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Questions and Answers
The distribution of z-scores has a mean of one.
The distribution of z-scores has a mean of one.
False
Transforming raw scores into z-scores changes the shape of the original distribution.
Transforming raw scores into z-scores changes the shape of the original distribution.
False
The number of standard deviations a z-score is from the mean is called the DEVIATION SCORE.
The number of standard deviations a z-score is from the mean is called the DEVIATION SCORE.
True
A raw score is the processed score obtained from a test or assessment.
A raw score is the processed score obtained from a test or assessment.
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Raw scores can be directly compared across different tests regardless of their difficulty.
Raw scores can be directly compared across different tests regardless of their difficulty.
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Empirical Probability relies on theoretical assumptions regarding outcomes.
Empirical Probability relies on theoretical assumptions regarding outcomes.
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A z-score of z = -1.50 corresponds to an X value that is below the mean.
A z-score of z = -1.50 corresponds to an X value that is below the mean.
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Transforming raw scores into z-scores allows for meaningful comparisons across distributions.
Transforming raw scores into z-scores allows for meaningful comparisons across distributions.
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Classical Probability involves outcomes that are assumed to be equally likely.
Classical Probability involves outcomes that are assumed to be equally likely.
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The average IQ score is set at 100, which is the peak of the normal distribution curve.
The average IQ score is set at 100, which is the peak of the normal distribution curve.
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Probability is primarily focused on analyzing past events.
Probability is primarily focused on analyzing past events.
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Subjective Probability is calculated through precise statistical methods.
Subjective Probability is calculated through precise statistical methods.
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Approximately 95% of the population scores between 85 and 115 on IQ tests.
Approximately 95% of the population scores between 85 and 115 on IQ tests.
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If a distribution has a mean of μ = 100 and a standard deviation of σ = 10, a score of X = 130 corresponds to a z-score of 3.
If a distribution has a mean of μ = 100 and a standard deviation of σ = 10, a score of X = 130 corresponds to a z-score of 3.
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The standard deviation in IQ testing is typically 10 points.
The standard deviation in IQ testing is typically 10 points.
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In random sampling, every individual must have an equal chance of being selected.
In random sampling, every individual must have an equal chance of being selected.
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A percentile indicates the percentage of individuals with scores at or above a particular value.
A percentile indicates the percentage of individuals with scores at or above a particular value.
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A z-score indicates whether a score is below or above the mean of a distribution.
A z-score indicates whether a score is below or above the mean of a distribution.
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Only about 1% of individuals score below 70 or above 130 on IQ tests.
Only about 1% of individuals score below 70 or above 130 on IQ tests.
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The null hypothesis is symbolized as H1.
The null hypothesis is symbolized as H1.
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Hypothesis testing aims to determine whether a treatment has any effect on individuals in a population.
Hypothesis testing aims to determine whether a treatment has any effect on individuals in a population.
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Sampling with replacement is necessary for the probabilities to remain constant in random sampling.
Sampling with replacement is necessary for the probabilities to remain constant in random sampling.
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The sample mean should align with the initial hypothesis if the hypothesis is correct.
The sample mean should align with the initial hypothesis if the hypothesis is correct.
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Percentiles measure only the absolute value of scores in a distribution.
Percentiles measure only the absolute value of scores in a distribution.
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The alternative hypothesis (H1) asserts that there is no relationship for the general population.
The alternative hypothesis (H1) asserts that there is no relationship for the general population.
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A sample mean that closely matches the population mean supports the null hypothesis.
A sample mean that closely matches the population mean supports the null hypothesis.
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An alpha level of 0.10 indicates a 10% risk of committing a Type I error.
An alpha level of 0.10 indicates a 10% risk of committing a Type I error.
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The alpha level represents the probability of making a Type II error.
The alpha level represents the probability of making a Type II error.
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A sample mean significantly different from the population mean suggests the null hypothesis may be wrong.
A sample mean significantly different from the population mean suggests the null hypothesis may be wrong.
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Setting an alpha level of 0.05 corresponds to a confidence interval of approximately 95%.
Setting an alpha level of 0.05 corresponds to a confidence interval of approximately 95%.
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The null hypothesis can be considered valid if the sample mean diverges significantly from the expected value.
The null hypothesis can be considered valid if the sample mean diverges significantly from the expected value.
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A right-tailed test assesses if a parameter is less than a specified value.
A right-tailed test assesses if a parameter is less than a specified value.
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A two-tailed test can detect both increases and decreases from a specified value.
A two-tailed test can detect both increases and decreases from a specified value.
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A Type I error occurs when a true null hypothesis is incorrectly accepted.
A Type I error occurs when a true null hypothesis is incorrectly accepted.
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In a one-tailed test, the entire alpha level is concentrated in one tail of the distribution.
In a one-tailed test, the entire alpha level is concentrated in one tail of the distribution.
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Cohen's d values are interpreted as small (0.5), medium (0.8), and large (0.2) effects.
Cohen's d values are interpreted as small (0.5), medium (0.8), and large (0.2) effects.
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Effect size indicates the magnitude of a research finding's practical significance.
Effect size indicates the magnitude of a research finding's practical significance.
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A two-tailed test requires less extreme results to achieve significance compared to a one-tailed test.
A two-tailed test requires less extreme results to achieve significance compared to a one-tailed test.
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A one-tailed test should be used when there is no prior evidence to suggest the direction of an effect.
A one-tailed test should be used when there is no prior evidence to suggest the direction of an effect.
