Inferential Statistics and Sampling Distributions Quiz

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Questions and Answers

What is the goal of a researcher when studying a population?

To collect data on every unit in the population

To use inferential statistics to draw conclusions about the population

To make descriptive statements about the entire set of persons, events, or other units included in the defined population (correct)

To generalize from a sample to the population with no degree of confidence

Why is it often impractical to study an entire population?

Because researchers prefer to use theoretical models instead

Because samples do not accurately represent the population

Because inferential statistics are not applicable to entire populations

Due to reasons such as time, cost, and feasibility (correct)

What do inferential statistics enable researchers to do?

Replace the need for studying samples

Generalize from a sample to the population with a specifiable degree of confidence (correct)

Guarantee that the generalizations are always correct

Make precise statements about the sample

What is the best way to minimize sampling error?

Selecting a sample that is most representative of the population through probability sampling Signup and view all the answers

What is the key feature of probability sampling?

Each unit in the population has the same chance of being included in the sample Signup and view all the answers

How can inferential statistics help with sampling error?

Estimate the amount of sampling error and make inferences about population parameters Signup and view all the answers

What does the example use to estimate the probability of various outcomes in 10 coin tosses?

Bar graph frequency distribution of the sample statistics Signup and view all the answers

What is the estimated probability of getting more than four heads in 10 coin tosses?

64% Signup and view all the answers

What does the example emphasize as crucial in the use of inferential statistics?

Understanding and accounting for sampling error Signup and view all the answers

What is the purpose of sampling distributions in inferential statistics?

To provide a solution for inferring population characteristics from a single sample Signup and view all the answers

What is the mathematical probability model used to determine the exact probability associated with each possible outcome in a series of trials?

Binomial distribution Signup and view all the answers

What is the assumption made about the two possible outcomes of a single coin flip in the text's example?

They are equally probable (50% probability each) Signup and view all the answers

What do sampling distributions allow in inferential statistics?

Making quantitative statements about the probabilities associated with getting a particular outcome in a single sample Signup and view all the answers

What is the purpose of inferential statistics?

To infer characteristics of a population from sample data Signup and view all the answers

Why is it impractical to use 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 Signup and view all the answers

In the example of 10 coin tosses, what does the horizontal axis of the bar graph represent?

Proportion of heads in any single sample of 10 tosses Signup and view all the answers

What is the key assumption made about the coin in the example of binomial distribution?

The coin is unbiased, with each flip having an equal chance of landing heads or tails Signup and view all the answers

What is the purpose of the binomial distribution as a sampling distribution in inferential statistics?

To determine the chances of obtaining specific outcomes in a series of trials with two possible outcomes Signup and view all the answers

Which type of inferential statistics assumes a normal distribution of variable values in a population?

Parametric inferential statistics Signup and view all the answers

What level of measurement is appropriate for parametric inferential statistics?

Interval or ratio level Signup and view all the answers

What is the key characteristic of nonparametric inferential statistics?

They do not assume a normal distribution of variable values in a population Signup and view all the answers

When are nonparametric inferential statistics the best choice?

When the variable is measured at the nominal or ordinal level and it is inappropriate to assume normal distribution in the population Signup and view all the answers

What do both parametric and nonparametric inferential statistics assume about the sample?

They assume a probability (usually a random) sample has been chosen Signup and view all the answers

What is the second difference between parametric and nonparametric inferential statistics?

Assumptions concerning the shape of the distribution of the values of the variable being studied Signup and view all the answers

What does the bar graph in Figure 10.2 represent?

Probabilities of obtaining different proportions of heads in 100 samples of 10 coin tosses Signup and view all the answers

What is the mean proportion in the distribution shown in Figure 10.3?

0.50 Signup and view all the answers

When is the binomial distribution essentially equivalent to the normal curve?

With a sample size over 50 Signup and view all the answers

What does the central limit theorem state about the sampling distribution of means for large samples?

It will be a normal curve with a mean equal to the mean of the scores for the population. Signup and view all the answers

What is the standard deviation of a theoretically derived normal sampling distribution called?

Standard error Signup and view all the answers

When the sampling distribution for standard deviations is not normal, what may be possible to determine its shape?

Application of other mathematical models Signup and view all the answers

What is a point estimate in statistics?

A single-value estimate of the population parameter based on a sample statistic Signup and view all the answers

Why are point estimates considered to have desirable simplicity and specificity?

They provide a single, specific value as an estimate of the population parameter Signup and view all the answers

What is the primary drawback of using point estimates for population parameter estimation?

The probability of the sample mean being the same as the population mean is relatively small Signup and view all the answers

Which mathematical symbol is used to make interval estimates for the population mean IQ score?

σx̄ Signup and view all the answers

What relationship does the standard error of the mean ($σx̄$) have with the standard deviation of the individual scores in the population?

It is smaller than the standard deviation of the individual scores in the population Signup and view all the answers

What does the sampling distribution for means show?

How the means of random samples of a fixed size are distributed Signup and view all the answers

What proportion of the area under the standard normal curve lies between $z = -1.96$ and $z = +1.96$?

