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Hypothesis testing March 20, 2000 Two-tailed tests • A two-tailed test means that an equal part of the standard error is in each “tail” of the distribution. Confidence interval • Using z scores, we can calculate the actual values represented by the boundary of the area in the tails. • For example, if we want a 95% confidence limit, we find the values for which 2.5% of scores (half of 5%) fall below and 2.5% fall above. confidence interval • Homework review Hypothesis testing with sample data Do two variables have a nonrandom relationship? • Four basic steps – State the null and research hypothesis – Select a significance level (alpha ()) – Select and compute a test statistic – Make a decision by comparing to critical value of the test statistic. Example - t test • Meier/Brudney, 11.12 • Change in toll booths from human collection to machine collection. • Human attendants (population) • = 1,253 cars per hour • New system; sample size of 100 hours • = 1,261 cars per hour st dev = 59 • State a hypothesis, null hypothesis, and evaluate them. What is your conclusion? • You are comparing the sample mean to the population mean. Example - t test • State the research and null hypothesis. • H1: The average number of cars passing through automated booths is higher than the average number of cars passing through booths with human attendants. • H0:There is no difference in the average number of cars passing through booths with automated collections and those with human attendant collection. • Note: this is a directional hypothesis. It states that one is higher or lower than the other. Use a one-tailed test. Example - t test • Compute the statistic X t s n 1261 1253 8 8 t 305 . . 59 100 59 10 59 Example - t test • Find the critical t • Because this problem has a directional hypothesis, we will use a one-tailed test. – The degrees of freedom are equal to the sample size minus 1. – Using the t table, find the critical t for = .05 and df = 99. • The critical t = 1.66 Example - t test • So, the calculated t = 3.05 • the critical t (t.05) = 1.66 • Because the calculated t score is greater than the critical t, we reject the null hypothesis. • There is a statistically significant difference in the mean number of cars passing through the automated tool booths and the mean number of cars passing through booths with human attendants. T-test • Practice problems One sample z test • We use the t-test when we want to compare the means of a sample to a population or the means of two samples if we know the mean and standard deviation for the samples, but only the mean of the population. • If, however, we also know the standard deviation of the population, we use a one-sample z test. One sample z test • Don’t confuse with z scores. – One-sample z test is an inferential statistic. – Z score is a descriptive statistic. Example - one sample z test • Over the course of many years, a Midwestern college has a mean law school acceptance rate of 50% per year. The standard deviation = 3.5 • The pre-law advisor developed a program intended to improve the rate of acceptance in 1999. • In 1999, 130 graduates applied to law school; 73 were accepted (56%). • Did law school acceptances improve following the program? Example - one sample z test • State the research and null hypotheses. • H1: Law school admission rates were higher for those participating in a pre-law advising program. • H0:There is no difference in law school admission rates between those who participate in an advising program and those who do not. • Note: This is a one-tailed hypothesis. One mean is higher than the other. Example - one sample z test • Select an . • Remember, in social sciences the general standard is = .05 (remember, this equates to what percent of the time you are likely to be wrong). • Choose a test statistic - since we are comparing the mean of a group (sample) to a known population mean and standard deviation, the one sample z test is an appropriate statistic. Example - one sample z test • Compute the statistic. Note: the denominator is the formula for a standard error. sample mean - pop mean pop st dev number of cases in sample z X n Example - one sample z test • Calculate z 56 50 6 6 z 19.35 . 114 . 0.31 35 . 130 35 Critical value of z (two-tailed test for non-directional Ho) Probability (level of significance) Critical value of z 0.05 1.96 0.01 2.58 0.001 3.29 Critical value of z (one-tailed test for directional Ho) Probability (level of significance) Critical value of z 0.05 1.65 0.01 2.33 0.001 3.09 Example - one sample z test • Calculated z = 19.35 • Critical z = 1.65 – (from z table, = .05) (z.05 = 1.65) • Since the calculated z is > z.05 , we reject the null hypothesis. The law school admission rate is statistically significantly higher following the advising program. one sample z test • Practice problems