Psychology Statistics #2
What is the purpose of constructing a confidence interval?
- because we cannot specify the population means value precisely; it would be desirable to specify the range of values in which the population mean can be expected to lie
Hypothesis
-a statement about a population that can be tested using a sample statistic
Difference score
-a subject/s score in one condition minus that same subject's (or related subject's) score in the other condition
standard score (z score)
-a variable whose value counts the number of standard deviations a score is above or below its mean
most of the time the population parameter or population mean lies within ________ of the sample statistic or sample mean
-about two standard errors - or 95% of the time the population parameter or population mean lies within 1.96 standard errors of the sample statistic or sample mean
In what way is a Type II error expensive to make?
-concluding that the null hypothesis should not be rejected when it should be -investigator will not report findings in journal -major expense is original investigator's time lost in the original investigation -there is no necessity of informing the community of a precious mistake because there was no report of findings in the first place -not as expensive to the scientific community as Type I error -cost in human terms is great though
Repeated measures design
-dependent-samples test -where the same individuals are measured twice, once before and once after some treatment
What factors affect t?
-the sample size -the width of the distribution -the distance between X- bar and mu
What is the critical value of a statistic?
-the value of a statistic that marks the boundary of a specified area in the tail of the distribution
Independent variable
-the variable whose value defines group membership -ex. Bloomer's study - the group - bloomers or others- to which the student is assigned
Pooled variance
-the weighted average of two variances; the weights depend on the respective sample sizes - a better point estimate of the population standard deviation than was wither of the two sample variances individually because the averaged pooled variance is based on more samples
What happens if the observed value of X-bar exceeds the critical value of X-bar or, equivalently if the observed value of the test statistic (zobs or tobs) exceeds the critical value of the test statistic (zcv or tcv)?
-then we reject the null hypothesis
The critical values for a 99% confidence level
plus or minus 2.58
Equation to obtain the standard error of the mean in a sample
s(x) = s/ square root of sample size
Reporting statistics
t(df) = tcalc, p<.05 -use p > .05 if there is no statistically significant difference between means - use p < .05 if there is a statistically significant different between means
Test statistic formula
test statistic = sample statistic - population parameter/ standard error of the sample statistic -population parameter = particular parameter specified by the null hypotheses -sample statistic = the statistic that is the best point estimate of that population parameter -standard error = standard error of the particular sample statistic specified in the numerator ( the one that point estimates the population parameter)
Where is the sample mean in a confidence interval?
the center
Null hypothesis
-the hypothesis that there is no effect or no difference in the population parameter -the hypothesis that we might reject -ex. living near power lines has no effect on IQ Ho: u = 100 (because IQ mean of general population is 100) -if null hypothesis is true, then the mean IQ of the population of all children living near power lines should be no different from the mean of the general population
T-distributions
-a family of distributions that, like the z distributions, are unimodal, symmetric, and asymptotic, but were sufficiently wider than z to account for the imperfection of using the standard error of the mean in a sample as a point estimate of the population standard error
The upper limit of a confidence interval is about ____ above the center of the confidence interval
- 2 s(x) -2 standard error's of the mean
Alternative hypothesis
-the hypothesis that there is in fact an effect or a difference -the one we might support -ex. living near power lines has an effect on IQ Ha: u =/ 100 -the mean IQ of all children living near power lines must have some value other than 100 -outcome of a study will be either that we reject the null hypothesis or that we fail to reject it
test statistics
-z or t test
Steps for constructing confidence interval when the population standard deviation is unknown
1. find the sample mean 2. compute the sample standard deviation 3. compute the standard error by using the standard deviation and dividing it by the square root of the sample size 3. determine t(cv) For a 95% confidence interval it is always 2.064 4. substitute into equation
The width of a confidence interval is based upon...
- s(x) -standard error of the mean
What advantage does a confidence interval have over a point-estimate?
