Estimate from percentile/frequency data

5 The lengths of the leaves of a particular type of tree are modelled by a normal distribution. A scientist measures the lengths of a random sample of 500 leaves from this type of tree and finds that 42 are less than 4 cm long and 100 are more than 10 cm long.

1. Find estimates for the mean and standard deviation of the lengths of leaves from this type of tree.
   The lengths, in cm , of the leaves of a different type of tree have the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\). The scientist takes a random sample of 800 leaves from this type of tree.
2. Find how many of these leaves the scientist would expect to have lengths, in cm , between \(\mu - 2 \sigma\) and \(\mu + 2 \sigma\).

6 The lengths of female snakes of a particular species are normally distributed with mean 54 cm and standard deviation 6.1 cm .

1. Find the probability that a randomly chosen female snake of this species has length between 50 cm and 60 cm .
   The lengths of male snakes of this species also have a normal distribution. A scientist measures the lengths of a random sample of 200 male snakes of this species. He finds that 32 have lengths less than 45 cm and 17 have lengths more than 56 cm .
2. Find estimates for the mean and standard deviation of the lengths of male snakes of this species.

6 The lengths of body feathers of a particular species of bird are modelled by a normal distribution. A researcher measures the lengths of a random sample of 600 body feathers from birds of this species and finds that 63 are less than 6 cm long and 155 are more than 12 cm long.

1. Find estimates of the mean and standard deviation of the lengths of body feathers of birds of this species.
2. In a random sample of 1000 body feathers from birds of this species, how many would the researcher expect to find with lengths more than 1 standard deviation from the mean?

4

1. It is given that \(X \sim \mathrm {~N} ( 31.4,3.6 )\). Find the probability that a randomly chosen value of \(X\) is less than 29.4.
2. The lengths of fish of a particular species are modelled by a normal distribution. A scientist measures the lengths of 400 randomly chosen fish of this species. He finds that 42 fish are less than 12 cm long and 58 are more than 19 cm long. Find estimates for the mean and standard deviation of the lengths of fish of this species.

6 The heights, in metres, of fir trees in a large forest have a normal distribution with mean 40 and standard deviation 8 .

1. Find the probability that a fir tree chosen at random in this forest has a height less than 45 metres.
2. Find the probability that a fir tree chosen at random in this forest has a height within 5 metres of the mean.
   In another forest, the heights of another type of fir tree are modelled by a normal distribution. A scientist measures the heights of 500 randomly chosen trees of this type. He finds that 48 trees are less than 10 m high and 76 trees are more than 24 m high.
3. Find the mean and standard deviation of the heights of trees of this type.

3 The random variable \(G\) has the distribution \(\mathbf { N } \left( \mu , \boldsymbol { \sigma } ^ { 2 } \right)\). One hundred observations of \(G\) are taken. The results are summarised in the following table.

1. By considering \(\mathrm { P } ( G < 40.0 )\), write down an equation involving \(\mu\) and \(\sigma\). [2]
2. Find a second equation involving \(\mu\) and \(\sigma\). Hence calculate values for \(\mu\) and \(\sigma\). [4]
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3. Explain why your answers are only estimates. [1]

4. A botanist believes that the lengths of the branches on trees of a certain species can be modelled by a normal distribution.
When he measures the lengths of 500 branches, he finds 55 which are less than 30 cm long and 200 which are more than 90 cm long.

1. Find the mean and the standard deviation of the lengths.
2. In a sample of 1000 branches, how many would he expect to find with lengths greater than 1 metre? \section*{STATISTICS 1 (A) TEST PAPER 7 Page 2}

6. Skilled operators make a particular component for an engine. The company believes that the time taken to make this component may be modelled by the normal distribution. They timed one of their operators, Sheila, over a long period. They find that when she makes a component, she takes over 90 minutes to make one \(10 \%\) of the time, and that \(20 \%\) of the time, a component was less than 70 minutes to make. Estimate the mean and standard deviation of the time Sheila takes to make a component.