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Terms in this set (40)When are t statistics used in lieu of z scores? The general rule of thumb for when to use a t-statistic instead of a z-score is when the sample: Why do we need a different measure of standard error for t tests than z tests? Because the population standard deviation is unknown, it is impossible to compute the standard error of M as we did with z-scores. A z-score can not be calculated when the population variance or standard deviation is unknown. What formula is used to compute the estimated standard error for the single-sample t? sM = s/√n (based on sample standard deviation) sM = √(s2/n) (based on sample variance) *Instead of sM, s x-bar can be used How does sample size influence the degree to which a t statistic approximates a z score? With a large sample, the estimation of the sample variance to the unknown population variance is very good, meaning the t statistic will be very similar to a z-score. With small samples, the t statistic will provide a relatively poor estimate of z. What are degrees of freedom? What abbreviation is used to represent degrees of freedom? How are degrees of freedom computed for each t statistic? Then number of scores in a sample that are independent and free to vary. Degrees of freedom describe how well the t statistic represents a z-score. It determines how well the distribution of t approximates a normal distribution. Abbreviation for degrees of freedom = df Formula: n-1 How does the shape of the t distribution change in relation to sample size? The exact shape of the t distribution changes with degrees of freedom. For large df values, the t distribution will be nearly normal. For small df values, the t distribution will be flatter and more spread out than a normal distribution. As sample size and df increase, the variability in the t distribution decreases, and it more closely resembles a normal distribution. As sample size and df decrease, the t distribution becomes flatter and more spread out than a normal distribution. What is the purpose of the single-sample t test? The goal is to use a sample from the treated population (a treated sample) as a basis for determining whether the treatment has an effect. The test is used when we want to know whether our sample comes from a particular population if we do not have full population information available to us. What formula is used to compute the single-sample t? t = M - µ/sM If the null hypothesis for the single-sample is true, what value do we expect for the sample mean? The same value as the known or hypothesized population mean. (No difference exists between the sample mean and population mean, the difference is equal to zero). State the null and alternative hypotheses for each t statistic. *See sheet How is Cohen's d computed for each t statistic? *See sheet What does r2 represent? Another measure of effect size (Cohen's d also measures effect size in terms of standard deviation) r2 measures the percentage of the variability that is accounted for by the treatment effect How is r2 computed for each t statistic? r2 = t2/t2 + df How do we evaluate r2? = 0.01-0.09 - small effect = 0.09-0.25 - medium effect > 0.25 - large effect How do interval estimates differ from point estimates? A point estimate is a single value and has the advantage of being very precise. An interval estimate consists of a range of values and has the advantage of providing greater confidence than a point estimate. (The interval estimate is less precise, but gives more confidence. For this reason, interval estimates are usually called confidence intervals). Define confidence interval. A confidence interval is a range of values that estimates the unknown population mean. The confidence interval uses the t equation, solved for the unknown mean. *See sheet for formulas for each t-statistic What is the purpose of the independent-measures t test? The general purpose of the independent-measure t test is to determine whether the sample mean difference obtained in a research study indicates a real mean difference between the two populations (or treatments) or whether the obtained difference is simply the result of sampling error. To evaluate the mean difference between two populations. Also called independent-measures or between-subjects design. Must have two separate or independent samples. Used to test for mean differences between two distinct populations (e.g. men versus women) or two different treatment conditions (e.g. drug versus no drug). Describe the procedure (three steps) to compute the independent-measures t test. 1. Find the pooled variance for the two samples. 2. Use the pooled variance to compute the estimated standard error. 3. Compute the t statistic. What formula is used to compute pooled variance? s2p = SS1 + SS2/df1 + df2 What formula is used to compute the estimated standard error for the independent-measures t? s(M1-M2) = √(s2p/n1 + s2p/n2) What formula is used to compute the independent-measures t? (M1 -M2)-(µ1 - µ2)/s(M1-M2) sample mean difference-population mean difference/estimated standard error What assumptions must be met for the independent-measures t test to yield a correct decision? 1. Samples are not related: samples are not correlated in
any way (they are distinct groups that we are comparing) (#3 and #4 are similar for many different tests so #1 and #2 are important to focus on to determine if we have an independent or related - samples t test) Related-samples t test side steps the variance problem entirely because each person serves as their own control group or participants are matched on a particular variable. Define homogeneity of variance. Why does this assumption matter? Requires that the two populations from which the samples are obtained have equal variances. Necessary to justify pooling the two sample variances and using the pooled variance in the calculation of the t statistic. If the assumption is violated, then the t statistic contains two questionable values. 1. The value for the population mean difference which comes from the null hypothesis and 2. The value for the pooled variance. The problem is that you cannot determine which of these two values is responsible for a t statistic that falls in the critical region. In particular, you cannot be certain that rejecting the null hypothesis is correct when you obtain an extreme value for t,. In which research situation is it appropriate to use a single-sample t test? Use the single-sample t test if we know the population mean, but don't have the population standard deviation. In which research situation is it appropriate to use an independent-measures t test? Use the independent-measures t test if we have 2 independent groups that we are comparing (comparing 2 separate populations or randomly assigning people to 1 of 2 different treatment conditions). In what two research situations is it appropriate to use a related-samples (correlated samples) t test? Which of these situations is more common? Use the related-samples (correlated samples) t test if the groups are not independent: 1.) if we have a repeated-measures (within-subjects) design: measure the same people
