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
At a medical centre, the length of a consultation with a doctor, $X$ minutes, is normally distributed with mean $m$ minutes and variance 9

The probability that a randomly chosen consultation lasts within $4k$ minutes of the mean is 0.9
(a) Find the value of $k$

(4 marks)

The time, $Y$ minutes, that a patient waits at the medical centre before their consultation is normally distributed with mean $\mu$ minutes and standard deviation $\sigma$ minutes. The mean of $Y$ is four times its standard deviation, and 11.51% of patients wait for less than 14 minutes.
(b) Find the value of $\mu$ and the value of $\sigma$

(5 marks)
Level 4 · 9 marksTry it
2
The time taken by a courier company to deliver a parcel, $Y$ minutes, is modelled by a normal distribution with mean $\mu$ and standard deviation $\sigma$ .

Given that $P(Y < 22) = 0.15$ and $P(Y > 35) = 0.2$ ,
(a) find the values of $\mu$ and $\sigma$ , giving your answers to 3 significant figures. (6 marks)
(b) A parcel is known to have taken more than $22$ minutes to deliver. Find the probability that it took less than $30$ minutes. (3 marks)
(c) The company's records show that a small number of parcels take several hours to deliver. Explain why this suggests that the normal distribution may not be a suitable model for delivery times. (1 mark)
Level 4 · 10 marksTry it
3
The time taken by a courier to complete a delivery round in a city, $T$ minutes, is modelled by a normal distribution with mean $\mu$ and standard deviation $\sigma$ . Records show that $\mathrm{P}(T < 40) = 0.0668$ and $\mathrm{P}(T > 58) = 0.1587$ .
(a) Determine the values of $\mu$ and $\sigma$ . (4 marks)

The courier company introduces a route-planning app and claims that it reduces the time taken to complete a round. After the app is introduced, the times taken for a random sample of $30$ rounds are recorded. The standard deviation of the times is assumed to be unchanged.
(b) Stating your hypotheses clearly, find the critical region for a test of the company's claim at the $5\%$ significance level, giving your answer in terms of the sample mean $\bar{T}$ . (4 marks)
(c) The mean time for the sample of $30$ rounds is $48.2$ minutes. Complete the test, stating your conclusion in context.

The company's analyst says that this result proves the app reduces the time taken to complete a round. Comment on the analyst's claim. (3 marks)
Level 4 · 11 marksTry it
4
The mass, $M$ 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 $10.56\%$ of loaves have a mass less than $780$ g and $3.04\%$ have a mass greater than $830$ g.
(a) Find the values of $\mu$ and $\sigma$ . (5 marks)

The bakery installs a new oven. A random sample of $40$ loaves baked in the new oven has a mean mass of $795.2$ g. It is assumed that the standard deviation of the masses is unchanged.
(b) Test, at the $5\%$ level of significance, whether the mean mass of loaves baked in the new oven is different from the value of $\mu$ found in part (a). (4 marks)
(c) The manager says, \"The test proves that the new oven has not changed the mean mass.\" Explain why this statement is not justified. (1 mark)
Level 4 · 10 marksTry it
5
$X \sim N(\mu, \sigma^{2})$ with $P(X < 10) = 0.05$ and $P(X < 70) = 0.99$ .

Find $\mu$ and $\sigma$ , giving your answers to 3 significant figures. (6 marks)
Level 4 · 6 marksTry it

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