5.05d Confidence intervals: using normal distribution

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Pre-U Pre-U 9795/2 2014 June Q3
8 marks Standard +0.8
A random sample of 400 seabirds is taken from a colony, ringed, and returned, unharmed, to the colony. After a suitable period of time has elapsed, a second random sample of 400 seabirds is taken, and 20 of this second sample are found to be ringed. You may assume that the probability that a seabird is captured is independent of whether or not it has been ringed and that the colony remains unchanged at the time of the second sampling.
  1. Estimate the number of seabirds in the colony. [1]
  2. Find a 98% confidence interval for the proportion of seabirds in the colony which are ringed. [5]
  3. Deduce a 98% confidence interval for the number of seabirds in the colony. [2]
CAIE FP2 2014 June Q10
Standard +0.3
10 The lengths of a random sample of eight fish of a certain species are measured, in cm, as follows. $$\begin{array} { l l l l l l l l } 17.3 & 15.8 & 18.2 & 15.6 & 16.0 & 18.8 & 15.3 & 15.0 \end{array}$$ Assuming that lengths are normally distributed,
  1. test, at the \(10 \%\) significance level, whether the population mean length of fish of this species is greater than 15.8 cm ,
  2. calculate a \(95 \%\) confidence interval for the population mean length of fish of this species.