- the point estimate of the sample mean is not likely to have exactly the precise value of the population mean -since we cannot specify the population mean's value precisely, it would be desirable to specify the range of values in which the population mean can be expected to lie
Directional (one-tailed) alternative hypothesis
- where one direction of change (an increase or a decrease based on prior empirical research) is specified in advance -whether there will be an increase or decrease in data -ex. based on prior empirical research, we have reason to believe that large doses of vitamin B increases intelligence. -we know in the general population that IQ scores are normally distributed with u=100 and o=15 -hypothesis would be: Ho: u<=100 Ha: u> 100 -we have reason to believe that vitamin B will increase IQ and alternative hypothesis focuses on that increase (u>100) -the hypotheses must cover all possibilities -therefore the null hypothesis in a directional experiment includes both those cases where vitamin B has no effect (u=100) and those cases that are contrary to our expectations - in the vitamin B study, where it decreases IQ (u<100) -one group is expected to have a larger or smaller mean compared to the other
The distance between the lower limit and the upper limit of a confidence interval is about...
-4 s(x) -4 standard error's of the mean
How to state a confidence interval?
-With 95% confidence we can say that the population mean is greater than 97.99 and less than 101.23 -when we say that we have 95% confidence, we are saying that if we were to follow the same procedure of drawing a sample, computing the sample mean, and preparing a confidence interval 1000 times, we would be preparing 1000 different confidence intervals, and the population mean would lie within the confidence interval in about 950 cases - in any given study, we do not know whether the population mean lies in our computed confidence interval
Observed value of the statistic
-a computed statistic of our sample
What is a point-estimate?
-a computed statistic that approximates a parameter -to point-estimate is to compute a statistic and accept that computed value as an approximation of the unknown value of the parameter -ex. the population mean is probably close to the sample mean of 95.68 pounds (not a perfect estimate) -to point estimate the mean is to compute the sample mean and accept that value as an approximation of the population mean -best point estimate of the population mean is the sample mean - a way of approximating a parameter from a statistic - almost never exactly correct
critical value of the statistic
-a criterion set in advance; the beginning of the rejection region. If the observed value of the statistic lies in the rejection region or is greater than this critical value, we reject the null hypothesis
95% Confidence interval
-a range of values that has a 95% probability of containing the actual value of the parameter or population mean -It is answering the question where's u? I don't know, but probably it is within an interval that is centered at the sample mean of the sampling distribution of means and extends about two standard errors in each direction -center of confidence interval is the sample mean and extends about two standard errors in each direction -the confidence interval limits are z(cv)o(x) below and above the sample mean
Relationship between the number of degrees of freedom and the critical value of t
-as the number of degrees of freedom (or n) decreases, the critical value of t increases (t distribution widens)
What does the central limit theorem specify about the shape of the sampling distribution of the means
-as the sample size n increases, the sampling distribution of the means of samples of size n approaches a normal distribution -the distribution of means of even a bimodal, non asymptotic distribution becomes normal as the sample size increases
The probability of making a Type II error
-called B (Greek Beta) -the compliment of power -B= 1 - power -The probability of mistakenly failing to reject Ho (B) is one minus the probability of correctly rejecting Ho (power) -ex. power = .8 so B = 1- .8 = .2 which implies that 20% of such experiments end in a Type II error
Steps when evaluating means of two independent samples
-check if nondirectional or directional -find null hypothesis -find alternative hypothesis -compute the standard deviation of each set of data -compute the pooled variance -compute the standard error of the differences between two means -sketch the distribution of the variable -find sample statistic -sketch the distribution of the sample statistic -find the test statistic -sketch the distribution of the test statistic -find level of significance -find how many degrees of freedom there are -look up the critical value(s) of the test statistic and enter them (it) on the distribution and shade the rejected regions -compute the critical value of the sample statistic - enter on the appropriate distribution and shade the rejected region -compute the observed value of the sample statistic - enter it on distribution -compute the observed value of the test statistic - enter it on the appropriate distribution -do the critical values, rejection regions, and observed values have the same position on the distribution? - is result statistically significant? -if result was statistically significant determine the raw effect size and the effect size index -describe the results, both statistically and practically
How much wider must the distribution of t be than the distribution of z
-depends on how efficient s is as an estimator of o -if s can be said to be almost identical to o, then t must be almost identical to z -the accuracy of s depends on the sample size -the larger the sample, the more precise the point-estimation -as sample size decreases, t distribution widens -when the sample size is larger than about 120, s is almost indistinguishable from o and therefore the t and z distributions are essentially identical -the smaller n is, the wider the t distribution must be because the efficiency of s as a point-estimator of o decreases -t distribution depends on n or as we prefer to say, it depends on the number of degrees of freedom -s is a moderately good point-estimator of o whenever df is greater than about 5, and s is an excellent point-estimator whenever df is greater than about 30
In what way is a Type I error expensive to make?
