7.2 Constructing a Confidence Interval for a Population Mean
5 min read•january 4, 2023
Josh Argo
Jed Quiaoit
Josh Argo
Jed Quiaoit
The t-distribution is a continuous probability distribution that is used to estimate population parameters when the is small and the is unknown. It is similar to the , but has , which means that it is more likely for observations to fall in the extreme tails of the distribution. This is because the t-distribution accounts for the additional uncertainty introduced by estimating the from the . 🚆
The degrees of freedom (df) in the t-distribution refer to the number of observations in the sample that are free to vary. In other words, it is the number of observations in the sample that are used to estimate the .
As the degrees of freedom increase, the t-distribution becomes more and more similar to the , and the area in the tails decreases. This is because with a larger , the is a more accurate estimate of the , and there is less uncertainty in the distribution of the sample mean.
Because σ (population standard distribution) is typically not known for distributions of quantitative variables, the appropriate
procedure for estimating the population mean of one quantitative variable for one sample is a one-sample t-interval for a mean.
Conditions for Inference
Before proceeding to calculate a , we have to check that our sampling distribution we are using meets some conditions:
(1) Random Sample
This reduces any bias that may be caused from taking a bad sample
When answering inference questions, it is always essential to make note that our sample was random, either by highlighting text on the exam, or by quoting the problem where it details its randomness. 💬
(2) Independence
This ensures that each subject in our sample was not influenced by the previous subjects chosen. While we are sampling without replacement, if our is not super close to our population size, we can conclude that the effect it has on our sampling is negligible. We can check this condition by questioning if it is reasonable to believe that the population in question is at least 10 times as large as our sample. 💙
A good way to state this when performing inference is to say, "It is reasonable to believe that our population (in context) is at least 10n"
For example, if we have a random sample of 85 teenagers math grades and we are creating a for what the average of ALL teenager math grades are, we could state, "It is reasonable to believe that there are at least 850 teenagers currently enrolled in a math class."
(3) Normal
This check verifies that we are able to use a curve to calculate our probabilities using either or z scores. We can verify that a sampling distribution is using the which states that if our is at least 30, we can assume that the sampling distribution will be approximately . Normality with our sampling distribution can also be assumed if it is given that the population distribution is normally distributed. 🔔
With our example with 85 teenagers, we can assume that the sampling distribution of 85 teenagers grades will be a because 85>30.
Formula
A is comprised of two parts: a point estimate and a .
Point Estimate ±
Point Estimate ± (t*) ()
Point Estimate
A point estimate is a single value that is used to estimate a population parameter. For example, if you are trying to estimate the mean of a population, the point estimate would be the sample mean (aka x̄).
The point estimate is the middle of the , and it is the best estimate of the population parameter based on the sample data. In this case, if you're trying to estimate the mean of a population using a sample of data, you would calculate the sample mean as the point estimate.
The would be calculated based on the sample mean and the of the mean, and it would be constructed so that there is a 95% (or another percentage value set by the person in charge of the statistical analysis) chance that the population mean falls within the interval.
Margin of Error
A can be thought about as a "buffer zone." It is the amount that we add and subtract to our sample mean to give some room for error in estimating our population mean. It is made up of two parts:
The is the t-score based on the mean and standard deviation of the sampling distribution, along with the degrees of freedom. Degrees of freedom can be calculated by taking the and subtracting one. Since we have a distribution that is only approximately , the degrees of freedom allow us to adjust our calculations based on how small or large our sample is. If we had an infinite , we would have a perfect curve (which would call for us to use a z-score). A can be calculated using either a calculator's or using the charts on the College Board provided formula sheet. 📄
Meaning of Confidence Interval
image provided by: rossmanchance.com
A is a range of values that we believe the true population mean will fall between. In the example above, we have a 95% when given a sample mean of 0, sample standard deviation of 10 and a of 100. The graphic shows this sampling distribution and how only 5% of the samples would fall outside of the (-2, 2) range. Hence, we can be 95% confident that the true population mean is somewhere between -2 and 2. 😎
Interpretation
On the AP exam, you are typically asked to create and interpret a . 🔨
When asked to do this for a population mean, interpret your interval using the following template:
"I am ___% confident that the true population mean of ______________ is between (___, ___)."
