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Hypothesis Testing. An understanding of the method of hypothesis testing is essential for understanding how both the natural and social sciences advance.
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Hypothesis Testing • An understanding of the method of hypothesis testing is essential for understanding how both the natural and social sciences advance. • In the science one begins with a theory, then collects data (hopefully under carefully controlled conditions) and asks the central question Does the data fit the theory?
Does the data fit the Theory/Model? • This question is not as easily answered as you might think. • As we know samples vary, measurements almost always contain small errors, so it is unreasonable to expect exact agreement with a theory/model based upon actual observations. • When can we say that the sample we have carefully collected does or does not fit the theory/model?
When can we say that the sample we have carefully collected does not fit the model? • There is no one answer, • rather we calculate the probability that a random sample would vary from that predicted by the theory/model by as much or more than the value we obtained. (This value is called the p-value.) • For example, • if we believe that the mean height of students at CSUMB is 5’6” we can collect a sample and calculate how likely it is that the average height of a sample of this size would vary from 5’6” by as much or more than that of our sample. • The average height we calculate from the sample is called the test statistic.
Ho determines the model. Small p-values indicate the sample data does not fit the model
What Model do we use? • We have seen that the average of almost all large samples (of size n) is modeled by a normal distribution with mean equal to the population mean and standard deviation • So if we know the population standard deviation we have the two parameters needed for our model and we can ask if the data fits the model. • If we do not know the population standard deviation we can use the sample standard deviation as long as the sample is large. (Generally this means >25.)
Hypothesis Testing about the mean • The model is defined by the parameters mean µ and standard deviation σ. • Since we can use the sample standard deviation in place of σ, we really only have one assumption: That we know the mean of the population. • This assumption µ = µ0 (a known value) is called the null hypothesis and is designated Ho
Hypothesis Testing about the mean • The model is defined by the null hypothesis Ho :µ = µ0 (in our example of student heights µ0 =5’ 6") • If the null hypothesis is not true then one of the following alternative hypotheses (HA) must be trueµ < µ0 , or µ > µ0 .If we have no idea which of these to expect we can state the alternative hypothesis as HA:µ ≠ µ0 although this is rarely used and I discourage you from ever using it in practice.
Inference: Null Hypothesis • The null hypothesis (H0) is the hypothesis/theory that is being tested. • H0 can never be proved, only disproved! This is how the sciences advance, by disproving a theory with data and suggesting an alternative theory that seems to agree with the data. • It is always a statement of the value of a population parameter. e.g., H0: = 0 signifies the population mean has the value 0. • H0 is presumed TRUE until there is sufficient evidence to reject it.
The Big Idea • The null hypothesis provides us with a model for the population from which the sample is selected. • A sample is collected, and the sample average (test statistic) compared to the population parameter. • In other words, we place the average of the sample on the model and ask how reasonable is it that we obtain a test statistic that varies from 0 by this much or more. • Generally values that are within 2 standard deviations of the (assumed) mean are considered reasonable.
Example • Suppose our model is the standard normal curve (mean = 0 and standard deviation = 1) • We take a sample of size 1 and obtain a sample average of 2.22 • We place the sample data on the model and, ask how unlikely is it we obtain a value this extreme if the model is correct.
Quantifying the improbable: p-value • p-value: The probability of observing, when the null hypothesis is true, a value of the test statistic that is as extreme or more extreme than the value observed. (memorize this!) • In the preceding example the value 0.139 is the p-value of the test.
Inference: Statistical significance • Traditionally, the decision to reject H0 was based upon selection of a level of significance () used to derive a critical value for the test statistic. The critical value set a gating value beyond/beneath which a test statistic must fall in order that H0 may be rejected. • Most technology tools produce p-values directly. The p-values carry more information about the test statistic, since they enable reporting the smallest possible significance level for which the results are statistically significant.
Inference: Conducting a Hypothesis Test (4 steps) • Identify the parameter to be tested and state the two hypotheses in symbolic terms. • Restate the hypotheses in terms of the variable being considered. • Analyze the sample data and decide if it contradicts the null hypothesis (i.e., can H0 be rejected?) • Based upon the outcome of the analysis, state the conclusion in terms of the variable being considered.
Example • Standards set by government agencies indicate that Americans should not exceed an average daily sodium intake of 3300 milligrams (mg). To find out whether Americans are exceeding this limit, a sample of 100 Americans is selected. The mean and standard deviation of daily sodium intake are found to be 3400 mg and 1100 mg respectively
Inference: Conducting a Hypothesis Test (State H0, HA) • H0: = 3300 mg • Americans’ average daily sodium intake is 3300 mg. • HA: > 3300 mg • Americans’ average daily sodium intake exceeds 3300 mg.
The Test Statistic • The test statistic is the value produced from the sample. We place this value on our model(a normal distribution with mean 3300 and standard deviation The p-value is the probability of getting a value of 3400 or larger on this normal curve. Using the normal worksheet you can check this value is about .1814
Conclusion • The p-values represents the chance of getting a value as high or higher than 3400 assuming the true average is 3300. • Thus, the p-value of .1814 means there is an 18.14% chance that whenever we conduct a similar experiment we would find a sample average of 3400 or higher. • We conclude that there is not enough evidence to show that Americans’ average daily sodium intake exceeds 3300 mg.
Inference: Guidelines & language of statistical significance
Confidence Levels • If p-value is less than 0.1 reject Ho at 90% confidence level, otherwise keep Ho • If p-value is less than 0.05 reject Ho at both 90% and 95% level, otherwise keep Ho • If p-value is less than 0.01 reject Ho at 90%, 95%, and 99% levels, otherwise keep Ho.
