![Hypothesis Testing with t: One-Sample
Case
• Similar formula for t test and z test
• Difference exist:
• Denominator in the t test is estimated standard
error of the mean (sX) [whereas]
• Denominator of the z test is the standard error of
the population (𝜎𝑋)
• T or z = observed sample mean – popul. mean
• estimated or known standard error
• Symbolically: t = X- 𝜇
• sX](https://image.slidesharecdn.com/thettestmeancomparison1-180327162020/85/The-t-test-mean-comparison-1-15-320.jpg)
The t test mean comparison 1
- 1. MEAN COMPARISON 1: THE t TEST THE t TEST Desmond Ayim-Aboagye, Ph.D.
- 2. THE t TEST • How to make inferences from sample data when the samples are small and the variability of the larger population is unknown.
- 3. William A Gosset (1876-1937) • Statistical test • 1. Enabling the brewery to detect differences between grains and hops • 2. Kegs of beer • Prior to this objective analysis of such problems was difficult • -- Is one strain of barley superior to another? • -- Is one batch of beer darker in color or richer in flavor than a separate batch? •
- 4. The t Test • Sample statistics to make inferences about population characteristics • A. Mean ˉX • B. Standard deviation s • T test examines differences existing between two means only
- 5. Three Variations of the t Test • 1. One variation of the t test is used for hypothesis testing about a sample mean when relevant population mean (𝜇) and standard deviation (𝜎) are unknown. • 2. It is specifically designed to detect significant differences between a control group and an experimental group in any classic two-group randomized experiment. • 3. The t test for dependent groups enables an investigator to demonstrate the presence of measurable change in the average attitudes or behavior of a group from one point in time (𝑡𝑖𝑚𝑒₁ ) to another time (𝑡𝑖𝑚𝑒₂)
- 6. U? Xs A One-Sample t Test Is X from a different population than 𝜇?
- 7. 𝜇1 𝜇2 X1 S1 X2 S2 Control G Experi mental G Is X1 different from X2? That is, is X1 from a different population than X2 after exposure to the indep. variable?
- 8. X1 S1 X2 S2 𝜇1 𝜇2 Treatme nt Is X1 different from X2? That is, following treatment, is X1 from a different population than X2?
- 9. T and Z Distributions: Any relationship? • 1. Use a Z test to detect mean differences when 𝜎 𝑖𝑠 known; otherwise, use one of the three t test variables. • 2. The Z distribution provides unreliable estimates of differences between samples when the number of available observations is less than 30.
- 10. The t Distribution • The T distributions are sampling distributions of means designed for use with small samples. Any t distribution has a mean of 0 and a standard deviation that decreases as the available degrees of freedom or number of observations increase. • T tests are used to compare one or two sample means– but not more than two. • Both the Z and T distributions test hypotheses involving either one or two sample means, but no more than two.
- 11. Assumptions Underlying the T test (Parametric test) • A. The populations the sample data are drawn from are normally distributed. • B. The data are either i. randomly sampled from a larger population or ii. Individually sampled from a larger population. In both cases, the sample data are used to generalize back to a population of origin. • C. Means can be calculated from the data, so that the dependent measures involved must be based on either interval or ratio scales. • D. When two independent samples are used to test a hypothesis, the samples are presumed to come from populations that have equal variances.
- 12. A Robust Statistical Test • A statistical test is described as robust when it provides reasonably valid portrayals of data (and relationships therein) when some of the test‘s assumptions are not met during its application.
- 13. Larger values of t, which point to significant mean differences • The difference between means is relatively large, and this difference serves as the numerator for calculating any t statistic • The standard deviation, which is used to estimate the standard error of the difference means, is relatively small. As the denominator for the t statistic, a smaller standard error will result in larger value of t. • As always, the larger sample sizes are desirable because they lead to smaller standard deviations, which in turn leads to a smaller standard error for the difference between the means.
- 14. Mean differences • A t test detects a significant difference between means when the difference is large, the sample standard deviation is small, and/or the sample size is large.
- 15. Hypothesis Testing with t: One-Sample Case • Similar formula for t test and z test • Difference exist: • Denominator in the t test is estimated standard error of the mean (sX) [whereas] • Denominator of the z test is the standard error of the population (𝜎𝑋) • T or z = observed sample mean – popul. mean • estimated or known standard error • Symbolically: t = X- 𝜇 • sX
- 16. One-Sample t test • The single or one-sample t test is used to compare the observed mean of one sample with a hypothesized value assumed to represent a population. One-sample t tests are usually employed by researchers who want to determine if some set of scores or observations deviate from some established pattern or standard.
- 17. Write Up the Result • “ A one-sample t test found that the training group of 20 students displayed a significantly higher recall for digits (M = 10.0, SD = 2.5) compared to the average recall, said to be around 7 digits, t (19) = 5.37, p < .05.“ • t (df) = t calculated, p < 𝛼. • No significant • t(df) = t calculated, p = p.
- 18. Confidence Intervals(One sample t test) • Computational formula X ±𝑡 𝑐𝑟𝑖𝑡 (sX) • Critical value of t at the .05, therefore 95% • Ie., training project (1- 𝛼 = 1- .05 = 95%) • Known sample mean 10, the two tailed critical value of t at .05 level 2.093, and the error of the mean .559 are all entered into the formula • 10 ±2.093 (.559) • Lower limit of confidence interval • 10 ±2.093 (.559) • 10 – 1.17 = 8.83 • Upper limit of confidence interval • 10 + 1.17 = 11.17 • Means representing mean digits appear interval ranging between 8.83 and 11.17. • Limitations: 1. unknown parent population, 2. small sample
- 19. Hypothesis Testing with Two independent Samples • The independent groups t test is ideal for hypothesis testing within experiments, as an experimental group can be compared to a control group.
- 20. Class Test on t test • 1. A Robust test is one that applies to many different types of data. TRUE or FALSE • 2. One of the assumptions of the t test is that means are based on interval or ratio scales of measurement. TRUE or FALSE • 3. The t tests are used to compare one or two sample means– but not more than two. TRUE or FALSE. • T test is parametric statistic. That is an inferential test that , prior to its use, assumes that certain specific characteristics are true of a population. TRUE or FALSE • 5. Briefly state or describe the essential characteristics of T distributions.