0.95 Signup and view all the answers

What role does sample size play in establishing confidence limits?

It leads to smaller confidence intervals for a particular confidence level Signup and view all the answers

What is the relationship between the standard error of the sampling distribution for means ($ ext{σ}_{ar{x}}$) and the standard deviation of the distribution of individual scores in the population ($S$)?

The standard error of the sampling distribution for means is the standard deviation of the distribution of individual scores in the population divided by the square root of the sample size Signup and view all the answers

What is the purpose of constructing a sampling distribution based on the null hypothesis assumption?

To determine the probability of a result occurring if the null hypothesis is correct Signup and view all the answers

What is the key difference between Type I and Type II errors in hypothesis testing?

Type I error occurs when a true null hypothesis is rejected, while Type II error happens when a false null hypothesis is accepted Signup and view all the answers

What does the calculation of z for the observed mean differences and referencing a z table help determine in hypothesis testing?

The probability of obtaining the observed difference by chance Signup and view all the answers

What is the purpose of using contradictory values and rejection region in hypothesis testing?

To reject the null hypothesis based on highly improbable values of a statistic within the rejection region Signup and view all the answers

In hypothesis testing, what does the Student’s t statistic help determine?

The probability that mean values of the same variable from two different randomly selected groups are from the same population Signup and view all the answers

What is the main purpose of indirectly testing the research hypothesis by testing the null hypothesis directly using inferential statistics?

To provide evidence against the null hypothesis and support the research hypothesis Signup and view all the answers

What is the significance level for a test of the null hypothesis if a researcher chooses a level of.05?

5% level Signup and view all the answers

What are the critical values that establish the boundary lines between the rejection and acceptance regions for a test of the null hypothesis?

$z = +1.96$ and $z = -1.96$ Signup and view all the answers

What does the significance level of.01 represent for a test of the null hypothesis?

It reduces the risk of making a type I error to one in 100 Signup and view all the answers

What is the critical value for a one-tailed test at a .01 significance level?

$z = 2.33$ Signup and view all the answers

Which test is used to test statistical significance of the difference between means?

Student’s t test Signup and view all the answers

What does a Z value of 2.17 correspond to in terms of probability?

0.035 Signup and view all the answers

What is the purpose of the t table in hypothesis testing?

To determine the probability of a calculated t occurring by chance Signup and view all the answers

In the context of the sampling distribution, what does the null hypothesis state?

$\bar{x}_1 - \bar{x}_2 = 0$ Signup and view all the answers

What does the t distribution closely approximate for large sample sizes?

Normal distribution Signup and view all the answers

In ANOVA, the null hypothesis states that:

The means are equal Signup and view all the answers

What does the F statistic in ANOVA follow?

F distribution Signup and view all the answers

What does the F ratio in ANOVA represent?

Ratio of explained to unexplained variance Signup and view all the answers

What is the between-group variance in ANOVA?

Variance of the mean scores from the grand mean Signup and view all the answers

What does ANOVA determine regarding the means of multiple groups?

Whether the means differ significantly Signup and view all the answers

How is the critical value of F determined in ANOVA?

Consulting the F table corresponding to the selected significance level Signup and view all the answers

In ANOVA, the F ratio is calculated as:

$F = \frac{\text{between-group variance}}{\text{within-group variance}}$ Signup and view all the answers

In ANOVA, when can we argue that the means differ significantly?

When the variance in y attributed to the independent variable is larger than the within-group variance Signup and view all the answers

What does the F table include values for?

$F \geq 1$ Signup and view all the answers

In ANOVA, the null hypothesis being tested is that:

The samples are drawn from the same population Signup and view all the answers

What is necessary to assume about the populations in ANOVA?

They are normally distributed for y and their variances are equal Signup and view all the answers

What does the F distribution vary according to?

The denominators Signup and view all the answers

What does ANOVA assume about population variances?

They must be equal Signup and view all the answers

What is the purpose of chi square in hypothesis testing?

To test for a relationship between two variables Signup and view all the answers

What does the chi square statistic determine?

If cell frequencies differ significantly from expected frequencies Signup and view all the answers

What is created by dividing the sums of squares by the degrees of freedom?

An unbiased estimate Signup and view all the answers

What is the formula to calculate degrees of freedom (df) for a chi-square test with a table of r rows and c columns?

$df = (r-1)(c-1)$ Signup and view all the answers

In the example with 2 rows and 2 columns, what are the degrees of freedom for the chi-square test?

1 Signup and view all the answers

What does a chi-square value occurring by chance less than 1% of the time lead to?

Rejection of the null hypothesis Signup and view all the answers

What does the chi-square test help determine about the relationship between variables?

Existence of a relationship Signup and view all the answers

When is the chi-square test most appropriate for use?

When both variables are measured at the nominal level Signup and view all the answers

Study Notes

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

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Test your knowledge of inferential statistics and sampling distributions with this quiz. Explore concepts such as sampling distributions, probability theory, and the use of inferential statistics to make inferences about populations from sample data.