In this part of the website, we review sampling distributions, especially properties of the mean and standard deviation of a sample, viewed as random variables. We look at hypothesis testing of these parameters, as well as the related topics of confidence intervals, effect size, and statistical power. For sufficiently large samples, it turns out that the mean of the sample is normally distributed (the Central Limit Theorem), and so the techniques described for the normal distribution can be used.
Topics
- Basic Concepts
- One-Sample Hypothesis Testing
- Standardized Effect Size
- Confidence Intervals
- Central Limit Theorem
- Hypothesis Testing using the Central Limit Theorem
- Comparing Means with Known Variances
- Simulation
- Sampling
- Sequential Randomness
- Statistical Power and Sample Size
- Power and Sample Size using Real Statistics
- Identifying Outliers and Missing Data
- Tolerance Intervals
Reference
Howell, D. C. (2010) Statistical methods for psychology, 7th Ed. Wadsworth. Cengage Learning
https://labs.la.utexas.edu/gilden/files/2016/05/Statistics-Text.pdf
Hello, I have sample of people who’s hearing ability is tested: 19 have under average, 64 have avrage and 17 people have good hearing ability. I need to make hypotheses testing if this distribution is normal given that average hearing ability is +/- 1 SD from average, and under average is everything less than that and good is everything above that. Please tell me what test to use? Thanks
Darko,
You have a sample of size 100. 64% of the sample is within one standard deviation of the mean and the two tails are roughly of equal size. This is consistent with the properties of a normal distribution, but you would need more detailed data to be able to test the likelihood that this data came from a normally distributed population.
One approach you can use is to generate a large number of samples of size 100 with the three values 64, 19 and 17 as you have described and determine how many of these samples pass some test for normality (e.g. Shapiro-Wilk). The percentage of those that pass would give you an idea of the probability that the sample came from a normally distributed population. THis is a Monte Carlo simulation approach.
Charles
Hello,
I have problem for which I need a suitable statistical test. I have data which is log normal distributed with a large upper tail containing low negative (up to -4) and large positive values (up to 50). I want to test the probability that the mean value of that data is different from a lognormal distribution with a mean of zero.
Which test would you recommend for that problem?
v/r Martin Theis
Hello Martin,
If I understand correctly, you are trying to see whether the sample data is coming from a lognormal population with zero mean. In this case, you can test whether the exponential of the data is coming from a normal distribution (e.g. by using the Shapiro-Wilks test). You should be able to test whether this population has zero mean by testing whether the exponential of the data is coming from a population with a mean of 1 (since exp(0) = 1).
Alternatively, you can use the KS or Anderson Darling test to see whether the exponential of the data is coming from a normal distribution with mean 1.
These tests are described on the Real Statistics website.
Charles
I need help with the following problem:
after conducting a survey to a specific group (n=75) it turned that many groups are over or under represenative. for example: 13 females, 60 males, and 2 other.
I am trying to find the relationship between gender and responses, how can I undersize or oversize my sample? any advice please.
Thank you
For many of the tests that you might use, it is acceptable to have different sized samples.
For tests where this is a problem, you have three main choices:
1. eliminate elements from the larger sample at random (the at random part is essential)
2. impute values for missing elements from the smaller sample (this is not such a great choice unless the smaller sample is almost the same size as the larger sample)
3. choose a different test which accepts unequal sample sizes
Charles