Statistics for the behavioral sciences exam III
• Assumptions for independent-samples t tests - Homogeneity of variance
- Dependent variable is scale. - Participants are randomly selected. - The population is normally distributed. - Independent variable is nominal - Homogeneity of variance means the variances of our populations are the same. σ²X =σ²Y
• Conceptual and computational understanding of 95% confidence intervals for one-sample t tests.
- Determine t statistic at each boundary. -Will always use critical values that correspond to a two-tailed test when creating CIs. Turn the t statistics back into raw means. Check to make sure that the confidence interval makes sense sM =s/√N
• How to use the t table to find critical values for hypothesis testing
- Must know degrees of freedom, alpha level, and whether we are using a one-tailed or two-tailed test. - Note the critical t values when the sample size is extremely large (infinite). They are (essentially) the same as the values from the z table
• Assumptions for the paired-samples t test
- The dependent variable is measured as a scale variable. - Participants are randomly selected. - The population of interest is normally distributed (based on the central limit theorem, if N is >30, you should be fine). - The independent variable is nominal. - We have pairs of related scores.
• Know when to use each of the following tests: - z test - One-sample t test - Paired-samples t test - Independent-samples t test
- z test if 1 sample and distribution of means - One-sample t test if 1 sample and distribution of means - Paired-samples t test if 2 samples (same participants) and distribution of mean differences - Independent-samples t test if 2 samples (different participants) and distribution of differences between means
• Hypothesis testing with independent-samples t tests - Calculating the standard error for independent- samples t tests
1) Calculate the corrected variance for each sample. 2) Pool the variances. 3) Convert the pooled variance to squared standard errors (for each group). 4) Add the squared standard errors (one for each distribution of sample means) to calculate the estimated variance of the distribution of differences between means. 5) Calculate the square root of the sum obtained from step 4. This will produce the estimated standard error of the sampling distribution of differences between means.
• Assumptions for the one-sample t test
1. Dependent variable is scale 2. Participants randomly selected 3. Population distribution is normal 4. Independent variable is nominal
• Conceptual and computational understanding of 95% confidence intervals for paired- samples t tests.
1. Draw a picture that will include the confidence interval, centered on sample mean difference. 2. Indicate the bounds of confidence interval on your drawing. 3. Determine the t statistics that fall at each boundary. We will always use critical values that correspond to a two-tailed test when creating CIs. 4. Turn the t statistics back into raw mean differences. Mlower = -t(sM) + M Mupper = t(sM) + M
• Conceptual understanding of the one-sample t test
A hypothesis test that compares a sample from which we collect data to a population for which we know the mean (or at least suspect a mean), but for which we do not know the population SD
What sampling distributions is used when conducting a paired-samplest test?
A sampling distribution of mean differences
• Degrees of freedom
DF = the number of scores that are free to vary when we estimate a population parameter from a sample. • Example: Imagine that there are 4 scores in a particular sample. The mean of the sample is 5. You know that 3 of the scores are 4, 6, and 7. What is the value of the final score? • You can determine that the final score must be 3. Otherwise, the mean would not be 5. • That is, only three of the scores are "free to vary." Once all but one of the scores are known, the value of the final score is determined. For a one-sample t test, degrees of freedom = N - 1
• Cohen's d for independent-samples t tests
Divide the numerator of our t statistic formula by the pooled standard deviation of the differences between means (sPooled). d = (MX − MY) − (μX − μY) / sPooled Spooled =√s²pooled
True or false We can not legitimately conduct an independent-samples t test if the sample sizes of our two groups are different
False
• Sampling distributions of differences between means
For an independent- sample t test, if we want to examine height differences between two different groups of people, we need to find a sampling distribution of differences between means: • Randomly select and measure the height of however many people in group 1 and compute the mean of those heights. Randomly select and measure the height of however many people in group 2 and compute the mean of those heights. Subtract the second mean from the first mean. • Record the difference between means.
What is an assumption of the independent-samples t test, but not an assumption of the one-sample t test?
Homogeneity of variance
Independent vs paired sample
Independent sample is two sets of observations that are independent of one another. That is, the observations should not be correlated between the two groups. When dealing with paired samples, we use a different approach (i.e., create difference scores) because the groups were correlated. With independent samples, the denominator is standard error (denoted as Sdifference). Specifically, it is the standard error of a sampling distribution of differences between means (calculated in step 3 of hypothesis test)
• Degrees of freedom for paired-samples test
Number of pairs - 1
One sample vs paired t test
One-sample t test: • Compare one sample to a population, testing the null hypothesis that there's no difference (between the mean of the population we sampled from and the comparison population's mean). Paired-samples t test: • Gather paired observations and compute a difference for each pair of scores. • The difference scores will be our data set. We will not use the original scores directly to compute our paired-samples t statistic. • Determine the probability of obtaining the mean difference you obtained in your sample, given that it came from a sampling distribution of mean differences centered on the value specified by your null hypothesis (typically 0).
