Hypothesis/chi square/student T

• Definition & Purpose of Hypothesis:

  • A hypothesis is a statement about one or more population set up to be discredited or approved
  • The purpose of hypothesis testing is to assist administrators, clinicians, and researchers in making decisions based on statistical analysis
• Test of Hypothesis:

  • Null Hypothesis:
  • Also known as the 'hypothesis of no difference'
  • The hypothesis to be tested
  • Set up to be discredited
  • Alternative Hypothesis:
  • Statement of the conclusion the researcher is trying to reach
• Test statistic:

  • Computed from the data sample
  • Serves as the decision maker for rejecting or not rejecting the null hypothesis
• Level of significance:

  • Probability of rejecting a true null hypothesis
  • Values like 0.01, 0.05, and 0.1 are commonly used
• Critical values or P-values:

  • Critical values separate rejection and non-rejection regions
  • P-value tells us how unusual our sample results are given the null hypothesis is true
• Decision rule:

  • Consists of rejecting or not rejecting the null hypothesis based on the test statistic falling in the rejection or non-rejection region
• Conclusion:

  • If the null hypothesis is rejected, we conclude that the alternative hypothesis is true
  • If the null hypothesis is not rejected, it does not mean it is true, just that it may be supported by the available data
• Types of errors:

  • Type I error:
  • Committed when a null hypothesis is rejected when it is actually true
  • Type II error:
  • Committed when a false null hypothesis is not rejected
• Methods:

  • Z-Test:
  • Used when the population variance is known and assumed to be normally distributed
  • Student t-test:
  • Used when the population variance is unknown, assumed to be normally distributed, and for small samples (n ≤ 30)
• A contingency table is used to show the classification of entities based on two criteria, with rows representing levels of one criterion and columns representing levels of the second criterion
• Chi-Square is a statistical technique used in the analysis of count or frequency data
• Uses of Chi-square include:

  • Test of dependence
  • Test of Homogeneity
  • Goodness of fit
• In a study by Stepanuk et al, researchers wanted to determine if preconception use of folic acid and Race are dependent
• Steps for testing the hypothesis include:

  • Null hypothesis
  • Alternative hypothesis
  • Significance level
  • Test statistic
  • Critical value
  • Decision rule and Decision
  • Conclusion
• To calculate the test statistic, expected frequencies are obtained by multiplying the total of the row by the total of the column and dividing by the grand total
• Decision rule: Reject the null hypothesis if the calculated test statistic is greater than the tabulated value, otherwise do not reject
• The conclusion from the study is that there is an association between the preconception use of folic acid and race, indicating that the two variables are dependent
• Researchers examined beliefs held by adolescents regarding smoking and weight, categorizing weight perception into underweight, overweight, or appropriate and smoking status into Yes or No
• The data from the telephone study of adolescents suggests a relationship between weight perception and smoking status in adolescents
• The t-test is a parametric method usually used for testing hypotheses about the mean of a small sample drawn from a normally distributed population when the population variance is unknown
• The student t-test was developed in 1908 by William Sealy Gosset, an English man who worked in a brewery
• The shape of the t-distribution depends on a value called 'degree of freedom', defined as the number of independent observations in the sample minus 1 (n-1)
• As the sample size (and thus the degree of freedom) increases, the t-distribution approaches the bell shape of the standard normal distribution
• Assumptions for t-tests:

  • The data are continuous
  • The sample data have been randomly sampled from a population
  • There is homogeneity of variance (i.e., the variability of the data in each group is similar)
  • The distribution is approximately normal (n ≤ 30)
• The one-sample t-test is used to determine whether an unknown population mean is different from a specific value
• In a study by Nakamura et al., they examined subjects with medial collateral ligament (MCL) and anterior cruciate ligament (ACL) tears
• In a study involving weight change after an exercise regime for 10 adults, the paired sample t-test was used to determine if the exercise regime resulted in a significant change in weight at the 5% level of significance
• The paired sample t-test compares the means of two measurements taken from the same individual or related units
• Other names for the paired sample t-test include dependent t-test, repeated measures t-test, and paired-t-test
• Conclusion for the weight change study: At a 5% significance level, the exercise regime resulted in a significant weight change among the adults
• Data is important in research because it is life and a necessity in building a strong research foundation
• Population: the largest collection of values of a random variable for which we have an interest at a particular time

  • Target population: the population from which a representative sample is desired
  • Sample: a representative part of a population chosen by probability or non-probability sampling designs
  • Confidence interval: displays the probability that a parameter will fall between a pair of values around the mean, measuring uncertainty or certainty in a sampling method
• Determining a sample is necessary because using an entire population is labor-intensive, time-consuming, and capital-intensive
• Sample size determination using Taro Yamane formula:

  n is the minimum sample size required

  N is the total population

  e is the sampling error

  Sample size calculation: N = 178, e = 0.05, yielding a sample size of 123. An additional 10% was added to account for non-response, making the final sample size 135
• Sample size determination:

  • Fischer’s method:

  n = minimum desired sample size

  Z = standard normal deviate (usually set at 1.96 for 95% confidence level)

  p = prevalence of behavior from a previous study (67%)

  q = Complimentary probability (1 - p)

  d = degree of accuracy desired (usually set at 0.05)
• A questionnaire is a research instrument comprising a series of questions set up to gather information from respondents
• Questionnaires can be carried out face to face, by telephone, computer, or by post
• Questionnaires are an effective means of measuring behavior, attitudes, preferences, opinions, and intentions of large numbers of subjects more cheaply and quickly than other methods
• Qualities of a good questionnaire:

  • The length should not be too long
  • The language used should be easy and simple
  • Questions should be arranged in an orderly way
  • Questions should be in an analytical form
  • Complex questions should be broken into filter questions
  • The questionnaire should be constructed for a specific period of time
  • Questions should revolve around the theme of the investigator
  • Answers should be short and simple
  • Answers should be appropriate to the problem
  • Answers should be clear to all respondents
• Questionnaire structure:

  • Questionnaires often use both open and closed questions to collect data
  • Closed-ended questions structure the answer by only allowing responses that fit into pre-decided groupings
  • Closed questions can provide nominal data or ordinal data
  • Advantages of closed-ended questions:
  • Economical
  • Easily converted into quantitative data
  • Standardized questions for reliability
  • Limitations of closed-ended questions:
  • Lack detail and may not reflect true feelings