Flashcards in Chapter 6 and 7 Deck (47):
Is the z italicized?
Note that as with all statistical symbols, the z is italicized.
What is the formula for z Score?
6-1: The formula for a z score is z = (X - μ) / σ
We calculate the difference between an individual score and the population mean, then divide by the population standard deviation.
How do we determine the raw score?
6-2: The formula to calculate the raw score from a z score is X = z(σ) + μ.
We multiply the z score by the population standard deviation, then add the population mean.
What is the formula for standard error?
6-3: The formula for standard error is:
σM = σ/ √N
We divide the standard deviation for the population by the square root of the sample size.
Why is sample size important in relation to the normal curve?
6.1: The distributions of many variables approximate a normal curve, a mathematically defined, bell-shaped curve that is unimodal and symmetric.
What is the benefit of calculating the z score?
6.2: z scores give us the ability to convert any variable to a standard distribution, allowing us to make comparisons among variables.
Note that as with all statistical symbols, the z is italicized.
What does a z score tell us?
6.3: z scores tell us how far a score is from a population mean in terms of the population standard deviation. Because of this characteristic, we can compare z scores to each other, even when the raw scores are from different distributions. We can then go a step further by converting z scores into percentiles and comparing percentiles to each other.
What two important principles does the central limit theorem demonstrates.
6.4: The central limit theorem demonstrates that a distribution made up of the means of many samples (rather than individual scores) approximates a normal curve, even if the underlying population is not normally distributed.
Which has a smaller standard deviation, a distribution of scores or a distribution of means?
6.5: A distribution of means has the same mean as a distribution of scores from the same population, but a smaller standard deviation. For a statistics test, we know that being above the mean is good; for anxiety levels, we know that being above the mean is usually bad. z scores create an opportunity to make meaningful comparisons
Does statistics have a notation?
It is important, however, to learn the notation and language of statistics.
What is standardization?
>Standardization is a way to create meaningful comparisons between observations from different distributions. It can be done by transforming raw scores from different distributions into z scores, also known as standardized scores.
What is a z score?
> A z score is the distance that a score is from the mean of its distribution in terms of standard deviations.
How do we transform z scores into raw scores?
> We also can transform z scores to raw scores by reversing the formula for a z score.
Explain z scores and percentiles?
> z scores correspond to known percentiles that communicate how an individual score compares with the larger distribution.
How do you calculate a particular z score?
Step 1: Determine the distance of a particular person’s score (X) from the population mean (m) as part of the calculation: X - μ
Step 2: Express this distance in terms of standard deviations by dividing by the population standard deviation, s. The formula for z based on the mean of a sample is:
z = (X - μ) / σ
We subtract the mean of the distribution of means from the mean of the sample, then
we divide by the standard error, the standard deviation of the distribution of means.
If you are at the mean, what is your z score?
What would your z score be if you fell exactly at the mean in your statistics class? If you guessed 0, you’re correct.
What is standard error?
Standard error is the name for the standard deviation of a distribution of means.
Standardization is a way to convert individual scores from different normal distributions
to a shared normal distribution with a known mean, standard deviation, and percentiles.
What is a z score?
A z score is the number of standard deviations a particular score is from the mean.
Explain the central limit theorem?
The central limit theorem refers to how a distribution of sample means is a more normal distribution than a distribution of scores, even when the population distribution is not normal.
What composes a distribution of means?
A distribution of means is a distribution composed of many means that are calculated from all possible samples of a given size, all taken from the same population.
What is the z distribution?
The z distribution is a normal distribution of standardized scores.
What is the standard normal distribution?
The standard normal distribution is a normal distribution of z scores.
What is an Assumption?
An assumption is a characteristic that we ideally require the population from which we are sampling to have so that we can make accurate inferences.
What is a Parametric Test
A parametric test is an inferential statistical analysis based on a set of assumptions about the population.
What is a Nonparemetric Test?
A nonparametric test is an inferential statistical analysis that is not based on a set of assumptions about the population.
What is a Two-Tailed Test?
A two-tailed test is a hypothesis test in which the research hypothesis does not indicate a direction of the mean difference or change in the dependent variable, but merely indicates that there will be a mean difference.
What is a Robust Hypothesis Test?