Rubrics generally include the following three aspects:
Confidence level
Context of problem
Demonstrates knowledge that we are inferring about the true population mean
🎥 Watch: AP Stats - Inference: Confidence Intervals for Means
Key Terms to Review (15)
Central Limit Theorem
: The Central Limit Theorem states that as the sample size increases, the sampling distribution of the mean approaches a normal distribution regardless of the shape of the population distribution.Confidence Interval
: A confidence interval is a range of values that is likely to contain the true value of a population parameter. It provides an estimate along with a level of confidence about how accurate the estimate is.Critical Value
: A critical value is a specific value that separates the rejection region from the non-rejection region in hypothesis testing. It is compared to the test statistic to determine whether to reject or fail to reject the null hypothesis.degrees of freedom (df)
: Degrees of freedom refers to the number of values in a calculation that are free to vary without violating any constraints.Empirical Rule
: The empirical rule, also known as the 68-95-99.7 rule, is a statistical guideline that states that for a normal distribution, approximately 68% of the data falls within one standard deviation of the mean, about 95% falls within two standard deviations, and roughly 99.7% falls within three standard deviations.Heavier Tails
: Heavier tails refer to the phenomenon where the probability of extreme events occurring in a distribution is higher than what would be expected in a normal distribution.Independence
: Independence refers to events or variables that do not influence each other. If two events are independent, knowing one event occurred does not affect our knowledge about whether or not the other event will occur.Inverse T Function
: The inverse T function, also known as the quantile function or percent-point function, is a mathematical operation that takes a probability as input and returns the value from a T-distribution that corresponds to that probability.Margin of Error
: The margin of error is a measure of the uncertainty or variability in survey results. It represents the range within which the true population parameter is likely to fall.Normal
: In statistics, normal refers to a distribution that follows a specific bell-shaped curve called the normal distribution. The normal distribution is symmetric and characterized by its mean and standard deviation.Normal Distribution
: A normal distribution is a symmetric bell-shaped probability distribution characterized by its mean and standard deviation. It follows a specific mathematical formula called Gaussian distribution.Population Variance
: Population variance measures how spread out or dispersed data points are from their mean value within an entire population. It provides information about the variability and diversity present in a population.Sample Size
: The sample size refers to the number of individuals or observations included in a study or experiment.Sample Variance
: Sample variance is a measure of how spread out the data points are in a sample. It quantifies the average squared deviation from the mean.Standard Error
: The standard error is a measure of the variability or spread of sample means around the population mean. It tells us how much we can expect sample means to differ from the true population mean.7.2 Constructing a Confidence Interval for a Population Mean
5 min read•january 4, 2023
Josh Argo
Jed Quiaoit
Josh Argo
Jed Quiaoit
The t-distribution is a continuous probability distribution that is used to estimate population parameters when the is small and the is unknown. It is similar to the , but has , which means that it is more likely for observations to fall in the extreme tails of the distribution. This is because the t-distribution accounts for the additional uncertainty introduced by estimating the from the . 🚆
The degrees of freedom (df) in the t-distribution refer to the number of observations in the sample that are free to vary. In other words, it is the number of observations in the sample that are used to estimate the .
As the degrees of freedom increase, the t-distribution becomes more and more similar to the , and the area in the tails decreases. This is because with a larger , the is a more accurate estimate of the , and there is less uncertainty in the distribution of the sample mean.
Because σ (population standard distribution) is typically not known for distributions of quantitative variables, the appropriate
procedure for estimating the population mean of one quantitative variable for one sample is a one-sample t-interval for a mean.
Conditions for Inference
Before proceeding to calculate a , we have to check that our sampling distribution we are using meets some conditions:
(1) Random Sample
This reduces any bias that may be caused from taking a bad sample
When answering inference questions, it is always essential to make note that our sample was random, either by highlighting text on the exam, or by quoting the problem where it details its randomness. 💬
(2) Independence
This ensures that each subject in our sample was not influenced by the previous subjects chosen. While we are sampling without replacement, if our is not super close to our population size, we can conclude that the effect it has on our sampling is negligible. We can check this condition by questioning if it is reasonable to believe that the population in question is at least 10 times as large as our sample. 💙
A good way to state this when performing inference is to say, "It is reasonable to believe that our population (in context) is at least 10n"
For example, if we have a random sample of 85 teenagers math grades and we are creating a for what the average of ALL teenager math grades are, we could state, "It is reasonable to believe that there are at least 850 teenagers currently enrolled in a math class."
(3) Normal
This check verifies that we are able to use a curve to calculate our probabilities using either or z scores. We can verify that a sampling distribution is using the which states that if our is at least 30, we can assume that the sampling distribution will be approximately . Normality with our sampling distribution can also be assumed if it is given that the population distribution is normally distributed. 🔔
With our example with 85 teenagers, we can assume that the sampling distribution of 85 teenagers grades will be a because 85>30.
