| Term | Definition |
|---|---|
| probabilities of errors | The likelihood or chance that errors will occur in statistical inference. |
| statistical inference | The process of drawing conclusions about a population based on data collected from a sample. |
| variation | Differences in data that occur by chance due to the random nature of sampling, rather than from systematic causes. |
| Term | Definition |
|---|---|
| confidence interval | A range of values, calculated from sample data, that is likely to contain the true population parameter with a specified level of confidence. |
| confidence interval procedure | A statistical method used to construct an interval estimate for a population parameter based on sample data. |
| critical value | A value from the standard normal distribution used to determine the margin of error for a given confidence level. |
| degrees of freedom | A parameter of the t-distribution that affects its shape; as degrees of freedom increase, the t-distribution approaches the normal distribution. |
| density curve | A graphical representation of a probability distribution showing the relative likelihood of different values. |
| independence | The condition that observations in a sample are not influenced by each other, typically ensured through random sampling or randomized experiments. |
| margin of error | The amount by which a sample statistic is likely to vary from the corresponding population parameter, calculated as the critical value times the standard error. |
| matched pairs | Paired observations where two measurements are taken on the same subject or on subjects that are matched according to specific criteria, used to analyze the mean difference between the paired values. |
| mean difference | The average of the differences between paired observations, denoted by μd, where the order of subtraction must be clearly defined. |
| normal distribution | A probability distribution that is mound-shaped and symmetric, characterized by a population mean (μ) and population standard deviation (σ). |
| one-sample t-interval | A confidence interval for a population mean constructed using the t-distribution when the population standard deviation is unknown. |
| outlier | Data points that are unusually small or large relative to the rest of the data. |
| population mean | The average of all values in an entire population, denoted as μ. |
| population means | The average values of two distinct populations being compared, denoted as μ₁ and μ₂. |
| population standard deviation | A measure of the spread or dispersion of all values in a population, denoted by σ, which is a parameter of the normal distribution. |
| random sample | A sample selected from a population in such a way that every member has an equal chance of being chosen, reducing bias and allowing for valid statistical inference. |
| randomized experiment | A study design where subjects are randomly assigned to treatment groups to establish cause-and-effect relationships. |
| sample mean | The average of all values in a sample, denoted as x̄, used as an estimate of the population mean. |
| sample size | The number of observations or data points collected in a sample, denoted as n. |
| sample standard deviation | The standard deviation calculated for a sample, denoted by s, using the formula s = √(1/(n-1) ∑(xᵢ-x̄)²). |
| sample statistic | A numerical value calculated from sample data that is used to estimate the corresponding population parameter. |
| sampling distribution | The probability distribution of a sample statistic (such as a sample proportion) obtained from repeated sampling of a population. |
| sampling without replacement | A sampling method in which an item selected from a population cannot be selected again in subsequent draws. |
| skewness | A measure of the asymmetry of a distribution, indicating whether data is concentrated more on one side of the center. |
| standard error | The standard deviation of a sampling distribution, which measures the variability of a sample statistic across repeated samples. |
| t-distribution | A probability distribution used when the population standard deviation is unknown and the sample standard deviation is used instead, characterized by heavier tails than the normal distribution. |
| tails | The extreme regions at both ends of a probability distribution's density curve where the t-distribution allocates more area than the normal distribution. |
| Term | Definition |
|---|---|
| confidence interval | A range of values, calculated from sample data, that is likely to contain the true population parameter with a specified level of confidence. |
| confidence level | The probability that a confidence interval will contain the true population parameter, typically expressed as a percentage such as 90%, 95%, or 99%. |
| margin of error | The amount by which a sample statistic is likely to vary from the corresponding population parameter, calculated as the critical value times the standard error. |
| matched pairs | Paired observations where two measurements are taken on the same subject or on subjects that are matched according to specific criteria, used to analyze the mean difference between the paired values. |
