Statistics vs non-statistics identification

Questions asking to identify whether given expressions are statistics (functions of sample data only, not unknown parameters) with justification.

4 questions

Edexcel S2 2021 January Q6
6. The owner of a very large youth club has designed a new method for allocating people to teams. Before introducing the method he decided to find out how the members of the youth club might react.
  1. Explain why the owner decided to take a random sample of the youth club members rather than ask all the youth club members.
  2. Suggest a suitable sampling frame.
  3. Identify the sampling units. The new method uses a bag containing a large number of balls. Each ball is numbered either 20, 50 or 70
    When a ball is selected at random, the random variable \(X\) represents the number on the ball where $$\mathrm { P } ( X = 20 ) = p \quad \mathrm { P } ( X = 50 ) = q \quad \mathrm { P } ( X = 70 ) = r$$ A youth club member takes a ball from the bag, records its number and replaces it in the bag. He then takes a second ball from the bag, records its number and replaces it in the bag. The random variable \(M\) is the mean of the 2 numbers recorded. Given that $$\mathrm { P } ( M = 20 ) = \frac { 25 } { 64 } \quad \mathrm { P } ( M = 60 ) = \frac { 1 } { 16 } \quad \text { and } \quad q > r$$
  4. show that \(\mathrm { P } ( M = 50 ) = \frac { 1 } { 16 }\)
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Edexcel S2 2007 January Q1
  1. (a) Define a statistic.
A random sample \(X _ { 1 } , X _ { 2 } , \ldots , X _ { \mathrm { n } }\) is taken from a population with unknown mean \(\mu\).
(b) For each of the following state whether or not it is a statistic.
  1. \(\frac { X _ { 1 } + X _ { 4 } } { 2 }\),
  2. \(\frac { \sum X ^ { 2 } } { n } - \mu ^ { 2 }\).
Edexcel S3 2022 June Q5
  1. A random sample of two observations \(X _ { 1 }\) and \(X _ { 2 }\) is taken from a population with unknown mean \(\mu\) and unknown variance \(\sigma ^ { 2 }\)
    1. Explain why \(\frac { X _ { 1 } - \mu } { \sigma }\) is not a statistic.
    2. Explain what you understand by an unbiased estimator for \(\mu\)
    Two estimators for \(\mu\) are \(U _ { 1 }\) and \(U _ { 2 }\) where $$U _ { 1 } = 3 X _ { 1 } - 2 X _ { 2 } \quad \text { and } \quad U _ { 2 } = \frac { X _ { 1 } + 3 X _ { 2 } } { 4 }$$
  2. Show that both \(U _ { 1 }\) and \(U _ { 2 }\) are unbiased estimators for \(\mu\) The most efficient estimator among a group of unbiased estimators is the one with the smallest variance.
  3. By finding the variance of \(U _ { 1 }\) and the variance of \(U _ { 2 }\) state, giving a reason, the most efficient estimator for \(\mu\) from these two estimators.
Edexcel S3 2014 June Q2
2. The weights of pears in an orchard are assumed to have unknown mean \(\mu\) and unknown standard deviation \(\sigma\). A random sample of 20 pears is taken and their weights recorded.
The sample is represented by \(X _ { 1 } , X _ { 2 } , \ldots , X _ { 20 }\). State whether or not the following are statistics. Give reasons for your answers.
    1. \(\frac { X _ { 1 } + 3 X _ { 20 } } { 2 }\)
    2. \(\sum _ { i = 1 } ^ { 20 } \left( X _ { i } - \mu \right)\)
    3. \(\sum _ { i = 1 } ^ { 20 } \left( \frac { X _ { i } - \mu } { \sigma } \right)\)
  2. Find the mean and variance of \(\frac { 3 X _ { 1 } - X _ { 20 } } { 2 }\)