Formula
A is comprised of two parts: a point estimate and a .
Point Estimate ±
Point Estimate ± (t*) ()
Point Estimate
A point estimate is a single value that is used to estimate a population parameter. For example, if you are trying to estimate the mean of a population, the point estimate would be the sample mean (aka x̄).
The point estimate is the middle of the , and it is the best estimate of the population parameter based on the sample data. In this case, if you're trying to estimate the mean of a population using a sample of data, you would calculate the sample mean as the point estimate.
The would be calculated based on the sample mean and the of the mean, and it would be constructed so that there is a 95% (or another percentage value set by the person in charge of the statistical analysis) chance that the population mean falls within the interval.
Margin of Error
A can be thought about as a "buffer zone." It is the amount that we add and subtract to our sample mean to give some room for error in estimating our population mean. It is made up of two parts:
The is the t-score based on the mean and standard deviation of the sampling distribution, along with the degrees of freedom. Degrees of freedom can be calculated by taking the and subtracting one. Since we have a distribution that is only approximately , the degrees of freedom allow us to adjust our calculations based on how small or large our sample is. If we had an infinite , we would have a perfect curve (which would call for us to use a z-score). A can be calculated using either a calculator's or using the charts on the College Board provided formula sheet. 📄
Meaning of Confidence Interval
image provided by: rossmanchance.com
A is a range of values that we believe the true population mean will fall between. In the example above, we have a 95% when given a sample mean of 0, sample standard deviation of 10 and a of 100. The graphic shows this sampling distribution and how only 5% of the samples would fall outside of the (-2, 2) range. Hence, we can be 95% confident that the true population mean is somewhere between -2 and 2. 😎
Interpretation
On the AP exam, you are typically asked to create and interpret a . 🔨
When asked to do this for a population mean, interpret your interval using the following template:
"I am ___% confident that the true population mean of ______________ is between (___, ___)."
Rubrics generally include the following three aspects:
Confidence level
Context of problem
Demonstrates knowledge that we are inferring about the true population mean
🎥 Watch: AP Stats - Inference: Confidence Intervals for Means
Key Terms to Review (15)
Central Limit Theorem
: The Central Limit Theorem states that as the sample size increases, the sampling distribution of the mean approaches a normal distribution regardless of the shape of the population distribution.Confidence Interval
: A confidence interval is a range of values that is likely to contain the true value of a population parameter. It provides an estimate along with a level of confidence about how accurate the estimate is.Critical Value
: A critical value is a specific value that separates the rejection region from the non-rejection region in hypothesis testing. It is compared to the test statistic to determine whether to reject or fail to reject the null hypothesis.degrees of freedom (df)
: Degrees of freedom refers to the number of values in a calculation that are free to vary without violating any constraints.Empirical Rule
: The empirical rule, also known as the 68-95-99.7 rule, is a statistical guideline that states that for a normal distribution, approximately 68% of the data falls within one standard deviation of the mean, about 95% falls within two standard deviations, and roughly 99.7% falls within three standard deviations.Heavier Tails
: Heavier tails refer to the phenomenon where the probability of extreme events occurring in a distribution is higher than what would be expected in a normal distribution.Independence
: Independence refers to events or variables that do not influence each other. If two events are independent, knowing one event occurred does not affect our knowledge about whether or not the other event will occur.Inverse T Function
: The inverse T function, also known as the quantile function or percent-point function, is a mathematical operation that takes a probability as input and returns the value from a T-distribution that corresponds to that probability.Margin of Error
: The margin of error is a measure of the uncertainty or variability in survey results. It represents the range within which the true population parameter is likely to fall.Normal
: In statistics, normal refers to a distribution that follows a specific bell-shaped curve called the normal distribution. The normal distribution is symmetric and characterized by its mean and standard deviation.Normal Distribution
: A normal distribution is a symmetric bell-shaped probability distribution characterized by its mean and standard deviation. It follows a specific mathematical formula called Gaussian distribution.Population Variance
: Population variance measures how spread out or dispersed data points are from their mean value within an entire population. It provides information about the variability and diversity present in a population.Sample Size
: The sample size refers to the number of individuals or observations included in a study or experiment.Sample Variance
: Sample variance is a measure of how spread out the data points are in a sample. It quantifies the average squared deviation from the mean.Standard Error
: The standard error is a measure of the variability or spread of sample means around the population mean. It tells us how much we can expect sample means to differ from the true population mean.
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