Term
| Single-sample formal tests of hypothesis: what two means are we given in each problem and what issues are we asked to decide? |
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Definition
| each single-sample formal test of hypothesis will provide the mean of a particular population (mu) and the mean of a sample (x-bar) and the test will allow us to decide whether the sample could have been drawn from the known population or if the sample must have been drawn from a different population |
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Term
| What is the known population? When/why do we use the phrases population from which the sample was drawn and different population? |
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Definition
| THe population for which we are told mu is the known population. Because we do not known until the end of the problem whether the sample comes from the known population or from a different population, we'll often refer to the "population from which the sample was drawn" before / during testing. At the end of the test, we'll have a conclusion about whether the population from which the sample was drawn is the known population or a different population |
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Term
| What are the three steps described in the method overview for the single-sample tests? |
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Definition
1. Use the mean of the known population (mu) to create a DSM for the known population.
2. Estimate the probability of the sample mean (x-bar) given in the problem having been drawn from the DSM of the known population.
3. If the sample given in the problem would be very rare in the DSM for the known population, then we will conclude that the sample is NOT from the known population, that it is actually from a different population. |
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Term
| Why is the language of inferential statistics full of probability statements? |
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Definition
| the language of inferential statistics is full of probability statements because we know that there is variability in the samples that we draw from populations, and that few samples drawn from a population will have a mean equal to the mean of the entire population. Thus, in a single-sample test of hypothesis, when we are given a sample mean and note that it does not equal the mean of the known population, this does NOT provide evidence that the sample comes from a different population. |
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Term
| We can't directly compare a sample mean to a population mean and conclude that the sample does not come from the known population because the means are not equal. Why not? |
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Definition
| Many samples we know to be drawn from the population will have means that vary from the population mean by chance alone. |
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Term
| When will we say that a sample mean could have differed from the mean of the known population by chance alone? If we say that it differs by chance alone are we saying that the sample came from the known population or form a different population? |
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Definition
| If a sample mean given in a particular problem resembles sample means common in the DSM for the known population, we will conclude that the given sample mean differs from the known population mean by chance alone. Thus, we will decide that we have insufficient evidence to conlcude that the sample was NOT drawn from the known population -- even though the sample mean does not equal the mean of the known population. |
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Term
| The method of formal hyptohesis-testing - how we conduct formal tests - does it involve directly comparing population means or sample means? |
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Definition
| The method of formal hypothesis testing - how we conduct the test - involves comparing the sample given in the problem to samples in the DSM we created for the known population. |
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Term
| Why is it true to say that a single-sample test of hypothesis is designed to compare two populations? |
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Definition
| A single-sample test of hypothesis is designed to compare two populations. When will we use a single-sample test of hypothesis? If we want to know if two populations differ and we know the mean of one of the populations, then we'll take a sample from what we believe may be a different population and conduct a single-sample test of hypothesis. |
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Term
| Why do we create null hypotheses? What 4 characteristics are true of null hypotheses? What characteristics are true of alternative hypotheses? |
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Definition
We want to show something is true, but our best statistical tools only know how to show something is false. So, we state the exact opposite of what we want to demonstrate is true (the null), disprove that, and what's left(the alternative) must be true.
Ha: what the researcher wants to demonstrate is true (differs from status quo); if the null hypothesis is disproved, then Ha is accepted as true; says that mu for the population from which the sample was drawn is >, < or different from the known population mu
Ho: the hypothesis that statistical tools can demonstrate is highly unlikely (can disprove); contradicts the Ha, taking into account every other possibility than the one the researcher is hoping to demonstrate is true;the hypothesis tentatively held to be true until evidence that it is very unlikely to be true is obtained; says that mu for the population from which the sample was drawn equals the population mu; always contains a version of equals sign |
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Term
| What are the four items to think through while you are reading a problem and writing hypotheses? |
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Definition
1. Name the two groups being compared and note which group is mentioned first in the most concise comparison sentence.
2. Note the relationship between groups predicted by the researcher.
3. Write the Ha to reflect the researcher's prediction and Ho to represent all other possibilities.
4. Determine for which group you are given the mu and insert that value into hypotheses. |
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Term
| What is the difference between difrectional and non-directional hypotheses? What are two types of additional hypotheses? What's another name for a non-directional hypothesis? |
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Definition
Directional hypotheses are for if we suspect that mu for the population from which the sample was drawn will be < or > mu. Lower tailed or Upper tailed.
