t-test12345678998745612337744231254478888888.pptx

statistics

1. Statistics

2. Introduction to t - Statistics The t-statistic, commonly referred to in the context of the Student's t-test, is a fundamental concept in statistics used to determine if there is a significant difference between the means of two groups or to assess the relationship between variables. It plays a critical role in hypothesis testing when the sample size is small, and the population standard deviation is unknown A z -test can be used if we know both the population mean [ μ ] and the population standard deviation [ σ ] – but it is rarely the case that we know the population standard deviation …

3. Introduction to t - Statistics When we do NOT know the population standard deviation, a t -test can be used A t -test uses the sample standard deviation [ s ] or the sample variance [ ] so that we can use a single sample to make our prediction To calculate the sample standard deviation [ s ], we use the equation: s = = SS = Sum of squares n = sample size n – 1 = degrees of freedom  

4. Hypothesis testing Null Hypothesis (H₀): Assumes that there is no effect or no difference. For example, it may state that the mean of two groups is equal Alternative Hypothesis (H₁): Assumes that there is an effect or a difference. For example, it may state that the mean of two groups is not equal.

5. Degrees of Freedom The degrees of freedom in a t-test refer to the number of independent observations in the data that are free to vary. It affects the shape of the t-distribution, which is used to determine the critical value for the test. If sample size is n d f = n - 1

6. Introduction to t - test When calculating the t -statistic, we use the estimated standard error of the sample [ ] » Provides an estimate of the standard distance between a sample mean M and the population mean The t -statistic is then calculated as: t=   t = used when population standard deviation unknown M = sample mean μ = population mean S M = Estimated standard error of the sample

7. t - Distribution The t-distribution is similar to the normal distribution but has heavier tails. It is used instead of the normal distribution when dealing with small sample sizes (typically n < 30). As the sample size increases, the t-distribution approaches the normal distribution. The shape of the distribution alters with the degrees of freedom [ df ] – the number of scores within the sample that are ‘free to vary’ independent of one another [ n – 1] … the larger the sample size, the narrower the distribution the smaller the sample size, the wider the distribution

8. t - Distribution

9. Critical Value and p-Value Critical Value: The threshold that the t-statistic must exceed to reject the null hypothesis. p-Value: The probability that the observed data would occur if the null hypothesis were true. A low p-value (typically < 0.05) indicates that the null hypothesis can be rejected.

10. t - Statistics Unit Normal Tables to locate proportions for t-statistics [just like with the z-scores …] Use critical values from the t -distribution table to compare to the t -statistic from the sample State hypotheses [ and ] about the population, and specify the alpha test level Determine the critical region for the test [based on the alpha level] Obtain a sample from the population and calculate the corresponding t -statistic Make a decision about , based on whether or not t is in the critical region, then form a conclusion related back to the population  

11. University students and IQ … T-test Table It is known that for the general adult population, IQ scores are normally distributed: – mean( ) = 100 / standard deviation ( ) = 15. Suppose we have a random sample of (n = 25) students and found their mean IQ was 112.  

12. Introduction to One-Sample T-Test Definition: A one-sample t-test compares the mean of a single sample to a known or hypothesized population mean . Purpose: To determine if the sample mean is significantly different from the population mean. Assumptions : The data is continuous and approximately normally distributed. The sample is randomly selected. The population standard deviation is unknown.

13. Testing the Average IQ Score

14. Average Weight of a Product

15. Average Test Score Comparison

16. Summary of One-Sample T-Test

17. Examples For example, imagine a company wants to test the claim that their batteries last more than 40 hours. Using a simple random sample of 15 batteries yielded a mean of 44.9 hours, with a standard deviation of 8.9 hours. Test this claim using a significance level of 0.05.