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The pooled standard deviation is used in the calculation of Cohen's d.
The pooled standard deviation is used in the calculation of Cohen's d.
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The critical region is defined as the set of values that would lead to accepting the null hypothesis.
The critical region is defined as the set of values that would lead to accepting the null hypothesis.
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If the test statistic falls outside the critical region, the null hypothesis must be rejected.
If the test statistic falls outside the critical region, the null hypothesis must be rejected.
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A Type I error occurs when a researcher correctly fails to reject a true null hypothesis.
A Type I error occurs when a researcher correctly fails to reject a true null hypothesis.
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A directional test, also known as a one-tailed test, allows testing for effects in both directions.
A directional test, also known as a one-tailed test, allows testing for effects in both directions.
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Setting an alpha level of 0.05 means that 95% of the distribution is in the critical region.
Setting an alpha level of 0.05 means that 95% of the distribution is in the critical region.
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In a left-tailed test, the critical region is placed entirely in the left side of the distribution.
In a left-tailed test, the critical region is placed entirely in the left side of the distribution.
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In psychological research, a hypothesis stating that training will improve performance is an example of a directional hypothesis.
In psychological research, a hypothesis stating that training will improve performance is an example of a directional hypothesis.
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A Type II error means concluding that a treatment does have an effect when it actually does not.
A Type II error means concluding that a treatment does have an effect when it actually does not.
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Study Notes
Raw Score
- A raw score is an unprocessed score from a test or assessment, reflecting an individual's performance without modifications.
- Raw scores are the starting point of statistical analyses.
- Raw scores lack context; a score of 75 on one test isn't directly comparable to a score of 75 on a different test with varying difficulty or scoring methods.
- Raw scores are often transformed (e.g., into z-scores) to facilitate meaningful comparison.
Z Score
- A raw score alone doesn't readily indicate position within a distribution.
- Raw scores are the direct outcome of measurement.
- Transforming raw scores into z-scores provides more insightful information about a score's position within a distribution.
- Z-scores help pinpoint the exact location of original scores within a distribution.
- Z-scores facilitate direct comparison between distributions.
Mean Score and Standard Deviation
- The average IQ score is 100, marking the peak of the bell curve.
- Standard deviation in IQ testing typically is 15 points.
- Approximately 68% of scores fall within one standard deviation of the mean (85-115).
- Approximately 95% of scores fall within two standard deviations of the mean (70-130).
Z-Score Transformation Process
- Z-scores effectively locate original scores within a distribution.
- Z-scores create a standardized distribution enabling comparison to other distributions.
- Each z-score precisely indicates an original X-value's position in the distribution.
Z-Score Properties
- Z-scores maintain the same shape as the original distribution.
- If the original distribution is normal, the z-score distribution will be normal.
- Z-score transformation does not change position in the distribution.
- Z-score distributions have a mean of zero and indicate deviation from the mean in standard deviation units.
Z-Score Formula
- Z-scores are calculated using the formula: z = (X - μ) / σ.
- This formula, called the deviation score, shows the distance between X and μ (mean), standardized by σ (standard deviation).
- The numerator represents the deviation from the mean.
- The denominator (σ) converts to standard deviation units.
Probability
- Probability quantitatively represents possible outcomes.
- Probability is calculated as a fraction or proportion of all possible outcomes.
Types of Probability
- Classical Probability: Assumes equally likely outcomes, for instance tossing fair coins.
- Empirical Probability: Observed data-driven probability based on experiments.
- Subjective Probability: Based on personal judgment or estimation.
- Axiomatic Probability: Defined by a set of principles.
Hypothesis Testing
- Hypothesis testing analyzes sample data to assess hypotheses regarding a population.
- Researchers state a hypothesis about a population (often about a parameter's value).
- They predict the characteristics the sample should have, based on the hypothesis.
- A random sample is drawn from the population to compare observed data to the prediction.
- If the sample data aligns with the prediction, the hypothesis is accepted as reasonable.
- Discrepancies between data and prediction indicate the hypothesis may need adjustment.
Steps in Hypothesis Testing
- State the Hypothesis: The null hypothesis (H0) posits no effect, while the alternative hypothesis (H1) proposes an effect or difference.
- Setting Criteria for Decision: Compare sample data to the hypothesis to determine whether or not evidence supports it.
Alpha Level
- The alpha level (α) represents the probability of a Type I error.
- Type I error is incorrectly rejecting a true null hypothesis.
- Commonly used values are 0.05 (5%), 0.01 (1%), or 0.10 (10%).
Critical Region
- The critical region (rejection region) comprises values for test statistics that lead to rejecting the null hypothesis.
- Data falling in the critical region suggests sufficient evidence against the null hypothesis (indicating an effect).
Types of Errors
- Type I Error: Incorrectly rejecting a true null hypothesis.
- Type II Error: Failing to reject a false null hypothesis.
Hypothesis for Directional Test
- A directional test (one-tailed test) specifies the expected direction of an effect or relationship.
- Useful when prior knowledge indicates the direction of the effect.
Effect Size
- Effect size is a numerical value quantifying the strength of a relationship or the difference between groups.
- A large effect size implies practical significance, while a small one suggests limited practical application.
- Cohen's d is a measure of the standardized difference between two means.
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Description
This quiz covers essential concepts in statistics and probability, including z-scores, transformations of raw scores, and various types of probability. It emphasizes the importance of comparing scores across different distributions and understanding the distribution of IQ scores. Test your knowledge on these fundamental statistical principles.