Non-directional hypotheses are for if we can't predict mu for the population from which the sample was drawn will be < or > mu, but we believe it will be different from the known population. Two tailed. |
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Term
| What is a rejection region? What is alpha? What does alpha have to do with the rejection region? |
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Definition
The rejection region is an area on the DSM for the known population that represents where the sample mean must fall before we reject Ho. Alpha is the value set before conducting a statistical test that indicates how rare a sample must be before we will decide that it does not come from the known population.
THe amount of area in the rejection region = alpha. |
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Term
| If the sample mean given in the problem falls in the rejection region of the DSM for the known population, what do we know about the probability (p) of having drawn that sample mean from the known population? Is the p less than, equal to, or greater than alpha? |
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Definition
| If the sample mean given in the problem falls in the rejection region of the DSM for the known population, the probability of the sample being drawn from the known population is less than alpha. YOu can reject Ho. |
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Term
| Where does the rejection region belong for 1. an upper-tailed test, 2. a lower-tailed test, and 3. two-tailed test? |
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Definition
1. Upper tailed hypothesis states that the population from which the sample was drawn has a mean higher than the kown population. Rejection region belongs in upper tail.
2. Lower-tailed hypothesis states that the population from which the sample was drawn has a mean lower than the known population. Rejection region belongs in the lower tail.
3. Two-tailed hypothesis states that the population from which the sample was drawn has a mean different than the known population. The test is 2-tailed and half of the rejection region is in the upper tail and half is in the lower tail. |
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Term
| Why do we divide alpha by 2 only when the hypothesis is non-directional? |
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Definition
| because it could be in the lower or the upper tail |
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Term
| What is the critical value? What does it indicate? L5 |
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Definition
| The critical value is a z- or t-score we get from a probability table that indicates how far the sample mean (x-bar) must be from the mean of the DSM (mu sub x-bar) before the sample is considered very rare. Thus, the critical value indicates the boundary of the rejection region -- if the z- or t-score for the sample mean is more extreme than the critical value, it will fall in the rejection region |
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Term
| When must we use the t-table to locate the critical value for a problem? |
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Definition
| To obtain accurate estimates of areas beneath the normal curve when a sample size is less than 30, we must use the t-table instead of the t-table. |
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Term
| Why is the t-distribution wider than the z-distribution? Does the t-distribution get wider or more narrow as sample size increases -- what does this have to do with what we learned about standard error in the last unit? |
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Definition
| The t-distribution is wider than the z because it reflects higher variability of sample means for smaller sample sizes. As sample size increases, the distribution gets more narrow. |
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Term
| How do we calculate degrees of freedom |
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Definition
| df = n-1. we do not need to calculate if N > 30 |
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Term
| What are decision rules? What do the decision rules have to do with the critical value? |
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Definition
| Decision rules state in words what has to be true for us to reject the null hypothesis -- what has to be true for our sample mean to fall in the rejection region. We state the decision rules before we calculate the z- or t- score so we are ready to make a decision once we calculate. |
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Term
| What is the calculated value? How do we obtain it? What does it indcate? |
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Definition
| The calculated value is the z- or t-score for the sample mean we were given ; it is obtained by using a formula; it indicates how far the sample mean (x-bar) actually is from the mean of DSM (mu sub x-bar) |
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Term
| What are assumptions? Are we guaranteed an accurate hypothesis-testing conclusion if assumptions are true? |
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Definition
| Assumptions are conditions that must be met in order for our hypothesis-testing conclusion to be valid. Testing errors can still occur even if the assumptions for the test are met. |
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Term
| Is the first assumption for our single-sample tests about the sample we're given or is it about the known population? Why is it important? |
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Definition
| Assumption 1: Individuals in the sample were selected randomly and independently, so our sample is highly likely to be representative of the larger population from which it was drawn. |
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Term
| Is the second assumption for our single-sample tests about the sample we're given or is it about the known population? Why is the second assumption important? |
