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Confidence Intervals / Tests of Significance & Power

Autor:   •  April 17, 2016  •  Essay  •  1,627 Words (7 Pages)  •  787 Views

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Confidence Intervals / Tests of Significance & Power

  • Confidence intervals
  • Purpose – to estimate an unknown parameter with an indication of how accurate the estimate is and of how confident we are that the result is correct
  • 68-95-99.7 rule says that the probability is about 0.95 that x bar will be within two standard deviations of the population mean score
  • Estimate +/- margin of error – form for confidence interval
  • Confidence interval for a parameter is an interval computed from sample data by a method that has probability C of producing an interval containing the true value of the parameter
  • Any normal distribution has probability about 0.95 within +/- 2 standard deviations of its mean
  • To construct a level C confidence interval:
  • Find the number z* such that any normal distribution has probability C within +/- z* standard deviations of its mean
  • If two tail, you find the z* using table A on the left side and on the right side, the area in the middle = C = our confidence interval
  • Probability C that x bar lies between (u – z*(std dev/sqr rt of n)) and (u + z*(std dev/sqr rt of n)):

=

  • M = (u +/- z*(std dev/sqr rt of n))
  • Choosing Sample Size
  • To obtain a desired margin of error m, plug in the value of std. dev. And the value of z* for your desired confidence level and solve for the sample size n
  • n =

  • Tests of Significance
  • Purpose: assess the evidence provided by the data in favor of some claim about the population parameters; compare observed data with a hypothesis whose truth we want to assess
  • Example – need to conclude whether or not the true means of the two random samples are different
  • Key steps:
  • Started with a question about the difference between two sample means and are trying to determine whether or not the data are compatible with no difference
  • Compare the means of the data and fine the difference between them
  • Turn the results into probabilities
  • If the probability is very small – we have observed something that is very unusual or the assumption (that there is no difference in the means) isn’t true
  • Null hypothesis – the statement being tested in a test of significance
  • Alternative hypothesis – the statement we hope/suspect to the true; is true when we reject the null
  • Test Statistic
  • Measures compatibility between the null hypothesis and the data
  • Purpose: use it for the probability calculation that we need for our test of significance
  • Z = (estimate – hypothesized value)/(standard deviation of estimate)
  • P-Values
  • Probability, assuming null hypothesis is ture, that the test statistic would take a value as extreme or more extreme than that actually observed
  • Smaller the P-value, the stronger the evidence that the null hypothesis is not true
  • To solve:
  • Find Z = (estimate – hypothesized value)/(standard deviation of estimate)
  • Check Table A for equivalent z score and find the percent/probability
  • Need to times the probability by two since it is on both sides of the curve and then you get the percentage of observing a difference as outside of the main area of the curve
  • Significance Level
  • If the p-value is as small or smaller than alpha, we say that the data are statistically significant at level alpha
  • Steps for Tests of Significance
  • State null hypothesis and determine alternative hypothesis
  • Calculate the value of the test statistic on which the test will be based – to measure how far the data are from the null
  • Find the p value for the observed data – this is the probability that the test statistic will weigh against the null
  • State a conclusion – conclude whether you can reject null or not
  • Tests for a Population Mean Summary
  • zTest for Pop Mean = z = (Xbar – mu0)/(std. dev./sq. rt. n)
  • Two-Sided Significance Tests / Confidence Intervals?
  • Two-sided tests rejects null hypothesis exactly when the value mu0 falls outside a level 1-mu confidence interval for the mu (mean)
  • P-values versus fixed alpha
  • P-value is the smallest level of alpha at which the data are significant
  • P-value gives us a better sense of how strong the evidence is
  • Power & Inference as a Decision
  • Power = probability that a fixed alpha significance test will reject the null when a particular alternative value of the parameter is true
  • Use when:
  • “is N subjects a large enough sample for this project?”
  • Calculate power in 3 steps:
  • State null and alternative hypotheses, as well as the significance level, alpha
  • Find values of Xbar that will lead us to reject the null
  • Use z test  z = (Xbar – mu0)/(std. dev./sq. rt. n)
  • Plug in the z score for z from Table A for whatever probability of test significance you choose
  • Solve for Xbar  as the unknown variable
  • Calculate probability of observing these values of Xbar when the alternative is true
  • Use same formula as in part 2, but solve for P instead of Xbar
  • High power is desirable
  • Ways to increase power:
  • Increase alpha
  • Increase sample size
  • Decrease standard deviation
  • Consider a new alternative hypothesis further away from mu0
  • Types of Error
  • Type I – reject null and accept alternative when null is true
  • Type II – accept null when alternative is true

Inference for Mean of a Population / Comparing Two Means

  • T distribution
  • Use when standard deviation is not known
  • When standard deviation is not known,         we estimate it with the sample standard deviation (s/sq. rt. n), which is equal to the standard error of the statistic
  • One-sample T statistic:
  • T =(Xbar – mu)/( s/sq rt n)
  • Degrees of freedom?
  • One-Sample T Confidence Interval
  • C = Xbar +/- t* (s/sq rt n)
  • Margin of error = t*(s/sq rt n)
  • One-Sample T Test
  • Same as with z test
  • t = (Xbar – mu0)/(s/sq. rt. n)
  • Calculate power of T test
  • We assume a fixed level of significance, often alpha = 0.05
  • Step 1 – decide on a standard deviation, significance level, whether the test is one-sided or two-sided and an alternative value of mu to detect
  • Write the event that the test rejects the null in terms of Xbar
  • Find probability of this even when population mean has this alternative value
  • Comparing Two Means
  • Use when you are comparing responses from two groups and the responses in each group are independent of those in the other group
  • Two Sample Z Statistic
  • Z =

  • Two Sample T Significance Test
  • T =

  • Two Sample T Confidence Interval
  • Questions:
  • When do you use t vs. s? Is it just when you don’t know the standard deviation?
  • How do you choose S in the power equation?

Comparing Two Proportions & Chi-Squared Tests

  • Two-way tables: organize data by two factors (e.g., group by age, then by education)
  • Marginal distributions – summarize each variable independently
  • Two-way table describes the relationship between variables
  • Chi-square test:
  • Purpose: to determine if the differences in sample proportions are likely to have occurred by just chance because of the random sampling
  • Null hypothesis in this test = no relationship between the variables
  • Compare actual counts with expected counts to test this
  • Expected count = (row total x column total) / n
  • Formula:

[pic 1]

  • Large values for x2  evidence against H0
  • When to use?
  • All individual expected counts are 1 or more (≥1)
  • No more than 20% of expected counts are less than 5 (< 5)

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