at two different times - give the participants some sort of measurement (this becomes the dependent variable), do a manipulation (where you are introducing an independent variable), and then measure the participants again, this allows you to determine if the manipulation changed the participants or their scores changed from time 1 to time 2 because of your intervention *A repeated-measures (within-subjects) design is more common than a matched-pairs (matched-subjects) design. How does a within-subjects design differ from a between-subjects design? Between-subjects design = each subject is assigned to only one condition (treatment), within-subjects design = each subject is exposed to all conditions (treatments). Between-subjects design (also known as an independent-measures research design) uses a separate group of participants for each treatment condition (or for each population). Within-subjects design (also known as repeated-measures design) has a dependent variable that is measured two or more times for each individual in a single sample. The same group of subjects is used in all the treatment conditions. The between-subjects design would use two separate samples (one in each treatment condition) and the within-subjects design would use only one sample with the same individuals participating in both treatments. What are the advantages of a repeated-measures design? Repeated-measures design, in general, has more advantages that the independent-measures design: The main advantage of a repeated-measures design is that it uses exactly the same individuals in all treatment conditions. Thus, there is no risk that the participants in one treatment are substantially different from the participants in another. What are the disadvantages of a repeated-measures design? Exposure to the first measurement can cause a change in participants that influences their later scores. In this case, results can be distorted by order effects, and this can be a serious problem in repeated-measures designs. Time-related factors: The primary disadvantage of a repeated-measures design is that the structure of the design allows for factors other than the treatment effect to cause a participant's score to change from one treatment to the next. Specifically, in a repeated-measures design, each individual is measured in two different treatment conditions, usually at two different times. In this situation, outside factors that change over time may be responsible for changes in the participants' scores (for example: participant's health or mood may change over time). To remedy this: counterbalancing = participants are randomly divided into two groups, with one group receiving treatment 1 followed by treatment 2, and the other group receiving treatment 2 followed by treatment 1. This distributes any outside effects evenly over the two treatments. Which scores are used for calculating the related-samples t? To calculate t-statistic: For t-statistic: t = MD-µD/SMD What formula is used to compute the estimated standard error for a related-measures t statistic? SMD = √(s2D/n) What formula is used to compute the related-measures t statistic? t = MD-µD/SMD Between vs. Within - Subjects Design --- which statistic to use for hypothesis testing? Between-Subjects Design, aka independent-measures research design = use independent-samples t test Within-Subjects Design, aka repeated-measures research design, or other non-independent group designs (groups are not distinct from one another) = use related-samples t test (aka t test for correlated groups)
Is a repeated-measures a true experiment or quasi-experiment? Quasi, because of the gap between testing times (we have lost experimental control). Don't want confidence interval to equal what? Zero. What happens if increase level of confidence (for example: from 95% to 99%)? Increases the width of the confidence interval. *To obtain greater confidence, you must use a wider range of t statistic values, which results in a wider interval. What happens to the width of the confidence interval if you increase the sample size? The width of the confidence interval will decrease. *A larger sample gives you more information and you can estimate the population parameter with more precision. How to control for individual differences in a research design such as an independent-samples design? Randomly assign participants to treatment groups. How to control for experimental error in a research design such as an independent-samples design? All procedures are held constant for all participants.
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STATISTICS A plane is missing and is presumed to have equal probability of going down in any of three regions. If a plane is actually down in region i, let $$ 1 - \alpha _ { i } $$ denote the probability that the plane will be found upon a search of the ith region, $$ i = 1,2,3 $$ . What is the conditional probability that the plane is in region 1, given that the search of region 1 was unsuccessful? Verified answer
STATISTICS Refer to the data below, which are total home game playing times (hours)for all Major League Baseball teams in a recent year (based on data from Baseball Prospectus). $$ \begin{matrix} \text{236} & \text{237} & \text{238} & \text{239} & \text{241} & \text{241} & \text{242} & \text{245} & \text{245} & \text{245} & \text{246} & \text{247} & \text{247} & \text{248} & \text{248}\\ \text{249} & \text{250} & \text{250} & \text{250} & \text{251} & \text{252} & \text{252} & \text{253} & \text{253} & \text{258} & \text{258} & \text{258} & \text{260} & \text{262} & \text{264}\\ \end{matrix} $$ Use the total game playing times to create a stemplot. What does the stemplot reveal about the distribution of the data? Verified answer
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What happens to the shape of the sample as the sample size increases?In other words, as the sample size increases, the variability of sampling distribution decreases. Also, as the sample size increases the shape of the sampling distribution becomes more similar to a normal distribution regardless of the shape of the population.
What is the effect of sample size on the tt-Distributions and Sample Size
The sample size for a t-test determines the degrees of freedom (DF) for that test, which specifies the t-distribution. The overall effect is that as the sample size decreases, the tails of the t-distribution become thicker.
What does the shape of the tThe smaller the sample size, the more it differs from the normal distribution. We usually talk about degrees of freedom, which are often denoted by ν, and equals n − 1 where n is the sample size. So if we have a sample size of 8, there are 7 degrees of freedom. The shape of the t-distribution depends on ν.
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How does the shape of the T distribution change as the sample size increases quizlet?
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