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Unformatted text preview: Testing Hypotheses in Research Research Review of Confidence Intervals Review
95% confidence interval for a population proportion: Sample proportion +/ 2*s.e. Where s.e.=sqrt(sample proportion*(1sample proportion)/n) 95% Confidence intervals for population mean: Sample value+/ 2 * s.e. Where s.e.=sample standard deviation/sqrt(n) 95% Confidence intervals for difference b/w means:
Mean 1 – Mean 2 +/ 2 * s.e. Where se= sqrt[sd12/n1 + sd22/n2] Where sd Review of CI (con.) Review
Distribution of sample means/proportions is Distribution normal normal CI Interpretation: 95% of 95% CI’s will include CI true population value true Can calculate 68% or 99.7% CI by changing “2” Can to 1 or 3, respectively to http://onlinestatbook.com/stat_sim/sampling_dist/i 68% confidence intervals exercise Have one member from your group draw the CI on Have the board the Write the 5 original heights next to the CI Mean Female Height in Sample: 65 inches Inferential Statistics Inferential Uses sample information to determine if there Uses is a relationship in the population is E.g. Does curriculum A result in higher levels of E.g. student knowledge than curriculum B? Do raids on crack houses have an effect on crime? Etc. on Remember chi squared test examples Review Review Step by Step guide for hypothesis testing 1. Determine Null and Alternative Hypotheses Hypotheses Null hypothesis: There is no relationship between Null a homosexual selflabel and a positive response in a request about a rental property request Alternative hypothesis: There is a relationship between a homosexual selflabel and a positive response in a request about a rental property response Hypothesis Example Hypothesis
Null hypotheses: The proportion of MSU Null students approving of Obama is the same as other Big Ten Universities. other Alternative Hypothesis: The proportion of MSU Alternative students approving of Obama differs from other Big Ten Universities. other Null Hypothesis Null
Why test the null hypothesis? Science starts from the assumption of no relationship. Science based on the principle of falsification. falsification. 2. Collect Data & Summarize in a Test Statistic Test Test statistic differs for different types of data E.g. chi squared stat Note that these data can result from an Note experiment, or from an observational study experiment, 3. Calculate pvalue 3. pvalue is the probability of rejecting the null pvalue IF THERE IS NO RELATIONSHIP in the population population [it is NOT the probability that the null hypothesis [it is true!!] is Calculate on a computer or look up in a table 4. Make a decision 4. Either: Choice 1: the pvalue is not small enough to rule out a Choice relationship in the sample occurring by chance; therefore we cannot reject the null hypothesis we Choice 2: the pvalue is small enough to rule out a Choice relationship in the sample occurring by chance; we reject the null hypothesis the Choice 2: the results are statistically significant Choice statistically Notes on pvalue Notes How small must the pvalue be to make How Choice 2? Choice Must be lower than a level of significance Must determined in advance by researchers determined By convention, usually .05 Some terms for hypotheses Some Researchers are interested in testing whether a Researchers population value equals a particular value; this value is called the null value null In a onesided test, values in the alternative hypothesis are In onesided values above the null value or below the null value only E.g. Is the proportion of smokers at MSU greater than .2? In a twosided test, values both above and below the null In twosided values value are included value E.g. Does the proportion of smokers at MSU differ from .2? Calculations necessary Calculations There are two values we need to calculate: test statistic pvalue Calculating Test Statistics Calculating This week: test statistics involving This proportions, means, and differences between proportions proportions We have already discussed relationships between We categorical variables (chi squared statistic) categorical 3 Types of Test Statistics Types 1. Compare a single observed proportion to a hypothetical 1. proportion (In a survey of 1000 adults, 55% approve of Obama. Do more than (In half of voters in the population approve of Obama?) half (We have a survey of 500 women drivers; they get in .2 accidents per year. Do women in the population get in fewer than 1 accident per year?) per (We have survey of 500 men and 500 women drivers; women get (We in .2 accidents per year (sd=.1) ; males get in .3 accidents per year (sd=1.1). Do males and female get in different numbers of accidents per year?) accidents 2. Compare a single observed mean to a hypothetical mean 3. Compare two sample means 3. mean or mean proportion of single population population Test statistic Test z=(Sample meannull value)/s.e. Where: se for proportions=sqrt(null value*(1null se value )/n) value se for means=sample standard se deviation/sqrt(n) deviation/sqrt(n) z=(sample mean 1sample mean 2)/ Sqrt[s12/n1 + s22/n2] (s1 is s.d. of sample 1, s2 s.d. of sample 2 (s n1 size of sample 1, n2 size of sample 2) Difference Difference of two means or proportions proportions Where do These Formulas Come From? From? Test statistics are zscores = Number of Test standard deviations from the null value standard Can use normal curve to find probability of Can selecting a sample with test statistic this large if null is true if Calculating Pvalues Calculating The zvalues can then be looked up in the table The displaying proportions under the normal curve to calculate pvalues to Note that all three of these test statistics Note assume that the sample size is large assume Calculating pvalues Calculating
