POLS 7000X STATISTICS IN POLITICAL SCIENCE CLASS 7 BROOKLYN COLLEGE-CUNY SHANG E. HA Leon-Guerrero and Frankfort-Nachmias, Essentials of Statistics for a Diverse Society

Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Chapter 7: Testing Hypotheses (Class 7 & 8)  Overview  Assumptions of Statistical Hypothesis Testing  Stating the Research and Null Hypotheses  Determining What is Sufficiently Improbable: Probability Values and Alpha  Five Steps in Hypothesis Testing: A Summary  Errors in Hypothesis Testing  Testing Hypotheses About Two Samples  The Sampling Distribution of the Difference Between Means  The t Statistic

Hypothesis Test vs. Confidence Interval  Confidence Interval: to estimate the population mean  Hypothesis Test: to evaluate whether a claim about the population mean is valid

An Example: One Sample t-Test  An insurance company is reviewing its current policy. When originally setting the premium rates they believed that the average claim amount was $1,800. If the true mean is actually higher than this, they could potentially lose a lot of money. On the other hand, if the true mean is actually lower than this, they could lower the premium rates for greater competitiveness in the market. They randomly select 100 claims, and calculate a sample mean of $1,910 with a standard deviation of $500

For the above example…  We may construct a confidence interval to estimate an average claim amount.  95% CI: 1,910 ± 1.96*(500/√100) = [1,812, 2,008]  We may conduct a hypothesis test to see if any average claim amount is different from $1,800.

Null hypothesis vs. Alternative/Research hypothesis  Null hypothesis; Ho: the default belief about the population mean (always contains [=]!!!)  μ = $1,800  Alternative/Research hypothesis; Ha (or Hresearch): the statement of a hypothesis that is opposite to the null hypothesis  μ ≠ $1,800 (two-tailed test)  μ > $1,800 (one-tailed test)  μ < $1,800 (one-tailed test)

Hypothesis Test (Two-tailed test)  Under the presumption that Ho: μ =$1,800 is true, the center of the sampling distribution of X is the value 1,800  The sample mean ($1,910) may be within the 95% confidence interval or may be outside of it  It the sample mean is outside of the 95% confidence interval, we are able to reject the null hypothesis

Hypothesis Test (two-tailed test)  Use t-score  |t| = |(1,910-1,800)/{(500/√100)}|= 2.2  As |t| increases, Ho is more likely to be rejected

Decision Rules I  We reject the null hypothesis if t-score is greater (in absolute value) than the critical value  In our case, the critical value at 95% confidence level with 99 degrees of freedom is  The critical value is determined by the level of confidence ( α )and degrees of freedom (n-1)

Decision Rules II  p-value is the probability that the test statistics at least as large in absolute value as the observed value if Ho is true  We reject the null hypothesis if p-value is less than α (significance level)

Things to Remember  Conclusions using (two-tailed) hypothesis tests are consistent with conclusions using confidence intervals  We say “Do not reject Ho" rather than “Accept Ho"  It is better to report the p-value than to indicate merely whether the result is “statistically significant" or not

Hypothesis Test (one-tailed test)  A different alternative hypothesis is sometimes used when a research predicts a deviation from Ho in a particular direction  By law, an industrial plant can discharge no more than 500 gallons of waste water per hour, on average, into a neighborhood lake. An environmental action group believes this limit is being exceeded and tests if the industrial plant indeed abides by the law. A random sample of 90 hours over a period of a month reports a sample mean of 525 gallons with a standard deviation of 110 gallon

One-tailed Test  In this example, we are only interested in if the industrial plant discharge more than 500 gallons of waste water per hour. Discharging less than 500 gallons per hour is basically the same result as discharging exactly 500 gallons per hour  Null hypothesis (Ho): μ = 500  Alternative hypothesis (Ha): μ > 500

One-tailed test  The test statistics is again the t-score:  t = ( )/(110/√90) = 2.16  The critical value at 95% confidence level with 89 degrees of freedom is 1.66  p-value = 0.017

Notes  For the same α, a one-tail test is easier to reject Ho than a two-tail test  In practice, two-tailed tests are more common than one-tailed tests  When deciding whether to use a one-tailed or a two-tailed test in a particular exercise, consider the research question posed  Whether μ has changed vs. whether μ has increased  Is there a theoretical reason to predict that the change should be in a particular direction?  You should not choose a one-tailed test simply because the sample mean is greater or less than the test value.

Summary  What is your research question? Are the sample data obtained using randomization? How large is n?  Based on your research question, explicitly state the null hypothesis and alternative hypothesis. Is a two-tailed test more appropriate than a one-tailed test?  Decide the α -level (the probability of error you are willing the take)  Calculate the test statistics (t-score)  Compare the test statistics to the critical value for the given α (or compare p-value to α )  Conclude and report your result