-expensive for the scientific community -ex. we report in a journal that doses of vitamin B increase IQ when it actually does not -as a result, our readers will alter their behavior, perhaps focusing on a Vitamin B diet while ignoring other avenues that may be affecting in raising IQ -later someone will conduct an experiment and find that vitamin B has no effect on IQ - they will demonstrate that we made a Type I error -we would want to find and contact the entire readership of our first report and inform them that our result was mistaken -expensive and difficult -cost in human terms is great
Type II error
-failing to reject the null hypothesis when it is in fact false -ex. Warren takes the cancer test and it says he does not have cancer cells, when in fact those cells are actually present -the null hypothesis is false (cancer cells are actually present), but we fail to reject it (the test did not say cancer) therefore this is a Type II error -blIIndness (missing something that is there) -we will not know if we have made a Type II error -unavoidable consequences of taking a random sample
Steps when evaluating mean of single samples
-find if test is directional or nondirectional -find the null and alternative hypothesis -find the variable in the problem -find the standard deviation of the variable in the problem -find the sample statistic in the problem -compute the sample statistic's standard error -sketch distribution of sample statistic -find the test statistic -sketch distribution of test statistic on distribution of sample statistic -find level of significance -find how many degrees of freedom -look up the critical value(s) of the test statistic -enter it (them) on the appropriate distribution and shade the rejection region -compute the critical value(s) of the sample statistic -enter it (them) on the appropriate distribution and shade the rejection region -compute the observed value of the samples statistic -enter it on distribution -compute the observed value of the test statistic -enter it on the distribution -check to see the critical values, rejection regions, and observed values have the same position -see if result is statistically significant -if it is determined the raw effect size and the effect size index d -describe results statistically and practically
Statistically significant
-if the outcome of an experiment leads to the rejection of the null hypothesis -they are not the kind of results we would expect by chance alone -or that there was a statistically significant difference - that is, a difference greater than would be expected by chance alone ex. for the bloomer's study the mean intellectual growth was 16.5 for bloomers and 16.4 for the other students, we would probably conclude that although these means were numerically different, they were not significantly different -ensures that the result is not merely a chance fluctuation
Practical significance
-if the result of an experiment is statistically significant, then the report of that experiment should include a discussion of its practical significance -the importance of a result -measures of effect size (a measure of the magnitude of a result) must be used in the discussion of practical significance -ex of effect size- the difference between the means of those taking vitamin B and those not taking the vitamin
How is the alternative hypothesis supported?
-if we reject the null hypothesis, then we support the alternative hypothesis
Why is a point-estimate referred to as an unbiased estimator?
-if we repeated the point-estimating process infinitely often, the same number of point estimates would be too high as too low -ex. if we take a large number of samples, each of size n, half the sample mean values will be larger than the population mean and the other half will be smaller -sample mean is the best unbiased point-estimate because in the long run, the sample mean will lie closer to the population mean more often than will either the median or the mode
Four Factors that affect the width of a confidence interval
-increasing n makes the confidence interval narrower -decreasing the population standard deviation or the sample standard deviation makes the confidence interval narrower -the confidence interval when the population standard deviation is known is generally narrower than when the population standard deviation is unknown (when we use s to estimate o we must use t instead of z and t is larger than z so the confidence interval using t is generally wider than the confidence interval using z) -decreasing the level of confidence makes the confidence interval narrower
Match scores variability
-match scores have less error/variability
Is it desirable to have a confidence interval wider or narrower
-narrower because the size of a confidence interval reflects our uncertainty about the actual value of the parameter u -the less uncertainty the better, so the narrower the confidence interval, the better
If others were independently to perform similar studies to ours would they construct identical confidence intervals to ours?
