# Hypothesis testing & Inferential Statistics

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Hypothesis testing & Inferential Statistics

Hypothesis Testing The process of determining whether a hypothesis is supported by the results of the study. Use inferential statistics to draw conclusions (make inferences) about the population based on data collected from a sample. Ex: cholesterol levels while on “an all fruit diet” vs. no diet at all. Hypothesis: individuals in the all fruit diet will have lower levels of cholesterol than general population.

Null and Alternate Hypothesis
Goal of science is to reject untrue information, thereby the information that is left over is assumed to be true. Statistically impossible to show that something is true. Statistically possible to show that something is false. Counter intuitive rationale we must reject a hypothesis as false in order to find support for the hypothesis we are seeking.

Null Hypothesis Ex: we want to show that an all fruit diet lowers cholesterol compared to no diet at all. Null hypothesis: no difference between the groups being compared. H0 : μ 0 = μ 1 H0 : μ of all fruit = μ of general population Goal of research study is to determine whether a null hypothesis is true or false. If the null hypothesis is rejected, then a possible true result remains.

Alternate Hypothesis Ex: we want to show that an all fruit diet lowers cholesterol compared to no diet at all. Alternate (or research) hypothesis: there is a significant difference between the groups being tested. Ha : μ 0 < μ 1 Ha : μ of all fruit < μ of general population Goal of research study is to support the alternate hypothesis by rejecting the null hypothesis.

Directional hypothesis or One-tailed
Directional hypothesis (one-tailed hypothesis) The experimenter predicts the direction of the expected results. Alternate directional hypothesis Ha : μ 0 < μ 1 Ha : μ of all fruit < μ of general population Null directional hypothesis= H0 : μ 0 ≥μ 1 H0 : μ of all fruit ≥ μ of general population

Nondirectional hypothesis or Two-tailed
Nondirectional hypothesis (two-tailed hypothesis) The experimenter predicts differences between the groups, but is unsure what the differences will be. Alternate directional hypothesis: Ha : μ 0 ≠ μ 1 Ha : μ of all fruit ≠ μ of general population Null directional hypothesis= H0 : μ 0 = μ 1 H0 : μ of all fruit = μ of general population

Type I Error Sometimes we can make mistakes in our research.
At times, we reject the null hypothesis when it should NOT have been rejected. Ex: we say the all fruit diet lowers cholesterol when it actually does not. Also known as a false positive: we say there was a difference between groups, when in reality there was no difference. We found a difference, but it was due to chance and the results are a fluke.

Type II Error In some cases, the null hypothesis SHOULD be rejected, but we fail to find a difference between the groups. The null hypothesis is false, but accept it anyway. Ex: we say the 2 groups have equal levels of cholesterol when in fact the all-fruit group has lower levels. Somehow, we failed to find a difference between the 2 groups.

Type I vs. Type II errors Type I error: Null hypothesis: not pregnant
Alternative hypothesis: pregnant If we reject the null (not pregnant) but we are incorrect, she is going to think she is pregnant when she is really not. Leads to actions: Freaking out Happiness – tell friends Buy baby clothes

Type I vs. Type II errors Type II error: Null hypothesis: not pregnant
Alternative hypothesis: pregnant if we fail to reject the null, we keep the null and she will think she is not pregnant when she really is, Leads to no actions: no prenatal care May go drinking and hurt the fetus.

Type I vs. Type II Which is worse for research?
Type I: saying a result is true when it is not true is more detrimental to research. But in other cases, maybe Type II can be worse. In order to avoid both errors, researchers try to replicate the results.

Statistical Significance
Ex: we find that the all-fruit diet group has lower levels of cholesterol than the rest of the population. Significantly lower! Statistical significance at .05 level (p ≤.05) We will get these results by chance only 5 times or less out of 100. 95% of the time, these results are due to our manipulation We can reject the null hypothesis because the pattern of the data are unlikely to have occurred by chance. probability of making a Type I error: 5 times or less out of 100 The field has established the alpha level at .05

Single-group design Research involving only one group and no control group. Simplest kind of hypothesis testing We compare the results of the group (sample) with the performance of the general population. Use parametric tests, a type of inferential statistics, that requires certain parameters about the population (i.e., mean, standard deviation) t test

Single sample t-test Parametric inferential statistical test of the null hypothesis. Used for a single sample when we know the population mean, but not the population standard deviation. T-tests compare differences between the mean of a sample and the mean of a population. However, we need to compare the sample mean with a distribution of sample means

Sampling Distribution
Distribution of sample means based on random samples, of fixed sizes, from a population. Ex: IQ scores of 1000 people (i.e., population) Take 100 samples of 10 people, plot the mean IQ of each sample. The mean of the population IQ will equal the mean of the distribution of means.

Sampling Distribution
Distribution of scores (pop.) and distribution of means have the same mean. The standard deviation (SD) of distribution of scores is bigger than the SD distribution of means.

Sampling Distribution
The standard deviation of a distribution of sample means has a new name. Standard error of the mean: the standard deviation of a distribution of means. sM = s √N Where s = ∑( X - X ) ² _________ N - 1

T-test To calculate a t-test, we need to know the sample mean, the population mean, and the standard error of the mean. t = (X - µM) sM s = 2.97 Pop. mean = 11 t = 1.05 Our sample mean falls standard deviations above the pop. mean. Is this difference large enough to be statistically significant? Standard error of the mean = 1.33

T-distributions We compare the t-value to a standardized distribution of t-values (i.e., t distributions). As the sample size increases, the t-distributions approaches a normal distribution. In t-distributions, we base sample size in terms of degrees of freedom or df

Degrees of freedom Degrees of freedom Ex: 2, 5, 6, 9, 11, 15 mean = 8
Number of scores in a sample that are free to vary. Ex: 2, 5, 6, 9, 11, mean = 8 In order to maintain the mean of 8, five scores are free to vary, except for the last one. df = N - 1

T-distributions

One-tailed t-Test Ha : μ of all fruit < μ of general population
We compare our t-value to a distribution of t-values. If our t-value is larger than the cut-off, we reject the null hypothesis. Directional hypothesis Ha : μ of all fruit < μ of general population H0 : μ of all fruit ≥ μ of general population

Two-tailed t-test If t-value falls in the region of rejection, we reject the null hypothesis. Nondirectional hypothesis Ha : μ of all fruit ≠ μ of general population H0 : μ of all fruit = μ of general population

T-table (appendix A.3) Which test is more conservative, one-tailed or two-tailed? Two-tailed test because it is more difficult to beat the critical value to reject the null hypothesis.

Statistical Power Probability that we can reject the null hypothesis and find significant differences when differences truly exist. To increase power Use one-tailed tests Increase the sample size: as sample size increases, the critical value decreases (i.e., easier to reject the null hypothesis when the null is false)

Steps to hypothesis testing
The t test The six steps of hypothesis testing 1. Identify mean of sample and mean of population of sample means. 2. State the hypotheses (null and alternate) 3. Characteristics of the comparison distribution Find standard error of the mean 4. Critical values 5. Calculate t-value 6. Decide to reject or sustain the null hypothesis.