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# Small sample test t test

Small sample test t test

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### Small sample test t test

1. 1. Small Sample Test T –Test & Paired Sample Test Sachin Nandakar
2. 2. What is the t-Test? t-Tests are a statistical way of testing a hypothesis when:  We do not know the population variance  The sample size is less than 30 (n<30).
3. 3. One-Sample t-Test We perform a One-Sample t-test when we want to compare a sample mean with the population mean. The difference from the Z Test is that we do not have the information on Population Variance/SD here. We use the sample standard deviation instead of population standard deviation in this case.
4. 4. Example We want to determine if on average girls score more than 600 in the exam. We do not have the information related to variance (or standard deviation) for girls’ scores. To a perform t-test, we randomly collect the data of 10 girls with their marks and choose our ⍺ value (significance level) to be 0.05 for Hypothesis Testing. In this example:  Mean Score for Girls is 606.8  The size of the sample is 10  The population mean is 600  Standard Deviation for the sample is 13.14
5. 5. Our tcal value is less than the critical value & falls in the acceptance region. Conclusion: Average girls score is less than or equal to 600 in the exam.
6. 6. Example: Two-Sample t-Test:
7. 7. Example An experiment is conducted to determine whether intensive tutoring is more effective than paced tutoring. Two randomly chosen groups are tutored separately and then administered proficiency tests. Use a significance level of α < 0.05.
8. 8. Solution: Null hypothesis: H0: μ 1 = μ 2 or H0: μ 1 – μ 2 = 0 Alternative hypothesis: Ha : μ 1 > μ 2 or Ha : μ 1 – μ 2 > 0
9. 9.  The degrees of freedom: This parameter is the smaller of (n1 – 1) and (n2 – 1). Because this is a one‐tailed test, the alpha level (0.05), DoF=9.  Look up t .05,9in the t‐table: 1.833  Since, tcal < tcritical, & the tcal value falls in the acceptance region => the Null hypothesis is accepted Conclusion: This test has provided statistically significant evidence that intensive tutoring is same as of the paced tutoring.
10. 10. Example
11. 11. Solution:  Ho: µA = µB  H1: µA != µB
12. 12. Tcal = -2.44 Tcritical: alpha=0.05, dof=21, = 2.080 Reject Null Hypothesis. Conclusion: The size of the tomato plants differ
13. 13. Example: Researchers at a pharmaceutical company want to test a new pain-relief medicine. Specifically, they want to see if its effect takes less time than their old medicine. They take a sample of people who suffer from chronic pain and randomly assign them into two groups. The first group receive the old medicine while the second group receive the new medicine (the participants don't know which medicine they have). Then, the participants are asked to measure the time since they take the medicine until their pain is gone. Here is a summary of the results:
14. 14. Paired Sample T-Test
15. 15. Paired sample t-Test: In paired sample hypothesis testing, a sample from the population is chosen and two measurements for each element in the sample are taken. Each set of measurements is considered a sample. Unlike the hypothesis testing studied so far, the two samples are not independent of one another. Paired samples are also called matched samples or repeated measures.
16. 16. When to Choose a Paired T Test ? Choose the paired t-test if you have two measurements on the same item, person or thing. You should also choose this test if you have two items that are being measured with a unique condition. For example, you might be measuring car safety performance in Vehicle Research and Testing and subject the cars to a series of crash tests. Although the manufacturers are different, you might be subjecting them to the same conditions. With a “regular” two sample t test, you’re comparing the means for two different samples. For example, you might test two different groups of customer service associates on a business-related test or testing students from two universities on their English skills. If you take a random sample each group separately and they have different conditions, your samples are independent and you should run an independent samples t test
17. 17. NOTE: The null hypothesis for the independent samples t-test is μ1 = μ2. In other words, it assumes the means are equal. With the paired t test, the null hypothesis is that the pairwise difference between the two tests is equal (H0: µd = 0).
18. 18. Test Statistics
19. 19. Example For example, if you want to determine whether drinking a glass of wine or drinking a glass of beer has the same or different impact on memory, one approach is to take a sample of say 40 people, and have half of them drink a glass of wine and the other half drink a glass of beer, and then give each of the 40 people a memory test and compare results. This is the approach with independent samples. Another approach is to take a sample of 20 people and have each person drink a glass of wine and take a memory test, and then have the same people drink a glass of beer and again take a memory test; finally we compare the results. This is the approach used with paired samples.