| population | The entire group of individuals or items from which a sample is drawn and about which conclusions are to be made. |
| population mean | The average of all values in an entire population, denoted as μ. |
| population means | The average values of two distinct populations being compared, denoted as μ₁ and μ₂. |
| random sample | A sample selected from a population in such a way that every member has an equal chance of being chosen, reducing bias and allowing for valid statistical inference. |
| sample | A subset of individuals or items selected from a population for the purpose of data collection and analysis. |
| sample size | The number of observations or data points collected in a sample, denoted as n. |
| width of a confidence interval | The range or span of a confidence interval, calculated as the difference between the upper and lower bounds of the interval. |
| Term | Definition |
|---|---|
| 10% condition | The requirement that sample size n is at most 10% of the population size N to ensure independence when sampling without replacement. |
| alternative hypothesis | The claim that contradicts the null hypothesis, representing what the researcher is trying to find evidence for. |
| approximately normal | A distribution that closely follows the shape of a normal distribution, allowing for the use of normal probability methods. |
| conditions for the test | The requirements that must be satisfied before conducting a hypothesis test for a population mean, including independence and normality of the sampling distribution. |
| independence | The condition that observations in a sample are not influenced by each other, typically ensured through random sampling or randomized experiments. |
| matched pairs | Paired observations where two measurements are taken on the same subject or on subjects that are matched according to specific criteria, used to analyze the mean difference between the paired values. |
| mean difference | The average of the differences between paired observations, denoted by μd, where the order of subtraction must be clearly defined. |
| null hypothesis | The initial claim or assumption being tested in a hypothesis test, typically stating that there is no effect or no difference. |
| one-sample t-test | A hypothesis test used to determine whether a population mean differs from a hypothesized value when the population standard deviation is unknown. |
| outlier | Data points that are unusually small or large relative to the rest of the data. |
| population mean | The average of all values in an entire population, denoted as μ. |
| population means | The average values of two distinct populations being compared, denoted as μ₁ and μ₂. |
| random sample | A sample selected from a population in such a way that every member has an equal chance of being chosen, reducing bias and allowing for valid statistical inference. |
| randomized experiment | A study design where subjects are randomly assigned to treatment groups to establish cause-and-effect relationships. |
| sample size | The number of observations or data points collected in a sample, denoted as n. |
| sampling distribution | The probability distribution of a sample statistic (such as a sample proportion) obtained from repeated sampling of a population. |
| sampling without replacement | A sampling method in which an item selected from a population cannot be selected again in subsequent draws. |
| significance test | A statistical procedure used to determine whether there is sufficient evidence to reject the null hypothesis based on sample data. |
| skewness | A measure of the asymmetry of a distribution, indicating whether data is concentrated more on one side of the center. |
| Term | Definition |
|---|---|
| degrees of freedom | A parameter of the t-distribution that affects its shape; as degrees of freedom increase, the t-distribution approaches the normal distribution. |
| matched pairs | Paired observations where two measurements are taken on the same subject or on subjects that are matched according to specific criteria, used to analyze the mean difference between the paired values. |
| null hypothesis | The initial claim or assumption being tested in a hypothesis test, typically stating that there is no effect or no difference. |
| p-value | The probability of observing a test statistic as extreme as or more extreme than the one calculated from the sample data, assuming the null hypothesis is true. |
| population mean | The average of all values in an entire population, denoted as μ. |
| population means | The average values of two distinct populations being compared, denoted as μ₁ and μ₂. |
| reject the null hypothesis | The decision made when the p-value is less than or equal to the significance level, indicating sufficient evidence against the null hypothesis. |
| sample mean | The average of all values in a sample, denoted as x̄, used as an estimate of the population mean. |
| sampling distribution | The probability distribution of a sample statistic (such as a sample proportion) obtained from repeated sampling of a population. |
| significance level | The threshold probability (α) used to determine whether to reject the null hypothesis in a significance test. |