-no, each is centered on its own sample mean and the sample mean differs from study to study -each is intended to provide an interval in which the population mean is likely to lie
Independent samples
-randomly select two separate, independent groups -give one group caffeine and the other group a placebo -then we would measure the reaction times in all subjects -the scores in these two groups are not related to each other: the reaction time of the first person in the first sample in no way depends on the reaction time of the first person in the second sample
Type I error
-rejecting the null hypothesis when it is in fact true -error does not imply that you did anything wrong; experiment may have followed entirely proper procedures and yet resulted in a Type I error because of the nature of random sampling -we will not know that we have made a Type I error because we are unable to ever know that the null hypothesis is true -ex. Sally takes cancer test and it says she has cancer although actually (an omniscient being would know) she does not. the null hypothesis is true (actually no cancer), but we reject it (the test says cancer) so this is a Type I error - gullIbility- seeing something that isn't there -unavoidable consequences of taking a random sample
Three common dependent samples designs
-repeated measures (same individuals are measured twice) -related samples (identical twins, husband and wife, two cars that are the same make) -matched pairs (where subjects are measured on some important variable and then paired off - the highest scorer is paired with the second highest scorer, the third highest with the fourth highest and so on. Then by random decisions one of each pair is assigned to be a member of the first group and the other a member of the second group) -one group would for example get the caffeine and the other would receive the placebo
How does the confidence interval change to account for the variabilities of both the population mean and population standard error
-the confidence interval must be somewhat wider to account for the imperfection of using sx as a point-estimate of ox
Raw effect size
-the magnitude of on experimental result measured in the scale of the original experiment -measures how far the observed results depart from those specified by the null hypothesis - X-bar(obs) - u - ex. it states that the average person in the trained group has a GRE score that is _____ higher than the average untrained person
What does the CLT specify about the mean of the sampling distribution of the means?
-the mean of the sampling distribution of the means is equal to the parent population mean
Dependent variable
-the measured outcome of interest that is of primary interest in the study ex. Bloomer's study - intellectual growth its value depends on the teachers expectations and the individual characteristics of the subject
Level of significance
-the probability (signified by a) of making a Type I error -the probability that we conclude that an effect exists when in fact there is none -the complement of the level of confidence -we want the probability that we make a Type I error to be small (typically .05 or .01) -ex. if the level of significance is .05 then out of every 100 experiments we perform where the null hypothesis is in fact true, about five will incorrectly conclude that the null hypothesis should be rejected
Power of a statistical test
-the probability of correctly rejecting the null hypothesis -it is desirable that power be high (close to 1) -ex. we say that a statistical test is powerful to the extent that it correctly identifies situations where the alternative hypothesis is true (and where the null hypothesis is correspondingly false) -ex. suppose that 100 different experimenters conducted 100 identical experiments to determine whether vitamin B increases IQ, and suppose that the statistical power of each of those experiments is .8. -We would expect that approximately 80% of those 100 experimenters would reject the null hypothesis and correctly conclude that vitamin B increases IQ whereas approx. 20% would fail to reject the null hypothesis and incorrectly conclude that vitamin B has no effect (result in a Type II error) -Power = 1 - B
effect size index
-the raw effect size divided by the standard deviation of the variable -effect size index d = .74 indicates that X-bar(obs) is .74 standard deviation above u -can be thought of as a measure of the amount of overlap between the distributions -if d= 0 then the two distributions overlap
What is the best point-estimator of the population standard deviation?
-the sample standard deviation (s)
What does the CLT specify about the variation of the sampling distribution of the means?
