Confidence Intervals for Population Means (College Board AP® Statistics)

Study Guide

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Naomi C

Written by: Naomi C

Reviewed by: Dan Finlay

One-sample t-interval for a mean

What is a confidence interval for a population mean?

  • A confidence interval for a population mean is

    • a symmetric range of values centered about the sample mean

    • designed to capture the actual value of the population mean

  • Different samples generate different confidence intervals

    • e.g. a sample mean of 5 may have a confidence interval of (4.5, 5.5)

How do I calculate a confidence interval for a population mean?

  • The confidence interval for a population mean is given by

    • sample space mean space plus-or-minus open parentheses critical space value close parentheses open parentheses standard space error space of space sample space mean close parentheses

  • Where:

    • The sample mean is calculated from the sample or is given to you

    • The critical value is the relevant t-value

      • The critical value depends on the confidence level C%

    • The standard error is an estimate of the population standard deviation from the data, fraction numerator s over denominator square root of n end fraction

Examiner Tips and Tricks

The general formula for confidence intervals (including a table of standard errors) is given in the exam: statistic space plus-or-minus open parentheses critical space value close parentheses open parentheses standard space error space of space statistic close parentheses.

You will need to apply it appropriately using the sample mean and the standard error of the sample mean.

What are conditions for a confidence interval for a population mean?

  • When calculating a confidence interval, you must show that:

    • items in the sample (or experiment) must satisfy the independence condition

      • by verifying that data is collected by random sampling

      • or random assignment (in an experiment)

      • and, if sampling without replacement, showing that the sample size is less than 10% of the population size

    • the population is approximately normally distributed

      • The distribution needs to be approximately symmetric

      • There should be no outliers

What is the margin of error?

  • The margin of error is the half-width of the confidence interval

    • margin space of space error space equals open parentheses critical space value close parentheses open parentheses standard space error space of space sample space mean close parentheses

  • The confidence interval is

    • sample space mean plus-or-minus margin space of space error

  • The total width of a confidence interval is 2 cross times margin space of space error

  • You may be given an interval and asked to calculate its margin of error

    • or another value, such as n

      • This involves forming and solving an equation

Examiner Tips and Tricks

You need to know that the width of a confidence interval increases as the confidence level increases, whereas it decreases as the sample size increases!

How do I interpret a confidence interval for a population mean?

  • You must conclude calculations of a confidence interval by referring to the context

    • Start by saying 'we can be C% confident that the interval from [lower limit] to [upper limit]...'

      • using the limits from the confidence interval

    • then end with it capturing the population mean in context

      • e.g. 'captures the actual population mean of the time taken by students in the school to run 100 m'

How do I use confidence intervals to justify a claim about a population mean?

  • If a population mean is claimed to be a specific value

    • check if that value lies in your confidence interval

  • If it does, the sample data provides sufficient evidence that the population mean is that value

    • If it does not, the sample data does not provide sufficient evidence that the population mean is that value

Worked Example

A factory produces pre-packaged noodles, where each packet is expected to contain 90 g of dried noodles. It is assumed that the distribution of the weights of dried noodles is approximately normal. A simple random sample of 24 packets, from a recent batch of 1000 packets, were selected to see if they contained the correct amount of dried noodles. The sample had a mean weight of 87 g and a standard deviation of 6 g. Based on this sample, what is a 95% confidence interval for the mean weight of dried noodles per packet?

State the type of interval being used and verify the conditions for the interval

The correct inference procedure is a one-sample t-interval with a 95% confidence level

  • The independence condition is satisfied, as

    • the sample of 24 packets was selected at random

    • and the sample size, 24, is less than 10% of the population, 1000 (24 < 100)

  • The distribution of weights is normal

  • The sample size is small (n equals 24, which is less than 30) and the population standard deviation is unknown, so the t-distribution can be used

Define the population parameter, mu, this is what we are trying to find

Let mu be the mean weight of noodles per packet that a factory produces

List the number of data items in the sample, n, the sample mean, x with bar on top, and the sample standard deviation, s subscript x

table row n equals 24 row cell x with bar on top end cell equals 87 row cell s subscript x end cell equals 6 end table

State the degrees of freedom (dof)

dof = 24 - 1 = 23

Using the t-table, find the t-value (critical value) for the sample mean, using dof = 23 and a confidence level of 95%

Remember that a confidence level of 95% is 5% in both tails combined, so use 2.5% for a single tail in the table (the row 'Confidence level C' at the bottom of the t-tables helps)

t-value = 2.069

Using the formula from the formula sheet, Confidence space interval equals statistic plus-or-minus open parentheses critical space value close parentheses space open parentheses standard space error space of space statistic close parentheses, calculate the confidence interval

table row CI equals cell x with bar on top plus-or-minus t asterisk times times fraction numerator s subscript x over denominator square root of n end fraction end cell row blank equals cell 87 plus-or-minus 2.069 times fraction numerator 6 over denominator square root of 24 end fraction end cell end table

State the confidence interval

open parentheses 84.41 comma space 89.59 close parentheses

Explain the confidence interval in the context of the question

We can be 95% confident that the interval from 84.41 g to 89.59 g captures the actual value of the population mean weight of dried noodles per packet

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Naomi C

Author: Naomi C

Expertise: Maths

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

Author: Dan Finlay

Expertise: Maths 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.