AP Statistics Flashcards: Constructing a Confidence Interval for a Population Mean
Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026
Review key ideas with interactive flashcards. This set includes 29 cards to help you master important concepts.
What is the 'critical value' in the context of a t-interval?
The critical value, denoted as t*, is a multiplier determined by the confidence level and degrees of freedom, used to calculate the margin of error.
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What is the 'critical value' in the context of a t-interval?
The critical value, denoted as t*, is a multiplier determined by the confidence level and degrees of freedom, used to calculate the margin of error.
What two values are needed to find the specific critical value t* from a table?
To find t* from a table, you need the desired confidence level and the degrees of freedom (n-1).
What distribution does the statistic (x-bar - mu) / (s / sqrt(n)) follow for a normally distributed variable?
For a normally distributed variable, this statistic follows a t-distribution with n-1 degrees of freedom.
Define 't-distribution'.
A t-distribution is a probability distribution that is used to estimate population parameters when the sample size is small and/or the population standard deviation is unknown.
When analyzing matched pairs data, what is the 'sample' that is used for the t-interval calculation?
For matched pairs, the data is first converted into a single sample of differences, and this sample of differences is used for the calculation.
What is the relationship between the standard error of the sample mean and the sample size?
The standard error (s / sqrt(n)) is inversely related to the square root of the sample size; as n increases, the standard error decreases.
In the context of a one-sample t-interval, what do the 'degrees of freedom' represent?
The degrees of freedom, calculated as n-1, determine the specific t-distribution to be used for finding the critical value t*.
If a sample has a mean (x-bar) of 50 and a margin of error of 5, what is the resulting confidence interval?
The confidence interval is constructed as the point estimate ± margin of error, so the interval would be 50 ± 5, or (45, 55).
How does a t-distribution compare to a normal distribution when the sample standard deviation (s) is used instead of the population standard deviation (sigma)?
When s is used instead of sigma, the resulting t-distribution has more area in the tails than a normal distribution.
What methods can be used to find the critical value, t*?
The critical value t* can be found using a t-distribution table or with statistical technology.
What is the effect of increasing the degrees of freedom on the shape of a t-distribution?
As degrees of freedom increase, the tails of a t-distribution get smaller, and the distribution approaches a normal distribution.
What is the standard error for a sample mean (x-bar)?
The standard error for a sample mean is calculated as s / sqrt(n), where s is the sample standard deviation and n is the sample size.
How do you calculate the degrees of freedom for a one-sample t-interval for a population mean?
The degrees of freedom for a one-sample t-interval are calculated as n-1, where n is the sample size.
What is the point estimate for a population mean (μ)?
The point estimate for a population mean is the sample mean, x-bar.
What is the general formula for a confidence interval for a population mean?
The confidence interval for a population mean is calculated as x-bar ± t*(s / sqrt(n)).
What is the first step in calculating a confidence interval for a population mean for matched pairs?
The first step is to calculate the difference for each pair of observations to create a single sample of differences.
What are the two primary conditions to verify before calculating a confidence interval for a population mean?
To estimate a population mean, one must check for the independence of observations and verify that the sampling distribution of the mean is approximately normal.
Why does the t-distribution approach the normal distribution as degrees of freedom increase?
As the sample size (and thus degrees of freedom) increases, the sample standard deviation (s) becomes a more reliable estimate of the population standard deviation (sigma), reducing the extra variability that characterizes the t-distribution.
What is the appropriate confidence interval procedure for a single population mean when the population standard deviation is unknown?
The appropriate procedure is a one-sample t-interval for a population mean.
A researcher wants to estimate the mean weight of a species of bird. They collect a sample of 25 birds, but do not know the population standard deviation. What procedure should they use?
Because the population standard deviation (sigma) is unknown, the researcher should use a one-sample t-interval for a mean.
How is the margin of error for a one-sample t-interval calculated?
The margin of error for a one-sample t-interval is calculated by multiplying the critical value (t*) by the standard error (s / sqrt(n)).
In the confidence interval formula x-bar ± t*(s / sqrt(n)), which part represents the margin of error?
In the formula, the term t*(s / sqrt(n)) represents the margin of error.
Why is it necessary to check that the sampling distribution is approximately normal when constructing a confidence interval for a mean?
This condition must be verified because the calculation of the t-interval relies on the properties of the t-distribution, which assumes an underlying normal model.
What does the 'independence' condition mean when checking requirements for a confidence interval for a mean?
The independence condition requires that the individual observations in the sample are independent of each other, often checked by ensuring random sampling and the 10% condition.
Under what specific condition is a one-sample t-interval for a mean the appropriate procedure?
When the population standard deviation (sigma) is unknown, the appropriate procedure is a one-sample t-interval for a mean.
For a matched pairs t-interval, how are the degrees of freedom calculated?
The degrees of freedom are calculated as n-1, where n is the number of pairs (or the number of differences).
In the confidence interval formula x-bar ± t*(s / sqrt(n)), which part represents the point estimate?
In the formula, the sample mean (x-bar) is the point estimate for the population mean.
How should matched pairs data be treated when constructing a confidence interval?
Matched pairs data can be analyzed as a single sample of the differences between pairs.
What is the primary reason for using a t-distribution instead of a normal distribution for mean inference when sigma is unknown?
Using the sample standard deviation (s) to estimate sigma introduces extra variability, which is accounted for by the wider tails of the t-distribution.