| significance test | A statistical procedure used to determine whether there is sufficient evidence to reject the null hypothesis based on sample data. |
| standard error | The standard deviation of a sampling distribution, which measures the variability of a sample statistic across repeated samples. |
| t-distribution | A probability distribution used when the population standard deviation is unknown and the sample standard deviation is used instead, characterized by heavier tails than the normal distribution. |
| test statistic | A calculated value used to determine whether to reject the null hypothesis in a hypothesis test, computed from sample data. |
| Term | Definition |
|---|---|
| approximately normal | A distribution that closely follows the shape of a normal distribution, allowing for the use of normal probability methods. |
| confidence interval | A range of values, calculated from sample data, that is likely to contain the true population parameter with a specified level of confidence. |
| confidence interval procedure | A statistical method used to construct an interval estimate for a population parameter based on sample data. |
| critical value | A value from the standard normal distribution used to determine the margin of error for a given confidence level. |
| degrees of freedom | A parameter of the t-distribution that affects its shape; as degrees of freedom increase, the t-distribution approaches the normal distribution. |
| difference of population means | The difference between the mean values of two distinct populations, calculated as μ₁ - μ₂. |
| independence | The condition that observations in a sample are not influenced by each other, typically ensured through random sampling or randomized experiments. |
| independent samples | Two or more separate groups of data where the values in one group do not influence or depend on the values in another group. |
| margin of error | The amount by which a sample statistic is likely to vary from the corresponding population parameter, calculated as the critical value times the standard error. |
| normal distribution | A probability distribution that is mound-shaped and symmetric, characterized by a population mean (μ) and population standard deviation (σ). |
| population means | The average values of two distinct populations being compared, denoted as μ₁ and μ₂. |
| population standard deviations | The measure of spread in each of two populations; when unknown, sample standard deviations are used as estimates. |
| quantitative variable | A variable that is measured numerically and can take on a range of values, allowing for mathematical operations and statistical analysis. |
| randomized experiment | A study design where subjects are randomly assigned to treatment groups to establish cause-and-effect relationships. |
| sample mean | The average of all values in a sample, denoted as x̄, used as an estimate of the population mean. |
| sample standard deviations | The measures of variability within each of the two samples, denoted as s₁ and s₂. |
| sample statistic | A numerical value calculated from sample data that is used to estimate the corresponding population parameter. |
| sampling distribution | The probability distribution of a sample statistic (such as a sample proportion) obtained from repeated sampling of a population. |
| sampling without replacement | A sampling method in which an item selected from a population cannot be selected again in subsequent draws. |
| simple random sample | A sample selected from a population such that every possible sample of the same size has an equal chance of being chosen. |
| skewed distributions | Distributions that are not symmetric, with data concentrated on one side and a tail extending to the other side. |
| standard error | The standard deviation of a sampling distribution, which measures the variability of a sample statistic across repeated samples. |
| t-distribution | A probability distribution used when the population standard deviation is unknown and the sample standard deviation is used instead, characterized by heavier tails than the normal distribution. |
| two-sample t-interval | A confidence interval procedure used to estimate the difference between two population means using sample data from two independent samples. |
| Term | Definition |
|---|---|
| confidence interval | A range of values, calculated from sample data, that is likely to contain the true population parameter with a specified level of confidence. |
| difference in sample means | The result of subtracting one sample mean from another sample mean, calculated as x̄₁ - x̄₂. |
| difference of population means | The difference between the mean values of two distinct populations, calculated as μ₁ - μ₂. |
| population means | The average values of two distinct populations being compared, denoted as μ₁ and μ₂. |
| random sampling | A method of selecting samples from a population where each member has an equal chance of being chosen, ensuring the sample is representative of the population. |
| sample size | The number of observations or data points collected in a sample, denoted as n. |
