Single sample z test

• Inferential statistics have two main areas: Estimating parameters and Hypothesis tests.
• Estimating parameters in inferential statistics involves taking a statistic from your sample data, such as the sample mean, and using it to say something about a population parameter, such as the population mean.
• Hypothesis tests in inferential statistics use sample data to answer research questions, for example, whether a new cancer drug is effective or if breakfast helps children perform better in schools.
• A research hypothesis is a formal statement or expectation about the outcome of a study, usually cast in terms of the dependent and independent variables, and needs to be concise and testable.
• In inferential testing, a hypothesis can be accepted or rejected based on the results of the test.
• Non-directional (two-tailed) hypothesis tests imply that participants will perform differently, meaning there would be an effect but you do not know what direction the effect would be.
• Numerous inferential tests are available, the selection of which depends on the shape of the distribution, the assumption of normality made, the study design (repeated measures or between subjects), and the type of data (interval/ratio; ordinal; nominal [frequencies]).
• Parametric = assumes data is normally distributed and the level of data measurement is often interval or ratio
• Non-parametric = does not assume data is normally distributed (it can be skewed) and the level of data measurement is often ordinal or nominal
• Non-parametric tests can be used even if the data distribution is normal if it is ordinal or nominal, but it is not easy to interpret the data.
• In single sample designs, data is collected from a single group of individuals, whom we suspect may be "different" in some way, from the population.
• In single sample designs, a measurement is taken from the sample (e.g., test score, reaction times, etc) and then compared to the population measurements.
• Single sample tests, either a z test or a t-test, are used to find out if the sample is/is not different (i.e., is/is not representative) of the population.
• If the test outcome is significant, it paves the way for further exploration as to why our sample might be different (i.e., not representative of) to the population.
• Single sample tests do not provide information about the causes of differences between the sample and the population.
• Does substance abuse during pregnancy lead to babies weighing differently to that of the  national average? Two tailed hypothesis
• Comparing a sample to a population is known as inter-population comparison.
• Hypothesis testing allows us to accept one of the following statements: the sample is representative of the population, in which case there is no significant difference in weights of babies born to mothers who have substance abuse problems, compared to those who do not, or the sample is not representative of the population, in which case there is a significant difference in weights of babies born to mothers who have substance abuse problems, compared to those who do not.
• A single sample z-test is used when we wish to compare a single sample against a population.
• If the z-test is significant, the sample is not representative of the population.
• If the z-test is not significant, the sample is representative of the population.
• Data must be normally distributed for a z-test to be valid.
• The population mean, population s.d., and sample N are known for a z-test to be valid.
• A one-tailed hypothesis predicts a specific direction of outcome, such as babies born to substance abuse mothers will weigh less than the national average.
• A two-tailed hypothesis does not predict a specific direction of outcome, simply stating that there will be a difference, such as weights of babies born to substance abuse mothers will not be the same as the national average.
• The choice between a one-tailed or a two-tailed hypothesis is important as it affects part of the Z-test procedure.
• A one-tailed hypothesis is usually used in situations where there is a compelling reason for doing so, such as when there is considerable previous evidence supporting a 1-tailed hypothesis or a one-tailed hypothesis is strongly supported by theory.
• For example, when testing a new drug supposed to improve patient health, a one-tailed hypothesis is more appropriate, as not merely looking for a difference, but the difference must be in the right direction.
• In most cases, a two-tailed (non-directional) hypothesis is sufficient.
• Null hypothesis (for info only not needed unless stated otherwise): the sample mean is equal to the population mean. Hence, the sample is representative of the population.
• Experimental (2-tailed) hypothesis: the sample mean is not equal to the population mean. Hence, the sample is not representative of the population.
• Alpha level (or the significance level) of a test reflects how stringent the testing criteria are.
• There are two commonly used alpha levels: 0.05 (for 95% level of significance) and 0.01 (for 99% level of significance) - the latter is more stringent.
• Unless specified otherwise, the default alpha level is ALWAYS 0.05.
• The selection of critical regions will be governed by whether a 1-tailed or a 2-tailed test is being performed.
• If alpha is 0.05 and two-tailed, that means the rejection region in each tail of the SND will be 0.025 (i.e half of 0.05).
• Critical regions for an alpha of 0.05 and a two tailed test are - / + 1.96.
• Now that you have your Z-obt (-6.56) and your Z-crit values of -1.96 to +1.96 it is time to make a decision about your hypothesis.
• Our Z-obt value of -6.56 falls OUTSIDE of the accepted range of -1.96 to +1.96, we ACCEPT the experimental hypothesis.
• The sample mean is not equal to the population mean, hence, the sample is not representative of the population.