Imagine a researcher conducts an experiment using a within-groups design. Specifically, the same participants provide related scores on a dependent variable at two different levels of an independent variable. What test?
Paired
• Cohen's d for paired-samples t test
Subtract population mean difference for the null hypothesis (typically 0), from the mean difference obtained from our samples. We divide by the sample standard deviation of the differences. d = (M −μ) / s
• The paired-samples t test - Computation of paired-samples t test
Subtract the mean of the sampling distribution of mean differences (μM) from the sample mean difference (M) and divide by the standard error (sM). **N = # of pairs And μM is typically = 0.
• Computational understanding of the one- sample t test and its use in hypothesis testing
The one-sample t test can be calculated by subtracting the mean of the sampling distribution (μM) from the sample mean (M) and dividing by the standard error (sM). The standard error for the one-sample t test (sM) can be calculated by dividing the sample standard deviation (s) by the square root of the sample size (N). If t stat is 3.0 and critical value (found on t table using # of tails, N and α), is 2.0 reject the null hypothesis. If t stat is 1.0 and critical value is 2.0, fail to reject
• Calculating standard deviations from samples when estimating the population standard deviation (i.e., calculating "s").
The standard deviation of a sample (sample standard deviation) can be calculated by subtracting the sample mean from each score in the data set, squaring all of those differences, summing the squared differences together, dividing that sum by the number of scores in the data set minus 1, and then taking the square root of the resulting quotient.
• Cohen's d for one-sample t tests
To calculate, subtract the population mean (μ) from the sample mean (M) and divide that difference by the sample standard deviation. D= (M −μ) / s
True or false: As the sample size increases, the corresponding t distribution more closely approximates the z distribution
True
True or false? If you compute a t statistic for 25 related pairs of scores using SPSS, the t statistic will have the same absolute value as a one-sample t statistic computed from the differences between those 25 pairs of related scores.
True
• Conceptual and computational understanding of 95% confidence intervals for independent- samples t tests.
Turn the t statistics back into raw differences between means Lower: -t (sdifference) + (MX - MY) Sample Upper: t (sdifference) + (MX - MY) Sample
• t distributions
Use t distribution if we don't know σ (SD). • t distribution is a family of distributions. • Different t distributions for different sample sizes. t distribution varies with sample size. Specifically, the t distribution is wider and flatter than the z distribution (especially will smaller sample sizes). • As N approaches infinity, the t distribution approaches the z distribution.
When do we use a t distribution, instead of the z distribution, to conduct a hypothesis test comparing two means
When SD is not known
• Reporting statistics in APA style
t (df) = test statistic, p < 0.05, d = 1.09 t (40) = 7.00, p < 0.05, d = 1.09 **round test statistic to 2 decimal places
• Computation of the independent samples t statistic
t = (MX − MY) − (μX − μY) / sDifference The subscripts "X" and "Y" denote the 2 samples that we are working with. Typically (μX - μY) is hypothesized to be = to 0 by the null hypothesis. "Sdifference" is the standard error (i.e., the standard deviation of the sampling distribution of differences between means).
• Conceptual understanding of independent- samples t test
t test used to compare the means of two groups when the scores between the groups are independent of one another (e.g., often when we have a between-subjects design)
• The paired-samples t test - When to use it
t test used to compare two means for within-groups design (i.e., an experimental design in which every participant experiences both levels of the independent variable). The paired- samples t test should also be used when the data are arranged in related pairs. • Related-samples tests capitalize on a known association (lack of independence) between paired observations. This can increase our power to detect differences. -
• The paired-samples t test - How to use the t table to find critical values for hypothesis
• Alpha level (0.10, 0.05, 0.01) • df = N-1 (N= # of pairs of scores). • For a two-tailed test, we need to find the t statistics that cut off the highest and lowest 2.5% of the distribution. - From the t table use 2 tail, alpha level and df to find critical values (+ AND - 0.213)
Paired t test (N, M, s, sM)
• N = number of pairs of observations • Mean (M) = mean of difference scores • Standard deviation(s) = standard deviation of difference scores • Standard error (sM) = standard error of difference scores
• Sampling distributions of mean differences
• Record the mean for the difference scores. • Repeat the process over and over. • The resulting distribution = sampling distribution of mean differences • A typical null hypothesis would predict that the comparison distribution will be centered on "0."
Examples of paired differences
• Same people, measured twice (measured in two conditions or at different times) -Longitudinal data • Different people who might provide related data for a particular variable (e.g., political views of siblings raised in the same household). Study design examples: • Measure depression, provide some form of therapy, then measure depression again for the same people. (We should have a control group for good internal validity). • Measure intent to quit smoking, introduce new packaging and warning labels on tobacco products, then measure intent to quit smoking again for the same people.