-the sampling distribution of means is narrower than the parent population by a factor of the square root of the sample size -the equation: standard error of the mean= standard deviation of population/ square root of sample size
Hypothesis evaluation
-the science of deciding whether inferences (called hypotheses) about populations and their parameters should be rejected or not rejected -main result of hypothesis tests are qualitative (yes, we should reject the hypothesis, or no we should not reject) - to test a hypothesis is to decide whether it is true or false, whether it can be supported or rejected -In all cases, hypothesis are about parameters -hypothesis testing is aimed at the null hypothesis -either we will reject the null or we won't -if we reject the null hypothesis then that rejection supports the alternative hypothesis
What are descriptive statistics
-the science of describing - convey the characteristics of- distributions of samples or populations -describe them in terms of their shape, central tendency, and variation -describe by specifying their statistics for a sample or their parameters for a population
What are inferential statistics?
-the science of using sample statistics to make inferences or decisions about population parameters -ex. the point-estimates and confidence intervals -when we use the sample mean to point estimate the population mean, we are making an inference about the population mean - namely that its magnitude is approximately equal to the sample mean -when we use a confidence interval, we are making an inference about a parameter - namely, that it has a 95% probability of being in the given interval -same, different, higher, lower
What two factors affect the magnitude of the standard error of the mean?
-the standard deviation and the sample size -the standard error will be small if the standard deviation is small or if the sample size is large (or both)
standard error of the mean
-the standard deviation of the sampling distribution of the means
What is the central limit theorem?
-the theorem that describes the shape, mean, and variation of the sampling distribution of means
Dependent samples
-two samples whose subjects are statistically related to each other -ex. we enlist 20 volunteers. For each, we measure their reaction time. Then we administer a dose of caffeine and then measure that same person's reaction time again -if someone has a naturally fast reaction time then the first score in both groups is likely to be fast -the score in the second sample depends on or is related to the score in the first sample (if you have a fast score in the first sample most likely to have a fast score in the second) -the order of the data in each sample is important - ex. if one member of a particular pair happens to be the seventh member of the first sample data set, then the other member of that same pair must also be the seventh member of the second sample data set (matched pairs) - there must be the same number of data points in each group
When do you use a z test and when do you use a t test
-use a z test when you know population standard deviation (o) -use a t test when you know sample standard deviation (s)
When to use nondirectional and directional tests
-use nondirectional test when prior evidence is inconclusive and you are looking whether there is a difference (we aren't looking at whether it specifically increases or specifically decreases just if it changes) -use directional test when prior evidence shows an increase or decrease in variable (looking at whether it specifically increases or specifically decreases)
Nondirectional (two-tailed) alternative hypothesis
-where we don't have previous empirical research telling us which way an increase or decrease will be seen -whether there will be any difference -where the hypothesized change can be either an increase or a decrease -the null and alternative hypotheses are of the form Ho=a and Ha=/a (does not equal) where a is some constant ex. a=100 -there is no difference vs. there is a difference
Can a result be statistically significant but have no practical significance?
-yes -Dr. able to reject the null and support the alternative hypothesis concluding that the people weigh less after using the device -We find that the effect size index is only .025 standard deviations or .2 pounds which has little to no weight loss at all -it is not practically significant
Why is a hypothesis easier to prove false than it is to prove true?
-you just need one data point to prove a hypothesis wrong but to prove it right you have to look at all data points possible
Hypothesis evaluation procedure
1. state the null and alternative hypotheses 2. set the criterion for rejecting the null hypotheses 3. collect a sample and compute the observed values of the sample statistic and the test statistic 4. Interpret the results
Confidence interval for the population mean when the population standard deviation is unknown
X- t(cv)(sx) <= u <= X + t(cv)(sx) - same as confidence interval when we do know the standard deviation -except that we substitute t for z and sx for ox
How to solve confidence interval for the population mean when the standard deviation is known
X- z(cv)(ox) <= u <= X + z(cv)(ox) -you are given the mean -z= 1.96 for 95% confidence intervals -solve for the standard deviation which is standard deviation (o)/ square root of the sample size - we think that probably the population mean is greater than or equal to the lower limit of the confidence interval; also we think that probably the population mean is less than or equal to the upper limit of the confidence interval
