A-level · Statistical distributions

Finding μ and σExam-style questions

Working backwards from given probabilities to find an unknown mean or standard deviation (or both) by standardising to a z-value and solving — including a pair of simultaneous equations when both are unknown.

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25 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 3 Level 4 Level 5

1
The mass, $X$ grams, of a loaf of bread produced by a bakery is modelled by a normal distribution with mean $\mu$ and standard deviation $\sigma$ . Records show that $2.28\%$ of loaves have mass less than $775$ g and that $91.04\%$ of loaves have mass between $775$ g and $831$ g.
(a) Find the values of $\mu$ and $\sigma$ . (5 marks)

The bakery installs a new oven. The baker believes that loaves baked in the new oven have a greater mean mass. It may be assumed that the standard deviation is unchanged. A random sample of $25$ loaves baked in the new oven has mean mass $812.5$ g.
(b) Test the baker's belief at the $5\%$ level of significance. State your hypotheses clearly. (5 marks)
(c) Explain how, if at all, your conclusion would change if the baker had believed only that the mean mass had changed, with no view as to the direction of the change. (2 marks)
(d) A second random sample of $n$ loaves from the new oven also has mean mass $812.5$ g. Find the smallest value of $n$ for which this sample would provide significant evidence, at the $5\%$ level, that the mean mass has changed. (2 marks)
Level 5 · 14 marksTry it
2
(a) The heights of a species of plant are normally distributed with mean $\mu$ cm and standard deviation $\sigma$ cm. It is found that $15\%$ of the plants are shorter than $20$ cm and $20\%$ are taller than $35$ cm.

Find the values of $\mu$ and $\sigma$ . (5 marks)
(b) A fair six-sided die is thrown $180$ times. Using a suitable approximation, find the probability that a six is obtained at least $40$ times. (3 marks)
Level 5 · 8 marksTry it
3
The weights of apples from an orchard, $W$ grams, are modelled by a normal distribution with mean $\mu$ and standard deviation $\sigma$ .

It is known that $10\%$ of the apples weigh less than $120$ g and that $5\%$ of the apples weigh more than $190$ g.
(a) Find the value of $\mu$ and the value of $\sigma$ , giving each answer to $3$ significant figures. (6 marks)

An apple is classed as large if it weighs more than $160$ g.
(b) Show that the probability that a randomly chosen apple is large is approximately $0.35$ . (2 marks)

A wholesaler buys a random sample of $200$ apples from the orchard.
(c) Using a suitable approximation, which you should justify, estimate the probability that at least $80$ of these apples are large. Use $\mathrm{P}(\text{large}) = 0.35$ . (5 marks)
Level 5 · 13 marksTry it
4
Component lifetimes are $N(\mu, \sigma^2)$ hours. $10\%$ last under $800$ and $5\%$ last over $1200$ .
(a) Find $\mu$ and $\sigma$ to 3 s.f.
(b) Hence find $P(X > 1000)$ , to 3 d.p.
Level 5 · 6 marksTry it
5
The lifetimes of a type of battery are modelled as $N(\mu, \sigma^{2})$ hours. $20\%$ of batteries last less than $1000$ hours and $10\%$ last more than $1500$ hours.

Find $\mu$ and $\sigma$ , giving your answers to 3 significant figures. (6 marks)
Level 5 · 6 marksTry it

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