Statistics II: Normal Distribution and Sampling

• Statistics II

Normal Distribution

• The normal distribution shows percentages within standard deviations from the mean.
• 68.2% of data falls within one standard deviation (1σ) of the mean (μ).
• 95.4% of data falls within two standard deviations (2σ) of the mean.
• 99.7% of data falls within three standard deviations (3σ) of the mean.

Statistical Concept: Sampling

• Parameters in a population (μ, σ2, σ) are estimated using statistics (x̄, s2, s) from a sample.

Variation in Between Sample Means

• Determining how well sample values estimate underlying population values is a key issue
• Confidence Interval (CI) is a range of values likely to contain the population mean with some degree of certainty: CI = [X̄ ± T × SEM]
• SEM (Standard Error) = SD/√n, where SD is the standard deviation and n is the sample size.
• T-value comes from the t-distribution and depends on sample number -1 (degrees of freedom or df) and confidence level (95%), this will be provided in the exam
• A 95% Confidence Interval means that can be 95% confident that the population mean falls within this interval.

Confidence Intervals

• Confidence intervals are useful for means and differences in means (e.g., change in blood sugar levels after treatment).
• A 95% confidence interval, estimated from a sample, shows the possible values of population with 95% confidence.
• This is valid if the sample is representative of the population and the study is unbiased.
• Repeating an experiment many times and constructing many CIs, expect the true population value to lie in 95% of these intervals.
• A confidence interval is a standard way to show how accurately sample means reflect population means.
• Narrower confidence intervals indicate better estimation of population parameters by sample means.

Comparing Two Groups

• Focus on one value: the average change.

Hypothesis Testing

• Core question: can observed differences be simply due to chance?
• Hypothesis Testing addresses this core question
• To answer this question, determine the probability of getting such a difference, assuming no difference exists: p-value.
• A scientific/medical question addresses whether the difference is biologically important.
• Bioscience research size of scientifically important difference is often not known, but the confidence interval is important.
• The general goal of hypothesis testing is to understand if sample to sample variations are plausible from a research study.
• Hypothesis testing can help determine whether a specific treatment affects individuals in a population.

The Steps of Hypothesis Testing

• Assume there is no difference (null hypothesis, H0) or that both samples belong to the same population.
• Consider samples from different population(alternative hypothesis, H1)
• Determine the p-value, or the probability of seeing the observed difference
• Accept H0 and reject H1 if there is no difference
• Reject H0 and accept H1 if there is a difference.

Statistical Significance

• A statistical test is significant with evidence of a group difference.
• Standard definition of "significant": p-value is less than significance level or α (0.05 or 5%).
• Researchers must state the hypothesis precisely.
• The null hypothesis (H0): no difference between population means, groups are the same, treatment had no effect.
• The alternative hypothesis (H1): difference between population means, implying treatment had an effect.
• The hypothesis test measures the probability of observed results if H0 is true.
• Smaller p-value indicates greater evidence against H0.
• P <0.05 is typically considered a significant result, so there is sufficient evidence to reject H0 and accept H1.
• A confidence interval should be included, but it is often omitted.

Student's t-test

• Gossett t-statistic and t-distribution compare 2 group means
• Suitable for comparing 2 sample means.
• Calculate if no difference is obtained
• Normal distribution is typically required, but can tolerate skewness in larger sample size.
• Types of Student's t-test
  • Unpaired t-test: independent samples, two-tailed or one-tailed
  • Paired t-test: related samples, two-tailed or one-tailed

Unpaired t-tests

• Compare 2 independent populations, by determining if there is a difference in the means of two groups
• Compare the means of the two groups after measuring each subject once.
• Example: blood pressure on different diets.

Unpaired t test requirements

• Independent observations are each dataset has different set of individuals.
• Drawn at random from normally distributed populations.
• Assume equal variance (corrected by Welch's Test if unequal).
• When comparing two groups in a glucose experiment, each student should only be in one group, making it an unpaired test

Using a t-test

• Are the groups different: two-tailed?
• Unpaired: each student in one group only?
• Null Hypothesis (H0): no difference between means at 30 minutes: μ (diet coke) = μ (coke).
• Alternative Hypothesis (H1): difference between means at 30 minutes: μ (diet coke) ≠ μ (coke) .

Unpaired t-test example

• Start by taking a significant level of 0.05(5%)
• Calculate that p = 0.0044
• Reject the null hypothesis since p-value <0.05
• There is statistically significant difference at 5% between mean blood sugar level with diet coke and regular coke.
• A low probability makes an effect unlikely, repeat in other groups and conditions if the sample size is small

Paired t-tests requirements

• It will be known if there is a difference between the means under two conditions.
• Differences are measured twice and should be independent

Paired t-test example

• Each student measured at both time points used to comparing glucose levels at 0 and 120 in the diet coke group.
• Test if sugar levels between base and 30 min statistically differ, if drinking diet coke raises blood sugar levels > 30 minutes

Paired t-test statistical analysis

• H0: is mean difference is zero, μα= 0
• H1: mean difference is not zero, μα ≠ 0
• Mean difference = - 0.27 mmol/L
• SD (differences) = 0.63 mmol/L
• SEM = 0.26 mmol/L

Paired t-test Conclusion?

• The test is a result of p=0.34
• Cannot reject the null hypothesis since p>0.05
• Accept the null hypothesis
• The mean difference is not significantly different above 0 in glucose above 30 min from drinking diet coke.
• Diet coke affects blood glucose levels above 30 minutes

One-tailed vs Two-tailed tests

• Two-tailed are in either direction
• One-tailed are in only one direction.
• A two-tailed test is better due to correspond confidence interval which is two tailed and always does
• Only use One-tailed in the following - exceptional circumstances or strong direction

Interpreting p-values

• p-value does not equal the chance of being true
• P says obtaining/greater difference is that if difference does not exist
• If p= 0.02, there is 2% chance of obtaining such or greater difference if there were no difference (i.e. under H0)

Summary

• Confidence intervals show how an accurate estimate might be
• Setup the null and alternative
• Prefer two tail tests
• Tests are P<0.05, remember the P value may not be enough and look for supporting evidence
• Small studies can important differences
• A larger study needed research