ch 7.1 stats quiz

smaller sample size

Decreased likelihood of a significant test -Smaller sample sizes, larger estimated standard error, more chance your results occurred by chance

large sample size

Increased likelihood of a significant test -Larger sample sizes, smaller estimated standard error, less chance your results occurred by chance

Confidence Intervals: Important Note

- we don't care about the sample, except that it tells us something about the population -Samples give us information that we can use to make inferences about populations

Confidence Intervals

-For a one-sample t-test, we are trying to estimate a population mean (μ) from a sample mean (x̅) -We can't just calculate a sample mean and assume the population mean will be the same because of sampling error -x̅ becomes our starting point for the estimate of μ -We then construct a band of values in between which we believe the true μ lies -We come up with upper and lower limits to create a confidence interval

Degrees of freedom (df)

-For t-tests we look up our critical value on a t-table, but we don't use sample size, we use df How to conceptualize df: -Given the mean of a distribution of n scores, n-1 of the scores are free to vary

The Baseball Example for Degrees of Freedom

-If a baseball coach is setting the batting line-up for 9 players, he only needs to make 8 decisions (n-1 = df) Why? -Once he has placed 8 of the players, the last player is a given --- he has no choice in who will be placed in the last spot -8 of the choices are "free to vary" and 1 is not

One Sample t-test Example STEP 4

-Measuring Effect Size for the t Statistic -Because the significance of a treatment effect is determined partially by the size of the effect and partially by the size of the sample, you cannot assume that a significant effect is also a large effect. -Therefore, it is recommended that a measure of effect size be computed along with the hypothesis test.

Family of t Distributions

-The family of t distributions is made up of several distributions -Each distribution is symmetrical and has a mean of 0 -The shape of the t distribution depends on the number of cases in the sample -When the sample size is small, the curve will be relatively flat (high variability) -As sample size increases, the middle portion of the curve will become more peaked and it will look more like a normal curve

The Estimated Standard Error and the t Statistic

-The goal for a hypothesis test is to evaluate the significance of the observed discrepancy between a sample mean and the population mean. -Whenever a sample is obtained from a population you expect to find some discrepancy or "error" between the sample mean and the population mean. -This general phenomenon is known as sampling error.

The t statistic (like the z-score) forms a ratio.

-The top of the ratio contains the obtained difference between the sample mean and the hypothesized population mean. -The bottom of the ratio is the standard error which measures how much difference is expected by chance.

how does df relate to the t distribution?

-The value of degrees of freedom, df = n - 1, is used to describe how well the t statistic represents a z-score. -Also, the value of df will determine how well the distribution of t approximates a normal distribution. -For large values of df, the t distribution will be nearly normal, but with small values for df, the t distribution will be flatter and more spread out than a normal distribution.

From z to t

-When computing a t statistic, we can't rely on the normal curve and sampling distribution of z (with our values of .05 = 1.96, .01 = 2.58) -We use an alternative to the normal curve distribution called the family of t distributions

Degrees of Freedom and the t Statistic

-You can think of the t statistic as an "estimated z-score." -The estimation comes from the fact that we are using the sample variance to estimate the unknown population variance. -With a large sample, the estimation is very good and the t statistic will be very similar to a z-score. -With small samples, however, the t statistic will provide a relatively poor estimate of z, there is a lot more variability in estimates resulting in more variability in the sampling distribution

Cohen's d

-measures the size of the treatment effect in terms of the standard deviation estimated Cohen's d: mean difference M-μ ────── = ────── standard deviation s

The hypothesis test attempts to decide between the following two alternatives:

1. Is it reasonable that the discrepancy between M and μ is simply due to sampling error and not the result of a treatment effect? 2. Is the discrepancy between M and μ more than would be expected by sampling error alone? That is, is the sample mean significantly different from the population mean?

t statistic formula

obtained difference M - μ t = ───────────── standard error sM

formula for estimated standard error

s sM = ── square root(n)

Cohen's Conventions for Effect Sizes

small: 0.2-0.49 medium: 0.5-0.79 large: 0.8<

ratio

A large value for t (a large ________) indicates that the obtained difference between the data and the hypothesis is greater than would be expected if the treatment has no effect.

