Glossary

8.2 A Single Population Mean using the Student t Distribution

2 min read•june 25, 2024

The Student's is crucial for small sample sizes and unknown population standard deviations. It allows for accurate confidence intervals and , adapting to different .

As sample size increases, the t-distribution approaches the . This flexibility makes it a powerful tool for in various real-world scenarios with limited data.

Confidence Intervals and the Student's t-Distribution

Confidence intervals using t-distribution

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  • Used when sample size is small (typically < 30) and is unknown
  • To calculate:
    1. Determine sample mean (xˉ\bar{x})
    2. Calculate (ss)
    3. Find ([t∗](https://www.fiveableKeyTerm:t∗)[t^*](https://www.fiveableKeyTerm:t^*)) based on desired confidence level and
    4. Use formula: xˉ±t∗s[n](https://www.fiveableKeyTerm:n)\bar{x} \pm t^* \frac{s}{\sqrt{[n](https://www.fiveableKeyTerm:n)}}
    • xˉ\bar{x} sample mean
    • t∗t^* critical t-value
    • ss sample
    • nn sample size
  • Example: Calculating 95% for mean height of students in a class (sample size 20)
  • The term sn\frac{s}{\sqrt{n}} in the formula represents the of the mean
  • The product t∗snt^* \frac{s}{\sqrt{n}} is known as the

Degrees of freedom in t-distribution

  • Represent number of independent pieces of information in a sample
    • For single sample, df = n - 1 (n sample size)
  • As degrees of freedom increase:
    • t-distribution becomes more similar to
    • Tails of distribution become thinner
    • Peak of distribution becomes lower
  • Lower degrees of freedom result in:
    • Thicker tails in distribution
    • Higher peak in distribution
    • Wider confidence intervals for given confidence level
  • Example: t-distribution with 5 degrees of freedom has thicker tails and higher peak than t-distribution with 20 degrees of freedom

T-distribution vs normal distribution

  • Similarities:
    • Both and
    • Both have mean of 0
    • Both used for statistical inference
  • Differences:
    • t-distribution has thicker tails and lower peak compared to normal distribution
    • t-distribution used when population standard deviation unknown and sample size small
    • Normal distribution used when population standard deviation known or sample size large (typically > 30)
  • Shape of t-distribution depends on degrees of freedom, while normal distribution has fixed shape
  • As degrees of freedom increase, t-distribution approaches normal distribution
  • Example: t-distribution with 2 degrees of freedom looks different from normal distribution, but t-distribution with 100 degrees of freedom looks very similar to normal distribution

Hypothesis Testing with t-Distribution

  • Used to make inferences about population parameters based on sample data
  • Steps in hypothesis testing:
    1. State null and alternative hypotheses
    2. Choose significance level
    3. Calculate test statistic ()
    4. Determine critical value or p-value
    5. Make decision and draw conclusion
  • The t-statistic is calculated using the formula: t=xˉ−μ0s/nt = \frac{\bar{x} - \mu_0}{s/\sqrt{n}}
  • The resulting is compared to critical values from the t-distribution to make decisions about the hypothesis

Key Terms to Review (23)

