Inferential Statistics and Sampling Distributions Quiz

Inferential Statistics and Sampling Distributions

• The text discusses the impracticality of using betting as a general procedure for determining the likelihood of particular outcomes in inferential statistics.
• Due to time and resource limitations, usually only one sample can be investigated from a population.
• Inferential statistics are used to infer characteristics of a population from sample data.
• Sampling distributions, derived from probability theory, provide a solution to inferring population characteristics from a single sample.
• Sampling distributions are derived mathematically and serve the same purpose as empirically derived distributions.
• A sampling distribution is created by specifying all possible outcomes of a measurement and assigning probabilities to each outcome.
• Theoretical or hypothetical distributions of a sample statistic are called sampling distributions.
• A sampling distribution allows making quantitative statements about the probabilities associated with getting a particular outcome in a single sample.
• The text illustrates constructing a mathematically derived sampling distribution using the example of flipping a hypothetical coin.
• It is assumed that the two possible outcomes of a single coin flip are equally probable (50% probability each).
• The probabilities associated with the possible results of a series of trials of coin flips are not equal, and a mathematical probability model known as the binomial distribution can be used to determine the exact probability associated with each possible outcome.
• Probability theory is used to provide insights into the expectations of outcomes in a series of trials, based on initial assumptions about the equal probabilities of each possible outcome in a single trial.

Understanding Sampling Distributions and Inference

• The standard deviation of a theoretically derived normal sampling distribution is called its standard error, representing a distribution of sampling error.
• Probability sampling involves taking a probability sample and determining the sampling distribution of the sample statistic to reduce uncertainties about sample representativeness.
• The sampling distribution indicates the probabilities of securing a particular result, and many probability sampling distributions have the shape of the normal curve.
• Probability sampling and inferential statistics help in reducing uncertainties associated with using samples to represent populations.
• The binomial distribution is useful for creating a sampling distribution when there are two equally probable outcomes in a single trial, producing a normal sampling distribution for large samples.
• The sampling distribution for sample means is also a normal distribution, and the central limit theorem provides the basis for its mathematical derivation.
• The central limit theorem states that the sampling distribution of means for large samples will be a normal curve, with a mean equal to the mean of the scores for the population.
• An unbiased estimate of the corresponding population parameter is referred to as an unbiased estimator, such as a sample mean.
• The mean of the sampling distribution of standard deviations (for large samples) is not equivalent to the population standard deviation; a sample standard deviation is therefore referred to as a biased estimate of the population standard deviation.
• Not all sampling distributions are normal distributions; for example, the sampling distribution for standard deviations is skewed for small samples (n ≤ 100).
• When the sampling distribution is not normal, it may be possible to determine the shape of the distribution through the application of other mathematical models or by using a computer simulation.
• Statisticians adjust the calculation formulas to improve the estimate when sample statistics are recognized as biased estimators.

Testing Research Hypotheses and Null Hypotheses

• Alternative hypothesis is formulated as a null hypothesis, assuming no changes in the dependent variable except those from random sampling error
• Sampling distribution constructed based on the null hypothesis assumption helps determine the probability of a result occurring if the null hypothesis is correct
• Research hypothesis is indirectly tested by testing the null hypothesis directly using inferential statistics
• Rejection of the null hypothesis when observed results have a low probability involves taking a risk, emphasizing the importance of replication studies
• Type I error occurs when a true null hypothesis is rejected, while Type II error happens when a false null hypothesis is accepted
• Estimating the probability of making a Type II error is complex, while estimating the risk of making a Type I error is less difficult
• Research hypothesis example: Children watching violent TV program will exhibit more violent play than those watching a nonviolent program
• Assertion about the mean number of violent acts during play being higher for children watching violent TV program is an assertion about a population parameter
• To test the null hypothesis, a sampling distribution for the differences between two sample means is needed
• Student’s t statistic used to determine the probability that mean values of the same variable from two different randomly selected groups are from the same population
• Calculation of z for the observed mean differences and referencing a z table helps determine the probability of obtaining the observed difference by chance
• Contradictory values and rejection region are used to reject the null hypothesis based on highly improbable values of a statistic within the rejection region

Analysis of Variance (ANOVA) and F Distribution

• ANOVA is a statistical test used to compare differences between three or more means and variances when the independent variable is nominal or ordinal.
• It is appropriate when the independent variable is measured at the nominal or ordinal level and the dependent variable is interval or ratio.
• The null hypothesis in ANOVA is that the means are equal, and the sample statistic of concern is the F statistic, which follows the F distribution.
• The sampling distribution of the F statistic is a ratio of explained to unexplained variance, with the between-group variance and within-group variance being the two variances in ANOVA.
• The F distribution varies according to the size of both samples whose variances comprise the fraction used for calculating F.
• To determine the critical value of F for a chosen significance level, the F table corresponding to the selected significance level is consulted, and the critical value of F is located in that table corresponding to the sample sizes involved.
• If the calculated F value is smaller than the critical value in the table, it can be inferred that the population variances do not differ. If it is larger, equal variances cannot be assumed, and an alternative t formula must be used.
• The F ratio, in slightly modified form, is considered again in the discussion of the analysis of variance in the next section.
• The total variance in ANOVA is partitioned into the sum of two other variances: the within-group variance and the between-group variance.
• The within-group variance is the variance for each group separately, and the between-group variance is the variance of the mean scores from the grand mean.
• The grand mean is derived from the individual scores from all groups, and the total variance is obtained from the squared deviations of the scores from the grand mean.
• ANOVA is used to answer research questions such as whether the means of multiple groups differ significantly, with the null hypothesis being that they do not differ, and any observed differences are due to sampling error.

Hypothesis Testing and Cautions in Inferential Statistics

• Degrees of freedom (df) must be calculated for each table in chi square test, with df = (r-1)(c-1)
• In the example, with 2 rows and 2 columns, the degrees of freedom are 1
• A chi square of this size would occur by chance less than 1% of the time, leading to rejection of the null hypothesis
• Chi square helps determine the existence of a relationship between variables, but not the strength of the relationship
• Chi square can be used with various table sizes and is distribution-free, suitable for non-normally distributed variables
• It is most appropriate when both variables are measured at the nominal level
• Chi square formula should only be used with raw frequency counts, not percentages or rates
• The magnitude of chi square is related to sample size, with large samples likely to yield statistically significant results
• Correlation coefficient can be used for hypothesis testing when both variables are at the interval or ratio level
• Significance levels and confidence intervals provide the probability of obtaining a result by chance, not its cause
• Random assignment in research design does not ensure typicality of the larger population
• Inferential statistics do not compensate for errors in research design, sampling procedures, or data recording