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Definition
| Assumption 2: The DSM for the known population is normal, either because the size of samples used to create the DSM is large (N > 30) or because, even thogh our samples were small, the variable we are analyzing was normally distributed in the known population |
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Term
| When do we say that a "statistically significant difference between populations" exits? What can we conclude about the sample mean we were given and the known population? What can we conclude about the means of the two populations of interest? |
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Definition
A "statistically significant difference between populations" is said to exist when it is highly unlikely that the mean of the population from which the sample was drawn equals the mean of the known population. If the difference between the sample mean and the known population is large (relative to standard error), then
1. it is highly unlikely that our sample mean (x-bar) differs from the known population mean (mu) by chance
2. it is highly unlikely that the sample mean (x-bar) was drawn from the known population
3. it is highly unlikely that the mean of the population from which the sample was drawn (mu) equals the mean of the known population (mu sub x-bar) |
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Term
| If the results of an inferential test of hypothesis are statistically significant then we conclude that we would get the same results with ________ and with __________. |
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Definition
| If the results of an inferential test of hypothesis are statistically significant then we conclude that we would get the same results with other samples drawn from the same population and with the entire population. |
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Term
| Tests can mislead us about what is true in the "real world" in two ways. They can lead to false alarms and to misses. THinking about the functioning of an in-home smoke detector, give an example of a false alarm and a miss. WHen it comes to pregnancy tests, smoke detectors, and air traffic control panels -- which error is most serious? |
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Definition
| IN terms of an in-house smoke detector, a false alarm would be when the alarm goes off but there is no fire. A miss is when the alarm does not go off and there is a fire. When it comes to pregnancy tests, smoke detectors, and air traffic control panels a miss is more serious. |
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Term
| Statistical tests are prone to false alarms and missed. Define and give examples of TYpe 1 and TYpe II errors. |
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Definition
| Type 1 errors are false alarms. They are when the researcher concludes -- on the basis of the sample data they have -- that there really is a difference between populations, when in truth there is no difference between the populations (the sample comes from the known population). Type 2 errors are missed. They are when the researcher concludes -- on the basis of teh sample data they have -- that there is no difference between populations , when in truth there is a difference between the populations ( the sample comes from a populations other than the known population) |
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Term
| What are the verbal formulas you can use to write Type I and II errors for particular hypotheses? |
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Definition
-A Type I error would be: If the test leads me to conclude "Ha" when, in fact, "Ho" is true.
-A Type II error would be: If the test leads me to conclude "Ho" when, in fact, "Ha" is true. |
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Term
| Why is incorrect to write "Accept Ho"? |
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Definition
| WE either Reject Ho or Do not reject Ho, we do not ever accept Ho. It is due to the nature of our method of hypothesis testing that we do not ever have the option of "accepting the null hypothesis". We create the null hypothesis in order to have a hypothesis to prove wrong, because proving a statement WRONG is the job that statistical tests do best (as opposed to proving something right) |
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Term
| Why are type 1 errors more serious than type II? |
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Definition
| Type I errors are generally more serious than TYpe II errors. Researchers generally want to demonstrate a difference between populations. Claiming a difference between populations when there really is no different between populations (a Type I error) is a serious mistake. If scientists are overeager to claim that their studies show statistically significant results and that they therefore, have a new fact to add to their discipline then science would accrue many "facts" that aren't really true. |
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Term
| When do Type I and Type II errors occur? |
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Definition
If I reject Ho, I might have made a Type I error, the probability of having done so is equal to alpha,
If I do not reject Ho, I might have made a Type II error. |
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Term
| How does alpha relate to TYpe I errors? |
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Definition
| Alpha is the value set before conducting a statistical tests that indicates how rare a sample must be before we will decide that it does not come from the known population. It is also equal the the probability of making a TYpe I error. Alpha determines the frequency of Type I errors. If we set alpha at .05, 5 times out of 100 when the null hypothesis is true (there is no difference between populations), you will reject the null hypothesis (claim the populations do differ) -- making a Type 1 error. |
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Term
| Why are Type I errors and Type II errors inversely related? |