Alternative Alternative hypothesis hypothesis Type of test Pvalue Proportion/mean is Onetailed Proportion/mean greater than null value value Proportion/mean is Onetailed Proportion/mean less than null value less Proportion/mean Twotailed Proportion/mean does not equal null value value Area above z Area below z 2*area above the 2*area absolute value of z Making A Decision Making The pvalue can then be compared to .05 (or The whatever level of significance you have chosen) chosen) If it is less than your level of significance, If reject the null reject Hypothesis Testing Example Hypothesis “Does Bill Clinton have the honesty and Does integrity you expect in a president” was asked in a May 1994 poll of 518 adults (Newsweek) in Do a majority of people doubt the integrity of Do the president? the 1. State your hypotheses. Do we want a one or twotailed test? 1. State your hypotheses. Null hypothesis: The proportion of the population Null who think that Clinton is honest equals or is greater than .5. than Alternative hypothesis: Fewer than .5 of the Alternative population think that Clinton is honest. population 2. Calculate Test Statistic 2. 45% said yes; sample proportion=.45, N 45% equals sample size The s.e. given that the null hypothesis is true is The sqrt (( .5 *.5)/518)=.022 sqrt So our standardized score=zscore=.45.5/.022=2.27 3. Use the test statistic to determine the pvalue. This is the probability of observing a zscore of 2.27 or less by chance. The pvalue score here equals .0116. here 4. Make a decision. 4. Our pvalue is less than .05, so we reject the Our null hypothesis. What did we just do? First, we assume that the null is true If we were to draw a large number of samples from the If population, if the null were true, we would get a distribution of samples that is normal with a mean of 50% distribution We drew a sample proportion of 45%. Given that the null is We true, how likely is it that we draw a sample with this proportion? That is, what is the probability of drawing a sample with a proportion of 45% or less? sample If this probability is small, we want to reject the null hypothesis, If because it is not likely given our data. 0 40 .05 Density .1 .15 .2 45 50 55 60 Hypothetical sample proportions Two types of errors Two
Type I error: We reject the null, but it is true Type (there is really no relationship) (there Type II error: The null is not rejected, but the Type null is false (there really is a relationship) null Type 1 error is seen as more serious Two Types of Errors (Con.) Two
Decision Decision Reject H0 Accept H0 Reject H0 Type I Error Type
(false positive) (false Right decision Truth Truth H1 Right decision Type II Error Right
(false negative) (false Two Types of Errors (Con.) Two If the null hypothesis is true, the probability of If making a type 1 Error is the level of significance (usually .05) significance If the null is true, it is impossible to make a If type 1 error type Type 1 error is serious, so we set the level of Type significance to .05 or some other low number significance Two Types of Errors (Con.) Two Type II error: the alternative hypothesis is true, Type but you fail to choose it (that is, fail to reject the null) the We do not know the probability of making a We type II error because it depends on how strong the relationship is the How are statistical tests reported in journals? journals? Higher smoking cessation rates were observed Higher in the active nicotine patch group at 8 weeks (46.7% vs 20%) (p<.001) and at 1 year (27.5% vs 14.2%) (p=.011). (Hurt et al., 1994, p. 595) vs Hypothesis Testing Notes Hypothesis When you test a research hypothesis, you are When really testing whether you can or cannot reject the null hypothesis. the However, it is common in some social science However, research NOT to see a formal statement of the null hypothesis. Statistical Significance Interpretation Interpretation Say we find support among the population for Bill Clinton is Say less than 50%; this does not mean that support for Clinton is “significantly less” than 50%. “significantly E.g. Does Bill Clinton have the honesty and integrity… sample proportion=.45, s.e.=.022 .45 +/ 2*.022 = (.406, .494) Between 40.6 and 49.4% think of Clinton as honest Statistical v. Substantive Significance Significance Statistical significance in itself does not reveal Statistical the size of the effect the The latter is the substantive significance or The real importance real E.g. If we detect a statistically significant E.g. difference in men’s and women’s annual salaries at MSU, but found that the difference was only $200, Sample Size & Hyp. Testing Sample
With a large enough sample size, almost any null can be rejected It is the size of the effect that matters If no relationship is found, it may be because of a small If sample size sample CI vs. Hypothesis Testing CI
Which statement is more informative: “95% confidence interval of the proportion thinking Clinton is 95% honest is .406 to .494” OR OR “The percentage of people who think of Clinton as honest is The less than 50%; this is statistically significant” less One v. twosided tests of significance One Is there a theoretical reason to conduct a onesided test? Note that the onesided test can be a more lax Note requirement requirement Onetailed versus Twotailed test Onetailed
16.socrates.berkeley.edu/~psy101/Lecture 12.ppt Research question  a question scholars answer with research in order to build the body of knowledge. knowledge. Hypothesis Testing Hypothesis Terms Typically twotailed
Does readability difficulty affect magazine readership? Hypothesis  a statement of a relationship Hypothesis between variables between Typically onetailed
As readability difficulty in magazines increases, magazine As readership decreases. readership Hypothesis versus RQ Hypothesis
Hypotheses are preferable when a body of Hypotheses research and/or theory exist about the relationship. relationship. Research questions are used when little is Research known about relationships. known Questions? ...
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 Fall '09
 TAMBORINI

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