| width of a confidence interval | The range or span of a confidence interval, calculated as the difference between the upper and lower bounds of the interval. |
| Term | Definition |
|---|---|
| alternative hypothesis | The claim that contradicts the null hypothesis, representing what the researcher is trying to find evidence for. |
| approximately normal | A distribution that closely follows the shape of a normal distribution, allowing for the use of normal probability methods. |
| difference of population means | The difference between the mean values of two distinct populations, calculated as μ₁ - μ₂. |
| independence | The condition that observations in a sample are not influenced by each other, typically ensured through random sampling or randomized experiments. |
| null hypothesis | The initial claim or assumption being tested in a hypothesis test, typically stating that there is no effect or no difference. |
| outlier | Data points that are unusually small or large relative to the rest of the data. |
| population means | The average values of two distinct populations being compared, denoted as μ₁ and μ₂. |
| quantitative variable | A variable that is measured numerically and can take on a range of values, allowing for mathematical operations and statistical analysis. |
| randomized experiment | A study design where subjects are randomly assigned to treatment groups to establish cause-and-effect relationships. |
| sampling distribution | The probability distribution of a sample statistic (such as a sample proportion) obtained from repeated sampling of a population. |
| sampling without replacement | A sampling method in which an item selected from a population cannot be selected again in subsequent draws. |
| significance test | A statistical procedure used to determine whether there is sufficient evidence to reject the null hypothesis based on sample data. |
| simple random sample | A sample selected from a population such that every possible sample of the same size has an equal chance of being chosen. |
| skewness | A measure of the asymmetry of a distribution, indicating whether data is concentrated more on one side of the center. |
| two-sample t-test | A statistical test used to determine whether there is a significant difference between the means of two independent population samples. |
| Term | Definition |
|---|---|
| degrees of freedom | A parameter of the t-distribution that affects its shape; as degrees of freedom increase, the t-distribution approaches the normal distribution. |
| difference in sample means | The result of subtracting one sample mean from another sample mean, calculated as x̄₁ - x̄₂. |
| difference of population means | The difference between the mean values of two distinct populations, calculated as μ₁ - μ₂. |
| normal distribution | A probability distribution that is mound-shaped and symmetric, characterized by a population mean (μ) and population standard deviation (σ). |
| null hypothesis | The initial claim or assumption being tested in a hypothesis test, typically stating that there is no effect or no difference. |
| p-value | The probability of observing a test statistic as extreme as or more extreme than the one calculated from the sample data, assuming the null hypothesis is true. |
| population means | The average values of two distinct populations being compared, denoted as μ₁ and μ₂. |
| quantitative variable | A variable that is measured numerically and can take on a range of values, allowing for mathematical operations and statistical analysis. |
| randomized experiment | A study design where subjects are randomly assigned to treatment groups to establish cause-and-effect relationships. |
| reject the null hypothesis | The decision made when the p-value is less than or equal to the significance level, indicating sufficient evidence against the null hypothesis. |
| sampling distribution | The probability distribution of a sample statistic (such as a sample proportion) obtained from repeated sampling of a population. |
| significance level | The threshold probability (α) used to determine whether to reject the null hypothesis in a significance test. |
| significance test | A statistical procedure used to determine whether there is sufficient evidence to reject the null hypothesis based on sample data. |
| simple random sample | A sample selected from a population such that every possible sample of the same size has an equal chance of being chosen. |
| standard error | The standard deviation of a sampling distribution, which measures the variability of a sample statistic across repeated samples. |
| statistical reasoning | The logical process of using sample data and significance test results to draw conclusions about populations and answer research questions. |
| t-distribution | A probability distribution used when the population standard deviation is unknown and the sample standard deviation is used instead, characterized by heavier tails than the normal distribution. |
| test statistic | A calculated value used to determine whether to reject the null hypothesis in a hypothesis test, computed from sample data. |
| two-sample test | A significance test used to compare the means of two different populations based on sample data from each population. |