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Definition
| Type I and Type II errors are inversely related because the more strict we make our criterion (standard) for accepting the research hypothesis ( to reduce the likelihood of a Type I error), the more likely it is that we will reject the research hypothesis when it is actually true. Statistical decisions are usually made so as to minimize the likelihood of making a Type I error, even at the risk of making lots of Type II errors. |
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Term
| What is power? How do sample size, precision of measurement, and alpha value selected influence power? |
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Definition
| Power is the "sensitivity" of the test -- it is the probability that a statistical test will lead to a decision to reject the Ho when Ho is indeed false. We want to have a high power: If there really is a difference between populations then we want to find it using statistics. Power increases as sample size and precision of measurement increase -- the larger our sample and the more precise our measurement -- the more likely likely it is that or statistical analysis will lead us to correctly reject the null hypothesis when our alternative hypothesis is correct. Power also increases as alpha increases. |
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Term
| When we decrease alpha, the probability of a Type I error declines and so does power -- why? |
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Definition
| The smaller alpha, the less likely it is that we will reject the null, and thus the less likely we will reject the null when we should reject it (power) |
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Term
| If alpha is 0.05 and we conclude that we can reject the Ho, then we know that p alpha -- again, why? |
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Definition
| If alpha is 0.05 and we conclude that we can reject the Ho, then we know that p alpha because the sample mean does not fall in the rejection region. |
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Term
| What is an exact p-value? |
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Definition
| an exact p-value is the exact probability of obtaining, from the known population, a sample as rare or more rare than the one we were given |
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Term
| What's the difference between a sample statistic and a population parameter? |
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Definition
| Sample statistics are descriptive statistics calculated for sample data. Population parameters are descriptive statistics calculated for data for an entire population |
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Term
| What's the difference between a point estimator and an interval estimator? |
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Definition
| A point estimator is a single number that represents a population parameter. An interval estimator is a range of values around a sample statistic that estimates a population parameter. |
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Term
| Do confidence intervals estimate the mean of the known population or the mean of the population from which the sample was drawn?? |
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Definition
| Confidence intervals estimate the mean of a population from which the sample was drawn. They do not estimate the mean of a known population |
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Term
| Why are confidence intervals a type of inferential statistic? |
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Definition
| Inferential statistics are mathematical techniques that allow us to make decisions, estimates, or predictions about a larger group of individuals (a population) on the basis of data collected from a much smaller group (a sample). Confidence intervals , therefore, are a type of inferential statistic because they estimate the mean of a population from which the sample was drawn |
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Term
| What is the definition of confidence interval? |
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Definition
| Confidence intervals estimate the mean of a population (mu) from which the sample was drawn by calculating a range of values around the sample mean (x-bar) that is highly likely to include the mean of the entire population (mu) -- with 0.90, 0.95, or 0.99 confidence. |
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Term
| What is the logic underlying how we calculate and interpret confidence intervals? |
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Definition
-We know that sample means (x-bar) tend to resemble the mean of the population (mu) from which they were drawn. The particular sample mean (x-bar) we have is likely to vary by chance from the mean of the population (mu) from which it was drawn--it is likely to be a bit higher or lower -- but it probably is not extremely different from the population mean (mu)
-We then calculate a range of values about the sample mean (x-bar) that describe other sample means (x-bar) common to the DSm from which the sample was drawn.
-Given that population means (mu) tend to resemble the most common sample means (x-bar) in a DSM, we conclude that the mean of the population (mu) from which the sample was drawn is highly likely to fall within the range we have calculated. |
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Term
| For confidence intervals, what does alpha represent? |
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Definition
| Alpha is the probability that the confidence interval does NOT enclose the population mean |
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Term
| For confidence intervals, what do critical values tell us? |
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Definition
| Critical values are z-scores or t-scores that are the upper and lower bounds of the confidence interval |
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