Absolute value of a residual: The absolute value of a residual is the non-negative difference between an observed value and the corresponding predicted value from a regression model. It measures the magnitude of prediction errors without considering their direction.
Bell-Shaped: A bell-shaped curve, also known as a normal distribution, is a symmetrical, unimodal probability distribution that is shaped like a bell. It is characterized by a single peak at the mean, with the data points tapering off evenly on both sides, creating a symmetrical, bell-like appearance. This distribution is widely observed in various natural and statistical phenomena, making it a fundamental concept in probability and statistics.
Confidence Interval: A confidence interval is a range of values used to estimate the true value of a population parameter, such as a mean or proportion, based on sample data. It provides a measure of uncertainty around the sample estimate, indicating how much confidence we can have that the interval contains the true parameter value.
Critical t-value: The critical t-value is a specific value from the t-distribution that is used as a reference point to determine whether a sample statistic is statistically significant in the context of hypothesis testing. It serves as the threshold for making decisions about the null hypothesis in a single population mean test using the Student's t-distribution.
Degrees of freedom: Degrees of freedom refer to the number of independent values or quantities that can vary in an analysis without breaking any constraints. In statistical calculations, they help determine the accuracy of variance estimates.
Degrees of Freedom: Degrees of freedom refer to the number of independent values or quantities that can vary in a statistical calculation without breaking any constraints. It plays a crucial role in determining the appropriate statistical tests and distributions used for hypothesis testing, estimation, and data analysis across various contexts.
Hypothesis Testing: Hypothesis testing is a statistical method used to determine whether a claim or hypothesis about a population parameter is likely to be true or false based on sample data. It involves formulating a null hypothesis and an alternative hypothesis, collecting and analyzing sample data, and making a decision to either reject or fail to reject the null hypothesis.
Margin of Error: The margin of error is a statistical measure that quantifies the amount of uncertainty or imprecision in a sample statistic, such as the sample mean or proportion. It represents the range of values above and below the sample statistic within which the true population parameter is expected to fall, with a given level of confidence.
Measures of the Spread of the Data: Measures of the spread of data refer to the statistical concepts that describe the dispersion or variability of a dataset. These measures provide information about how the data points are distributed around the central tendency, which is crucial for understanding the characteristics of a dataset and making informed statistical inferences. The key measures of the spread of data are particularly relevant in the context of topics such as 2.7 Measures of the Spread of the Data, 7.1 The Central Limit Theorem for Sample Means (Averages), and 8.2 A Single Population Mean using the Student t Distribution.
N: The variable 'n' is a fundamental concept in probability and statistics, representing the number of trials or observations in a given experiment or sample. It is a crucial parameter that appears in various statistical distributions and theorems, providing crucial information about the size and structure of the data being analyzed.
Normal distribution: A normal distribution is a continuous probability distribution that is symmetrical around its mean, with a characteristic bell-shaped curve. In a normal distribution, most of the data points are concentrated around the mean, and the probabilities for values further away from the mean taper off equally in both directions.
Population Standard Deviation: The population standard deviation is a measure of the amount of variation or dispersion in a set of values from the mean of that population. It provides insight into how spread out the values are within a complete population, helping to understand the consistency of data points relative to their mean. This concept connects with various statistical principles, including the use of sampling techniques, measures of data spread, the behavior of distributions, and how these concepts are applied when estimating population parameters.
Sample Standard Deviation: Sample standard deviation is a measure of the amount of variation or dispersion in a set of sample data points. It quantifies how much the individual data points differ from the sample mean, providing insight into the spread and reliability of the sample data. A smaller sample standard deviation indicates that the data points are closer to the mean, while a larger value suggests greater variability in the data.
Standard deviation: Standard deviation is a measure of the dispersion or spread of a set of data points around its mean. It quantifies how much the individual data points deviate from the mean value.
Standard Error: Standard error is a statistical term that measures the accuracy with which a sample represents a population. It quantifies the variability of sample means from the true population mean, helping to determine how much sampling error exists when making inferences about the population.
Standard normal distribution: The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. It is used to standardize scores from different normal distributions for comparison.
Standard Normal Distribution: The standard normal distribution, also known as the Z-distribution, is a special case of the normal distribution where the mean is 0 and the standard deviation is 1. It is a fundamental concept in statistics that is used to model and analyze data that follows a normal distribution.
Statistical Inference: Statistical inference is the process of using data analysis and probability theory to draw conclusions about a population from a sample. It allows researchers to make educated guesses or estimates about unknown parameters or characteristics of a larger group based on the information gathered from a smaller, representative subset.
Symmetric: Symmetric refers to a balanced and equal distribution of data, where the left and right sides of a graph mirror each other. In this context, a symmetric distribution indicates that the mean, median, and mode are all located at the center, creating a visually appealing shape that is often associated with normal distributions. When analyzing data, recognizing symmetry helps in understanding the overall behavior and characteristics of the dataset.
T-distribution: The t-distribution is a continuous probability distribution that is used to make inferences about the mean of a population when the sample size is small and the population standard deviation is unknown. It is closely related to the normal distribution and is commonly used in statistical hypothesis testing and the construction of confidence intervals.
T-score: The t-score is a statistical measure that represents the number of standard deviations a data point is from the mean of a population. It is used when the population standard deviation is unknown, and the sample size is small. The t-score is central to understanding various statistical concepts, including hypothesis testing and confidence intervals.
T-statistic: The t-statistic is a statistical measure used to determine the probability that the difference between two sample means is due to chance. It is commonly employed in hypothesis testing to assess the significance of the difference between a sample mean and a hypothesized population mean, or the difference between two sample means.
T^*: t^* is a test statistic used in statistical inference, specifically in the context of estimating a single population mean using the Student's t-distribution. It represents the standardized difference between the sample mean and the hypothesized population mean, taking into account the sample size and the standard deviation of the sample.
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