Direct expected frequency calculation

Given a normal distribution with known mean and standard deviation, and a sample size n, calculate the expected number of observations satisfying a single condition (e.g., greater than a value, within a range) using P(condition) × n.

7 questions · Moderate -0.7

2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation
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CAIE S1 2021 June Q2
5 marks Moderate -0.8
2 A company produces a particular type of metal rod. The lengths of these rods are normally distributed with mean 25.2 cm and standard deviation 0.4 cm . A random sample of 500 of these rods is chosen. How many rods in this sample would you expect to have a length that is within 0.5 cm of the mean length?
CAIE S1 2019 June Q2
4 marks Easy -1.2
2 The volume of ink in a certain type of ink cartridge has a normal distribution with mean 30 ml and standard deviation 1.5 ml . People in an office use a total of 8 cartridges of this ink per month. Find the expected number of cartridges per month that contain less than 28.9 ml of this ink.
CAIE S1 2010 November Q3
7 marks Moderate -0.3
3 The times taken by students to get up in the morning can be modelled by a normal distribution with mean 26.4 minutes and standard deviation 3.7 minutes.
  1. For a random sample of 350 students, find the number who would be expected to take longer than 20 minutes to get up in the morning.
  2. 'Very slow' students are students whose time to get up is more than 1.645 standard deviations above the mean. Find the probability that fewer than 3 students from a random sample of 8 students are 'very slow'.
Edexcel S1 2003 November Q3
9 marks Moderate -0.8
3. Cooking sauces are sold in jars containing a stated weight of 500 g of sauce The jars are filled by a machine. The actual weight of sauce in each jar is normally distributed with mean 505 g and standard deviation 10 g .
    1. Find the probability of a jar containing less than the stated weight.
    2. In a box of 30 jars, find the expected number of jars containing less than the stated weight. The mean weight of sauce is changed so that \(1 \%\) of the jars contain less than the stated weight. The standard deviation stays the same.
  2. Find the new mean weight of sauce.
Edexcel S1 Q7
15 marks Moderate -0.3
7. The volume of liquid in bottles of sparkling water from one producer is believed to be normally distributed with a mean of 704 ml and a variance of \(3.2 \mathrm { ml } ^ { 2 }\). Calculate the probability that a randomly chosen bottle from this producer contains
  1. more than 706 ml ,
  2. between 703 and 708 ml . The bottles are labelled as containing 700 ml .
  3. In a delivery of 1200 bottles, how many could be expected to contain less than the stated 700 ml ? The bottling process can be adjusted so that the mean changes but the variance is unchanged.
  4. What should the mean be changed to in order to have only a \(0.1 \%\) chance of a bottle having less than 700 ml of sparkling water? Give your answer correct to 1 decimal place.
Edexcel Paper 3 Specimen Q5
8 marks Moderate -0.8
5. The lifetimes of batteries sold by company \(X\) are normally distributed, with mean 150 hours and standard deviation 25 hours. A box contains 12 batteries from company \(X\).
  1. Find the expected number of these batteries that have a lifetime of more than 160 hours. The lifetimes of batteries sold by company \(Y\) are normally distributed, with mean 160 hours and \(80 \%\) of these batteries have a lifetime of less than 180 hours.
  2. Find the standard deviation of the lifetimes of batteries from company \(Y\). Both companies sell their batteries for the same price.
  3. State which company you would recommend. Give reasons for your answer.
WJEC Unit 4 2018 June Q4
8 marks Moderate -0.8
Arwyn collects data about household expenditure on food. He records the weekly expenditure on food for 80 randomly selected households from across Wales.
Cost, \(x\) (£)\(x < 40\)\(40 \leqslant x<50\)\(50 \leqslant x<60\)\(60 \leqslant x<70\)\(70 \leqslant x<80\)\(80 \leqslant x<90\)\(x \geqslant 90\)
Number of households51116181596
  1. Explain why a normal distribution may be an appropriate model for the weekly expenditure on food for this sample. [1]
Arwyn uses the distribution N(64, 15²) to model expenditure on food.
  1. Find the number of households in the sample that this model would predict to have weekly food expenditure in the range
    1. \(60 \leqslant x < 70\),
    2. \(x \geqslant 90\). [4]
  2. Use your answers to part (b)
    1. to comment on the suitability of this model,
    2. to explain how Arwyn could improve the model by changing one of its parameters. [2]
  3. Arwyn's friend Colleen wishes to use the improved model to predict household expenditure on food in Northern Ireland. Comment on this plan. [1]