find the probability that a randomly chosen piston will not fit into a randomly chosen cylinder.
WJEC Further Unit 5 2019 June Q1
  1. A coffee shop produces biscuits to sell. The masses, in grams, of the biscuits follow a normal distribution with mean \(\mu\). Eight biscuits are chosen at random and their masses, in grams, are recorded. The results are given below.
    \(\begin{array} { l l l l l l l l } 32 \cdot 1 & 29 \cdot 9 & 31 \cdot 0 & 31 \cdot 1 & 32 \cdot 5 & 30 \cdot 8 & 30 \cdot 7 & 31 \cdot 5 \end{array}\)
    1. Calculate a 95\% confidence interval for \(\mu\) based on this sample.
    2. Explain the relevance or otherwise of the Central Limit Theorem in your calculations.
    3. The continuous random variable \(X\) is uniformly distributed over the interval \(( \theta - 1 , \theta + 5 )\), where \(\theta\) is an unknown constant.
    4. Find the mean and the variance of \(X\).
    5. Let \(\bar { X }\) denote the mean of a random sample of 9 observations of \(X\). Find, in terms of \(\bar { X }\), an unbiased estimator for \(\theta\) and determine its standard error.
    6. The rules for the weight of a cricket ball state:
      "The ball, when new, shall weigh not less than \(\mathbf { 1 5 5 . 9 ~ g }\), nor more than \(\mathbf { 1 6 3 ~ g }\)."
      A company produces cricket balls whose weights are normally distributed. It wants \(99 \%\) of the balls it produces to be an acceptable weight.
    7. What is the largest acceptable standard deviation?
    The weights of the cricket balls are in fact normally distributed with mean 159.5 grams and standard deviation 1.2 grams. The company also produces tennis balls. The weights of the tennis balls are normally distributed with mean 58.5 grams and standard deviation 1.3 grams.
  2. Find the probability that the weight of a randomly chosen cricket ball is more than three times the weight of a randomly chosen tennis ball.
OCR Further Statistics 2018 September Q9
9 The continuous random variable \(C\) has the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\). The sum of a random sample of 16 observations of \(C\) is 224.0 .
  1. Find an unbiased estimate of \(\mu\).
  2. It is given that an unbiased estimate of \(\sigma ^ { 2 }\) is 0.24. Find the value of \(\Sigma c ^ { 2 }\).
    \(D\) is the sum of 10 independent observations of \(C\).
  3. Explain whether \(D\) has a normal distribution. The continuous random variable \(F\) is normally distributed with mean 15.0, and it is known that \(\mathrm { P } ( F < 13.2 ) = 0.115\).
  4. Use the unbiased estimates of \(\mu\) and \(\sigma ^ { 2 }\) to find \(\mathrm { P } ( D + F > 157.0 )\). \section*{OCR} \section*{Oxford Cambridge and RSA}
AQA Further Paper 3 Statistics 2022 June Q5
5 The mass, \(X\), in grams of a particular type of apple is modelled using a normal distribution. A random sample of 12 apples is collected and the summarised results are $$\sum x = 1038 \quad \text { and } \quad \sum x ^ { 2 } = 90100$$ 5
  1. A 99\% confidence interval for the population mean of the masses of the apples is constructed using the random sample. Show that the confidence interval is \(( 81.7,91.3 )\) with values correct to three significant figures.
    5
  2. Padraig claims that the population mean mass of the apples is 85 grams. He carries out a hypothesis test at the \(1 \%\) level of significance using the random sample of 12 apples. The hypotheses are $$\begin{aligned} & \mathrm { H } _ { 0 } : \mu = 85
    & \mathrm { H } _ { 1 } : \mu \neq 85 \end{aligned}$$ State, with a reason, whether the null hypothesis is accepted or rejected.
    5
  3. Interpret, in context, the conclusion to the hypothesis test in part (b).