A robust hypothesis test is one that produces fairly accurate results even when the data suggest that the population might not meet some of the assumptions.
What is a One Tailed Test?
A one-tailed test is a hypothesis test in which the research hypothesis is directional, positing either a mean decrease or a mean increase in the dependent variable, but not both, as a result of the independent variable.
What is a Critical Value?
A critical value is a test statistic value beyond which we reject the null hypothesis; often called a cutoff.
What is a Critical Region?
The critical region is the area in the tails of the comparison distribution in which the null hypothesis can be rejected.
What is an ALPHA?
The probability used to determine the critical values, or cutoffs, in hypothesis testing is a level; often called alpha.
What is a Statistically Significant Finding?
A finding is statistically significant if the data differ from what we would expect by chance if there were, in fact, no actual difference.
What are the Six Steps of Hypothesis Testing?
The Six Steps of Hypothesis Testing
We use the same six basic steps with each type of hypothesis test.
1. Identify the populations, distribution, and assumptions, and then choose the appropriate hypothesis test.
2. State the null and research hypotheses, in both words and symbolic notation.
3. Determine the characteristics of the comparison distribution.
4. Determine the critical values, or cutoffs, that indicate the points beyond which we will reject the null hypothesis.
5. Calculate the test statistic.
6. Decide whether to reject or fail to reject the null hypothesis.
When do we conduct a One-Tailed Test?
7.3: We conduct a one-tailed test if we have a directional hypothesis, such as that the sample will have a higher (or lower) mean than the population. We use a two-tailed test if we have a nondirectional hypothesis, such as that the sample will have a different mean than the population does.
What are the three assumptions of a z test?
7.2: When we calculate a parametric statistic, ideally we have met assumptions regarding the population distribution. For a z test, there are three assumptions:
1. The dependent variable should be on a scale measure,
2. the sample should be randomly selected, and
3.the underlying population should have an approximately normal distribution.
What are the three assumptions for conducting analyses?
Assumption 1: The dependent variable is assessed using a scale measure. If it’s clear that the dependent variable is nominal or ordinal, we could not make this first assumption and thus should not use a parametric hypothesis test.
Assumption 2: The participants are randomly selected. Every member of the population of interest must have had an equal chance of being selected for the study. This assumption is often violated; it is more likely that participants are a convenience sample. If we violate this second assumption, we must be cautious when generalizing from a sample to the population.
Assumption 3: The distribution of the population of interest must be approximately normal. Many distributions are approximately normal, but it is important to remember that there are exceptions to this guideline (Micceri, 1989). Because hypothesis tests deal with sample means rather than individual scores, as long as the sample size is at least 30 (recall the discussion about the central limit theorem), it is likely that this third assumption is met.
What is the difference between a Parametric Statistic and an Nonparametric statistic?
> Parametric statistics are those that are based on assumptions about the population distribution; nonparametric statistics have no such assumptions. Parametric statistics are often robust to violations of the assumptions.
What information do we need to conduct a z test?
> We conduct a z test when we have one sample and we know both the mean and the standard deviation of the population.
How do we decide between a one tailed test and a two tailed test.
> We must decide whether to use a one-tailed test, in which the hypothesis is directional, or a two-tailed test, in which the hypothesis is nondirectional.
Are one tailed tests common?
> One-tailed tests are rare in the research literature.
What are the three ways that Dirty Data shows up?
> The problem of dirty data can show up in three ways: missing data, misleading data, and outliers. A variety of techniques can be used to address dirty data, and researchers should report whatever techniques they chose to use when reporting their data.
What are the three ways to describe a score within a normal distribution?
> Raw scores, z scores, and percentile rankings are three ways to describe the same score within a normal distribution.
Explain how to use a z table for determining percentages?
> If we know the mean and the standard deviation of a population, we can convert a raw score to a z score and then use the z table to determine percentages below, above, or at least as extreme as this z score.
Can the z table be used in reverse?
> We can use the z table in reverse as well, taking a percentage and converting it into a z score and then a raw score.
Can the same conversions be performed on a sample mean as compared to a sample score?
> These same conversions can be conducted on a sample mean instead of on a score. The procedures are identical, but we use the mean and the standard error of the distribution of means, instead of the distribution of scores.