20. 20. Example A clinic provides a program to help their clients lose weight and asks a consumer agency to investigate the effectiveness of the program. The agency takes a sample of 15 people, weighing each person in the sample before the program begins and three months later to produce the table given below. Determine whether the program is effective.
21. 21. Example 2: Calculate a paired t test by hand for the following data:
22. 22. Solution:  Ho: µ1 = µ2  H1: µ1 != µ2
23. 23. If you don’t have a specified alpha, use 0.05 (5%). For this example problem, with df = 10 (n-1), the t-value is 2.228. Since, tcal > tcritical, reject Null Hypothesis. Conclusion: there is difference between means of two scores.
24. 24. Practice Problem Twelve cars were equipped with radial tires and driven over a test course. Then the same 12 cars (with the same drivers) were equipped with regular belted tires and driven over the same course. After each run, the cars’ gas economy (in km/l) was measured. Is there evidence that radial tires produce better fuel economy? (Assume normality of data, and use alpha = 5%)
25. 25. What is P-value ?
26. 26. P-Value  The P value, or calculated probability, is the probability of finding the observed, or more extreme, results when the null hypothesis (H0) of a study question is true.  In statistics, the p-value is the probability of obtaining results at least as extreme as the observed results of a statistical hypothesis test, assuming that the null hypothesis is correct
27. 27. Interpretation  Let, alpha = 5% If p = 0.02, then p-value < alpha => Reject Ho If p = 0.089 then p-value > alpha => Don’t Reject Ho
28. 28. Estimation of p-value ?
29. 29. Calculating a p-value: The p-value is calculated using the sampling distribution of the test statistic under the null hypothesis, the sample data, and the type of test being done (lower-tailed test, upper-tailed test, or two-sided test). The p-value for:  a lower-tailed test is specified by: p-value = P(TS <= ts | H0 is true) = cdf(ts)  an upper-tailed test is specified by: p-value = P(TS >= ts | H0 is true) = 1 - cdf(ts)  assuming that the distribution of the test statistic under H0 is symmetric about 0, a two-sided test is specified by: p-value = 2 * P(TS > |ts| | H0 is true) = 2 * (1 - cdf( |ts| ) ) Where: P -Probability of an event TS -Test statistic ts -observed value of the test statistic calculated from your sample cdf() -Cumulative distribution function of the distribution of the test statistic (TS) under the null hypothesis
30. 30. Example:  Given: n=25, µ=260, X(bar)=330.6, SD(σ)=154.2 Two-tailed, Z-test:
31. 31. Solution:  Given: n=25, µ=260, X(bar)=330.6, SD(σ)=154.2 Two-tailed, Z-test: Z = x(bar) - µ / SDerror = 2.2922 = Area from µ = 0.4890 (from Z-table) Thus, P-value = 2 x (0.5- 0.4890) P-value = 0.022
32. 32. Key Terms in Hypothesis Test:  Null hypothesis (H0): The null hypothesis states that a population parameter (such as the mean, the standard deviation, and so on) is equal to a hypothesized value.  Alternative Hypothesis (H1): The alternative hypothesis states that a population parameter is smaller, greater, or different than the hypothesized value in the null hypothesis.  Two-sided: Use a two-sided alternative hypothesis (also known as a non- directional hypothesis) to determine whether the population parameter is either greater than or less than the hypothesized value.  One-sided: Use a one-sided alternative hypothesis (also known as a directional hypothesis) to determine whether the population parameter differs from the hypothesized value in a specific direction.
33. 33. Key Terms  Confidence Interval: A confidence interval is a range of values, derived from sample statistics, that is likely to contain the value of an unknown population parameter.  Confidence level: The confidence level represents the percentage of intervals that would include the population parameter if you took samples from the same population again and again.  Point Estimate: The single value estimates a population parameter by using your sample data.  Interval Estimate: The range of values estimates a population parameter by using your sample data.
34. 34. Key Terms:  Test Statistic: A test statistic is a random variable that is calculated from sample data and used in a hypothesis test. You can use test statistics to determine whether to reject the null hypothesis. The test statistic compares your data with what is expected under the null hypothesis. The test statistic is used to calculate the p-value.  Critical value? A critical value is a point on the distribution of the test statistic under the null hypothesis that defines a set of values that call for rejecting the null hypothesis. This set is called critical or rejection region.

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• #### KetevanGentschLalias

Apr. 27, 2021
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May. 11, 2021

Small sample test t test

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