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Mean of a Sample & Confidence Intervals (AQA Level 3 Mathematical Studies (Core Maths)): Revision Note

Exam code: 1350

Naomi C

Written by: Naomi C

Reviewed by: Dan Finlay

Updated on

Mean of a Sample

What is the sample mean?

  • For any given population it can often be difficult or impractical to find the true value of the population mean, bold italic mu

    • The population could be too large to collect data using a census

    • Collecting the data could compromise the individual data values and therefore taking a census could destroy the population

  • The population mean can be estimated by taking the mean from a sample from within the population

    • The mean of a random sample is known as the point estimate for the mean of the population

    • A point estimate is a 'best guess' of the mean of the population

  • A bigger sample size will most likely lead to a more accurate point estimate

    • A sample size of 1, for example, will not give a very good estimate of the mean of the population

What is the distribution of the sample means?

  • If a number of samples are taken from a population with a normal distribution, then the means of these samples will themselves have a normal distribution

  • For a random variable, X, whose population has a distribution straight N open parentheses mu comma space sigma squared close parentheses, the normal distribution of the sample means, top enclose X, will be straight N open parentheses mu comma fraction numerator space sigma squared over denominator n end fraction close parentheses

    • The distribution of the population and the sample means have the same mean, mu

    • The distribution of the sample means has variance sigma squared over n , where n is the sample size

    • Therefore, the standard deviation of the sample means is fraction numerator sigma over denominator square root of n end fraction, which is the standard deviation of the population divided by the square root of the sample size

  • The standard deviation of the sample means is also known as the standard error of the mean

Worked Example

The weights of bags of sugar are normally distributed with mean 500 g and variance 25 g.

(a) What is the probability that the weight of a bag of sugar chosen at random is less than 496 g?

Find straight P open parentheses X less than 492 close parentheses, given straight N open parentheses 500 comma space 25 close parentheses using your calculator

Lower bound equals negative 99999999
Upper bound equals space 496
mu equals 500
sigma equals square root of 25 equals 5

straight P open parentheses X less than 496 close parentheses equals 0.211855...

(Alternatively you can find the z-value of 496 g and use the statistical tables)

Convert to a percentage and round to 3 significant figures

bold 21 bold. bold 2 bold percent sign (3 s.f.)

Bags of sugar are shipped to supermarkets in boxes. Each box contains 8 bags of sugar.

(b) What is the probability that for a randomly selected box, the mean weight of a bag of sugar is less than 496 g?

The distribution of the sample means will be straight N open parentheses mu comma fraction numerator space sigma squared over denominator n end fraction close parentheses

straight N open parentheses 500 comma space 25 over 8 close parentheses

So using the calculator, find straight P open parentheses top enclose X less than 496 close parentheses

Lower bound equals negative 99999999
Upper bound equals space 496
mu equals 500
square root of sigma squared over n end root equals square root of 25 over 8 end root

straight P open parentheses X less than 492 close parentheses equals 0.011825...

Convert to a percentage and round to 3 significant figures

bold 1 bold. bold 18 bold percent sign (3 s.f.)

Confidence Intervals

  • A range of possibilities is often given rather than a single value, this is because we cannot always be exact when dealing with probabilities

  • The range of possibilities around a measurement is called the confidence interval

    • A confidence interval describes how precise the measurement is

    • A confidence interval of 95% means that 95% of the data falls within the given range

Examiner Tips and Tricks

When tackling an exam question, think about the context and the impact of a result being outside a given range.

For example, when we need to ensure that something is accurate, such as the safety of a piece of equipment, we would expect a high confidence interval (99%), but for something less vital, such as the results of a class test, a lower confidence interval (90%) would be acceptable.

  • A confidence interval is described using the notation open square brackets a comma space b close square brackets, where a is the lowest values in the range and b is the greatest value in the range

  • If a percentage of the data falls within a number of standard deviations of the mean for a normal distribution

    • Then for samples from that distribution, the same percentage of the data falls within the same number of standard errors of the mean

  • You can work out the confidence interval for a population with a normal distribution

    • Find the z-value associated with the given percentage for the confidence interval

    • Add/subtract the product of the z-value and the standard deviation to/from the mean

  • The confidence interval for a sample can be found in a similar way

    • Find the z-value associated with the given percentage for the confidence interval

    • Add/subtract the product of the z-value and the standard error to/from the mean

  • The confidence interval is always assumed to be symmetrical about the mean for a normal distribution

  • The confidence level required (%) and the sample size will always be given to you in an exam

Examiner Tips and Tricks

It can be useful to remember the key confidence intervals for speed in an exam:

  • Confidence interval for 99% of the distribution, open square brackets negative 2.58 comma space 2.58 close square brackets

  • Confidence interval for 95% of the distribution, open square brackets negative 1.96 comma space 1.96 close square brackets

  • Confidence interval for 90% of the distribution, open square brackets negative 1.64 comma space 1.64 close square brackets

However, you can use the statistical tables in the data booklet to help you work them out if you can't remember them.

Worked Example

The reaction times of a population is normally distributed with mean mu milliseconds and standard deviation 75 milliseconds.

A sample of 60 people had a total reaction time of 24360 milliseconds.

(a) What is the point estimate for the sample?

Divide the total reaction time by the number of people in the sample to find the point estimate

24360 over 60 equals 406

406 milliseconds

(b) Construct a 98% confidence interval for the mean reaction time for mu.

Using the statistical tables find z for straight capital phi open parentheses straight z close parentheses equals 0.98

straight capital phi open parentheses 2.0537 close parentheses equals 0.98

Remember, the point estimate for the sample is the same as the mean for the population

Find the confidence interval for the population by multiplying the z-value by the standard deviation and adding to/subtracting from the mean

406 minus 2.0537 cross times 75 equals 251.9725 406 plus 2.0537 cross times 75 equals 560.0275

Round appropriately

stretchy left square bracket 560 comma space 252 stretchy right square bracket

(c) Construct a 98% confidence interval for the mean reaction time for the people in the sample.

Find the standard error of the sample, fraction numerator sigma over denominator square root of n end fraction

fraction numerator 75 over denominator square root of 60 end fraction

Find the confidence interval for the mean reaction time in the sample using the same z-value

406 minus 2.0537 cross times fraction numerator 75 over denominator square root of 60 end fraction equals 386.11513... 406 plus 2.0537 cross times fraction numerator 75 over denominator square root of 60 end fraction equals 425.88486...

Round appropriately

stretchy left square bracket 386 comma space 426 stretchy right square bracket

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    Author
    Reviewer
    Naomi C

    Author: Naomi C

    Expertise: Maths Content Creator

    Naomi graduated from Durham University in 2007 with a Masters degree in Civil Engineering. She has taught Mathematics in the UK, Malaysia and Switzerland covering GCSE, IGCSE, A-Level and IB. She particularly enjoys applying Mathematics to real life and endeavours to bring creativity to the content she creates.

    Dan Finlay

    Reviewer: Dan Finlay

    Expertise: Maths Subject Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.