hypothesis testing (1 sample)

• Hypothesis Testing: One Sample Case involves comparing a sample statistic to a population parameter to detect significant differences.
• The logic of hypothesis testing involves the Five-Step Model.
• Hypothesis testing for single sample means includes the z test and t test.
• Testing sample proportions is a part of hypothesis testing.
• One-vs-Two-tailed tests are a type of hypothesis testing.
• Two-tailed tests are used to determine if there is a significant difference between two groups.
• Type I error, also known as alpha error, is the probability of rejecting a true null hypothesis.
• Type II error, also known as beta error, is the probability of failing to reject a false null hypothesis.
• One-tailed tests are used when the researcher wants to determine if the sample mean is greater than or less than a specific value.
• Hypothesis testing is designed to detect significant differences, which are differences that did not occur by random chance.
• In the “one sample” case, we compare a random sample (from a large group) to a population.
• In hypothesis testing, we compare a sample statistic to a population parameter to see if there is a significant difference.
• The education department at a university has been accused of “grade inflation” so education majors have much higher GPAs than students in general.
• The average GPA for all students is 2.70, which is a parameter.
• To the right is the statistical information for a random sample of education majors: μ = 2.70, X = 3.00, s = 0.70, n = 117.
• If the H 0 were true, a sample outcome of 458 would be unlikely, therefore, the H 0 is false and must be rejected.
• The curve of the t distribution is flatter than that of the Z distribution but as the sample size increases, the t-curve starts to resemble the Z-curve.
• Is the obtained t score significantly different from the population average (μ = 440)?
• The formula for one-sample t-test is identical to z-test, but uses a different distribution.
• A random sample of 26 economics graduates scored 458 on the GRE advanced economics test with a standard deviation of 20.
• The curve of the t distribution varies with sample size, and in using the t-table, degrees of freedom are based on the sample size.
• When the sample size is small, the Student’s t distribution should be used.
• If the test statistic is not in the Critical Region (at α=.05, is between +1.96 and - 1.96), then fail to reject the H 0.
• The obtained t score fell in the Critical Region, so we reject the H 0 (t (obtained) > t (critical).
• For a one-sample t-test, df = n - 1.
• The test statistic is known as “t”.
• In hypothesis testing, the Null Hypothesis (H 0) always states there is “no significant difference” and always contradicts the Alternative hypothesis (H 1), which always states there is a significant difference.
• In hypothesis testing, the probability of getting the sample mean (3.00) if the H 0 is true and all education majors really have a mean of 2.70 is calculated.
• The method for testing proportions is the same as the one sample Z-test for means.
• Use the formula for proportions and 5-step method to solve.
• In a recent provincial election, 55% of voters rejected lotteries.
• In a one-tailed test, the researcher predicts the direction (i.e. greater or less than) of the difference.
• A two-tailed test splits the critical region equally on both sides of the curve.
• The formula for proportions is: P s is the sample proportion, P u is the population proportion, and Z = (P s - P u) / (n - 1).
• If the data are in % format, convert to a proportion first.
• If the sample is large, use the Z distribution.
• The alpha (α) level is usually set at .05.
• All of the critical region is placed on the side of the curve in the direction of the prediction.
• When your variable is at the nominal (or ordinal) level, the one sample z-test for proportions should be used.
• The first step in hypothesis testing is to determine if a random sample was obtained.