Confidence Intervals for Population Means (College Board AP® Statistics)
Study Guide
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
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,
Examiner Tips and Tricks
The general formula for confidence intervals (including a table of standard errors) is given in the exam: .
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
The confidence interval is
The total width of a confidence interval is
You may be given an interval and asked to calculate its margin of error
or another value, such as
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 (, which is less than 30) and the population standard deviation is unknown, so the t-distribution can be used
Define the population parameter, , this is what we are trying to find
Let be the mean weight of noodles per packet that a factory produces
List the number of data items in the sample, , the sample mean, , and the sample standard deviation,
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, , calculate the confidence interval
State the confidence interval
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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