Application of Statistics to Research Design

Hypothesis Testing

For every research project, there are two hypothesis or predicted outcomes
- null hypothesis - inherently assumes that there are no statistically significant differences between the sample mean and the population mean
  - statistically, this means that the group of subjects recruited for the project (sample) and the infinite number of people to whom the results are generalized (population) performed similarly
- alternative hypothesis - opposing view of the null hypothesis and assumes that there are statistically significant differences between the sample and population
  - statistically, findings from the alternative hypothesis indicate that the group of subjects recruited for the project (sample) performed differently than the infinite number of people to whom the results are generalized (the population)

Hypotheses are written as One or Two-Tailed

one-tailed hypothesis - the null hypothesis includes both
- the no difference prediction
- the prediction that the data from the project will yield results in the opposite direction
  - example of a one tailed null and alternative
    - null: the use of a directional microphone in an open-ear hearing aid DOES NOT improve the reception threshold for sentences (RTS) relative to the measured RTS for an omnidirectional microphone in an open-ear hearing aid, yielding an INCREASE in hearing aid returns
    - alternative: the use of a directional microphone in an open-ear hearing aid DOES improve the RTS relative to the measured RTS for an omnidirectional microphone in an open-ear hearing aid, yielding a DECREASE in hearing and returns
- two-tailed hypothesis - does not specify direction, it modifies the one tailed hypothesis to
  - null - the use of a directional microphone in an open-ear hearing aid DOES NOT improve the RTS relative to the measured RTS for an omnidirectional microphone in an open-ear hearing aid
  - alternative: the use of a directional microphone in an open-ear aid DOES improve the RTS relative to the measured RTS for an omnidirectional microphone in an open-ear hearing aid

Hypothesis Testing

In statistics, the goal is to reject the null hypothesis in favor of the alternative hypothesis
A statistical analysis of the data is required to determine whether the investigator has met this objective.
If the analysis indicates that the significance level or alpha level has not been met (p > or = 0.05) then the null hypothesis is accepted and the alternative hypothesis is rejected
- suggesting that both the sample and population performed similarly
if the analysis indicates that the level of significance has been met (p < 0.05) then the alternative hypothesis is accepted and the null hypothesis is rejected, indicating that the sample performed differently than the population
hypothesis testing is based on either accepting or rejecting the null hypothesis
- the decision of the investigator is either correct or incorrect, leading to four possible outcomes
  - two of these outcomes are correct
  - two involve error

Z-test

written as the sample mean minus the population mean divided by the standard deviation of the the mean
standard deviation is calculated from alpha/sqrtn where alpha is the population standard deviation and n is the sample size
the resulting data from the z-test provides the investigator with a z-score
- a z-score is the number of standard deviation units a value deviates from the mean
the resultant z score lets the investigator know whether the sample group's performance rejects (or accepts) the null hypothesis
- the investigator compares the calculated z score to a latter number which stems from converting the lower bound of the 95% significance values (two standard deviations) to a z-score
if the z-score is more than the value of the two standard deviation calculated z score then the performance of the sample group is not significantly different (p > 0.05) from the performance of the population
therefore, the investigator accepts the null hypothesis and rejects the alternative hypothesis
to calculate the statistical power of this example, the investigator looks up the probability (Beta) value of the z-score from a z table
the probability (B) value under the normal curve is ...

Calculations

T-Test Calculator - http://www.socscistatistics.com/tests/tsinglesample/Default.aspx

Type I error:
the error of rejecting H
0
when it is true
(Alpha):
the probability of a Type I error

Type II error:
the error of not rejecting H0 when it is false
(Beta): the probability of a Type II error