2220 exam II

Standardizing

(computing z scores for each question) equalizes the scales. A legitimate way to compare apples to oranges, converting everything to mangoes

Imagine that a researcher conducts a one-tailed z test with an alpha level (also called a p-level) of 0.05 (5%). What is the absolute value of the critical value for the corresponding hypothesis test? When answering, please provide at least two decimal places (if relevant)

1.645

A score of z = -1.96 is

1.96 standard deviations below the mean

Assumptions of the z test

The dependent variable is measured as a scale variable. •i.e., NOT nominal or ordinal Participants are randomly selected. The population of interest is normally distributed. •Remember, based on the central limit theorem, if your N is >30, you are probably alright. The independent variable is nominal.

Which of the following statements are true for a distribution of means? (When answering, please assume that the sample size for each mean is equal to 80 and that all scores are drawn from the same population of scores).

The distribution of means has less variability than the corresponding population of scores. The distribution of means has the same mean (average) as the population of scores. Correct! If the population of scores is not normally distributed, the distribution of means should be more normally distributed than the population of scores.

Distributions of means

a distribution of values, where each value is the mean (average) of values from another distribution. -When we compute the average of a sample, we're effectively drawing one observation (a mean) from a distribution of means. -We can create a distribution of means and compute z scores for these means just like we did for other raw scores. --These distributions are often referred to as sampling distributions of the mean.

standard normal distribution

a specific version of the normal distribution. It is a normal distribution that is defined to have a mean of 0 and a standard deviation of 1.

Bedford's Law

expects 30.1% of numbers in a list of financial transactions to begin with 1. Each successive digit should represent a progressively smaller proportion.

high variance scores vs low variance scores

high--> really high and really low scores, low---->scored near the mean

Parametric Tests

inferential statistical tests based on assumptions about a population •Are there assumptions about the characteristics of the population? If yes, it is a parametric test. Most (if not all) of the tests we will talk about in this class are parametric tests.

Nonparametric Tests

inferential statistical tests not based on assumptions about the population

Deviations from a central tendency are more often

small than large, and the deviations occur randomly in either direction.

Meeting the assumptions improves

the quality of the research. ---However, not meeting the assumptions does not necessarily invalidate the research. Some tests are fairly robust (i.e., pretty accurate even if you violate the assumptions).

standard error, σM,

the standard deviation of a distribution of means for a specific sample size, N.

As N increases,

the standard error decreases; variability in the distribution of means decreases as N increases. 39

If the distribution of scores is normal,

then the distribution of means will also be normal.

If the distribution of scores is not normal

then the distribution of means will be increasingly normal with higher sample sizes.

However, the central limit theorem will often allow us

to work with approximately normal distributions even when the population distribution is not normal.

To convert to a standard normal distribution

we will need to take our raw scores and standardize them.

A particular normal distribution has a mean of 50 and a standard deviation of 4. Compute a z score for each of the following raw scores. 46 52 55 50

(46 - 50) / 4 = -1 (52 - 50) / 4 = 0.5 (55 - 50) / 4 = 1.25 (50 - 50) / 4 = 0, This is the mean. It should always have a z score of 0.

A score of z = 1

1 standard deviation above the mean.

Critical value

A test statistic value beyond which we reject the null hypothesis (also called a cutoff)

What are the mean, median, mode, standard deviation, and variance of the standard normal distribution?

Mean = 0 Median = 0 Mode = 0 Standard deviation = 1 Variance = 1

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 normally distributed.

Critical region:

The area in the tail(s) of the comparison distribution in which the null hypothesis can be rejected.

Consider a standard deviation of a population and a corresponding standard error for the distribution of means. Which should be larger? Why?

The standard deviation should be larger. The standard error is equal to the population standard deviation divided by the square root of the sample size. As long as N > 1, the standard error must be smaller.

Which of the following correctly describes a standard error?

The standard error is what we call the standard deviation for a distribution of means (from a particular population and with a specific sample size).

Imagine that a researcher obtains a mean value of 40 on variable X, and that the corresponding distribution of means has the following property: µM = 44. Based on this information, what can we know about the corresponding z statistic?

The z statistic must be a negative z statistic.

Imagine that you randomly sampled 10,000 values of the variable Y from a population where Y is positively skewed. You then calculated and recorded the mean of that distribution. Next, you repeated the process again, and again until you had many thousands of means for which you create a graph. What would you call this distribution, what shape would the distribution have? Draw a rough sketch of the distribution below.