One-sample t test

An inferential statistical procedure that uses the mean for one sample of data for either estimating the mean of the population or testing whether the mean of the population equals some claimed value.

size

Both the sample ______ and the sample variance influence the outcome of a hypothesis test.

One Sample t-test Example STEP 5

Cohen's D= 17.16-20/7.22 = absolute value [-.39] - small/medium treatment effect (in SD units)

Confidence Intervals Example

Computations: (Test value + Lower)AND (Test value + Higher) 20 + (-4.44) AND 20 + (-1.24) 15.56 AND 18.76 We are 95% confident that the population mean of anxiety scores for students doing therapy falls between 15.56 and 18.76.

One Sample t-test Example STEP 1

Express hypotheses in symbols H0 : μ = 20 The average test anxiety in the sample of college students will not be statistically significantly different than 20. H1 : μ ≠ 20 The average test anxiety in the sample of college students will be significantly different than 20.

are

If p<.05 your results _____ statistically significant, i.e., you are 95% confident your treatment was effective

are not

If p>.05 your results ______ ____statistically significant, e.g., you are not confident that your treatment was effective and likely could just be a result of chance/sampling error

One Sample t-test Example STEP 3 Part 2

Make Statistical and Substantiative Conclusions -Statistical conclusion: Reject the null b/c our obtained p=.001 < critical p=.05. -Substantiative conclusions (say your results in words): Students enrolled in the therapy program had significantly lower test anxiety scores (M = 17.16) compared to students who were not in the reading program (μ = 20) (t(80) = -3.54 , p =.001).

one sample t-test example by Identifying population, IV (levels), DV, and test to use

Population - college students IV - therapy No therapy (population) Therapy (treatment group) DV - test anxiety score Single sample t test because we have a treated sample what we are comparing to a population mean

One Sample t-test Example STEP 3

Record SPSS Output of one-sample t-test n = 81 M = 17.16 s = 7.22 sM = 0.80 *Test value = 20 *t = -3.54 df = 80 *p = .001 Mean Difference = -2.84

An example of the effect of sample size

SAMPLE SIZE OF 25 -Mean = 7.8 -SE = .75/ sqrt (25) = .15 -Calculate test statistic (turn M into a z value) Z = x̅ - μ / σM Z = 8.1 - 7.8/.15 = 2.00 (Critical z, +/- 1.96) SAMPLE SIZE OF 100 -Mean = 7.8 -SE = .75/sqrt (100) = .075 -Calculate test statistic (turn M into a z value) Z = x̅ - μ / σM Z = 8.1 - 7.8/.075 = 4.00 (Critical z, +/- 1.96)

SPSS and Confidence Intervals

To construct a 95% CI (we are 95% confident that the actual μ lies between the upper and lower values) use the t value at the .05 level

sample;population

We must use the _________ standard deviation, s, in place of the unknown __________ standard deviation, σ

One Sample t-test Example

You are conducting an experiment to see if a given therapy works to reduce test anxiety in a sample of college students. A standard measure of test anxiety is known to produce a μ of 20. A sample of 81 students was given the new therapy for 6 weeks. Use a two-tailed alpha level of .05μ= 20 σ= ? n= 81 Is this difference statistically significant at the .05 level?

The t Statistic

You need: A treated sample (M and s) μ - known or a reasonable hypothesis σ - completely unknown -All that is required for a hypothesis test with t is a sample and a reasonable hypothesis about the population mean.

The critical first step for the t statistic hypothesis test is to

calculate exactly how much difference between M and μ is reasonable to expect. -However, because the population standard deviation is unknown, it is impossible to compute the standard error of M as we did with z-scores -Therefore, the t statistic requires that you use the sample data to compute an estimated standard error of M.

One Sample t-test Example STEP 2

input t-test in SPSS