This would be a distribution of means (also called a sampling distribution of the mean, or just a sampling distribution). Despite the positive skew in the population of scores, the distribution of means would be approximately normally distributed.

Why compute z scores?

Translating questions to put them on the same scale.

The Normal Curve

Unimodal Bell-shaped Symmetric It is also defined mathematically

Standard Normal Distribution:

a normal distribution with a mean of 0 and a standard deviation of 1. •Notation: N(0,1)

A score of z = 0

at the mean, because Mz = 0.

What would the z statistic be (for the same mean), if N = 81?

σM = 9/√81 = 9/9 = 1 z = (17 - 20) / 1 = -3/1 = -3

If you randomly sampled 10,000 values of the variable Y from a population where Y is positively skewed, what type of distribution would you expect to have for the sample of Y scores? Draw a rough sketch of the distribution below.

Given that the population distribution is positively skewed, the sample distribution of scores should also be positively skewed.

Indicate three properties of a distribution of means.

-The distribution of means has the same average as the distribution of scores. -The distribution of means has a less variability than the distribution of scores. This also reduces the range of observed values. -Assuming a sufficient sample size, the distribution of means becomes more normally distributed (even if the population distribution is not normal).

6) Convert the following z scores into raw scores for a distribution with a mean of 100 and a standard deviation of 5.

1 -2 0.5 1.5 1 x 5 + 100 = 105 -2 x 5 + 100 = 90 0.5 x 5 + 100 = 102.5 1.5 x 5 + 100 = 107.5

Normal distribution:

A distribution of values having a specific shape that is symmetric, unimodal, and bell-shaped. Also defined mathematically.

What is the difference between a standard deviation and a standard error?

A standard deviation is a measure of variability for a distribution of scores in a single sample or in a population of scores. A standard error is the standard deviation in a distribution of means of all possible samples of a given size from a particular population of individual scores.

Imagine that a researcher appropriately collects data from 100 people from a particular population for variable X. He obtains a mean of 19.2 from these 100 people. He also knows that the population mean for variable X is 21.05 (μ = 21.05) with a population standard deviation of 10 (σ = 10). If the researcher uses a two-tailed hypothesis test with an alpha level (or p-level) of 0.05 (5%), what decision should the researcher make when following the rules of hypothesis testing? (When answering, please assume that any relevant assumptions are met for this hypothesis test).

Fail to reject the null hypothesis

If you randomly sampled 10,000 values of the variable X from a population where X was normally distributed, what type of distribution would you expect to have for the sample of X scores? Draw a rough sketch of the distribution below.

Given that the population distribution is normally distributed, the sample distribution of scores should also be normally distributed.

11) A population has a mean of 20 with a standard deviation of 9. Based on a corresponding distribution of means, what is the z statistic for a mean of 17 when N = 9?

σM = 9/√9 = 9/3 = 3 z = (17 - 20) / 3 = -3/3 = -1

One-tailed tests

•Also called "directional" tests •Critical region in one tail of the distribution •α = .05, put all 0.05 (5%) in the one tail

Two-tailed tests

•Also called "nondirectional" tests •Critical region divided between the two tails of the distribution •α = .05, put 0.025 (2.5%) in each tail. •More conservative. Reduces power for a particular tail (relative to a one-tailed test). 21

10) A population standard deviation is 10. What is the standard error when:

•N = 16 •N = 25 •N = 100 •N = 400 √16 = 4 10 / 4 = 2.5 √25 = 5 10 / 5 = 2 √100 = 10 10 / 10 = 1 √400 = 20 10 / 20 = 0.5

With the z table, we can examine our data in different, but equivalent, ways. Once we have z scores, we can examine:

•Percentages under the curve for a particular z score •Find the raw score that would correspond to a certain percentile

Three properties for a distribution of means:

•The distribution of means has the same average as the population distribution of scores. •The distribution of means has a less variability than the distribution of scores. This also reduces the range of observed values. •Assuming a sufficient sample size, the distribution of means becomes more normally distributed (assuming that the parent population is not normal already). •This last point is REALLY important.

We need to set our "acceptable" risk of a Type I error when we run a study and conduct analyses. By convention, this is usually 0.05 or 5%.

•This is usually referred to as an alpha (α) or p level. •The alpha that we set, along with the type of test (one-tailed or two-tailed) lets us determine our critical value(s) and critical region(s).