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1. The graph of the t- distribution with 15 degrees of freedom is shown below. Find the value of t1 such that the: a. shaded area on the left is equal to 0. 05 b. shaded area on the right is 0.1 c. Total shaded area is 0.02 d. Area to the left of ti = 0.95 2. Find the 5'h percentile of a t-distribution with 28 degrees of freedom. 3. The area to the left of 11 is 0.995, what is the value of 11? ExampE'2: The total shaded area of a curve is 0.1 with df = 18. Which percentile is h and what is its value? P = (1 -c1: )100% = (1 - ) move-m is divided by 2 because 0.1 is the 2 total shaded area at the left and right P = 0.95 Thus, t1 represents the 95'\" percentile. To find the value, proceed to the right until the "1 0 t1 column headed 0.05. The result is 1.734 LESSON NO. 7: THE T- DISTRIBUTION INTRODUCTION The T Distribution (and the associated t scores), are used in hypothesis testing when you want to figure out if you should accept or reject the null hypothesis. In general, this distribution is used when you have a small sample size (under 30) or you don't know the population standard deviation. For practical purposes (i.e. in Rejection the real world), this is nearly always the case. So, unlike in your elementary statistics class, you'll likely be using it in real life situations more than the normal Region distribution. If the size of your sample is large enough, the two distributions are practically the same. - 1.96 1.96 INTERACTION The T distribution is a family of distributions that look almost standard normal identical to the normal distribution curve, only a bit shorter and distribution fatter. The t distribution is used instead of the normal distribution t distribution when you have small samples. The larger the sample size, the more the t distribution looks like the normal distribution. -2 T Distribution Formula each corresponding curve is bell-shaped and are symmetric about 0 t = ( X - H) Where: rope x = Sample mean H = Population mean s = standard deviation of the sample mean n = sample size Table of the Student's f-distribution The table gives the values of ta;, where 1.337 1.746 2.120 2.583 2.921 3.686 4.015 Pr(Tv> cv) = a, with v degrees of freedom 1.333 1.740 2.110 2.567 2.898 3.646 1.965 1.330 1.734 2.101 2.552 2.878 3.610 3.922 2.09 2.539 2.861 3.579 3.883 0.1 0.05 0.025 0.01 0.005 0.001 0.0005 1.328 1.729 20 1.325 1.725 2.086 2.528 2.845 3.552 3.850 V 3.078 6.314 12.076 31.821 63.657 318.310 636.620 1.323 1.721 2.080 2.518 2.831 3.527 3.819 1.886 2.920 4.303 6.965 9.925 22.326 31.598 2.074 2.508 2.819 3.505 3,792 1.638 2.353 3.182 4.541 5.841 10.213 12 924 1.321 1.717 1.319 1.714 2.069 2.500 2.807 3.485 3.767 UIA WN . 1.533 2.132 2.776 3.747 4.604 7.173 8.610 1.318 1.711 2.064 2.492 2.797 3.467 3.745 1.476 2.015 2.571 3.365 4.032 5.893 6.869 1.316 1.708 2.060 485 2.787 3.450 3.725 1.440 1.943 2.447 3.143 3.707 5.208 5.959 2.056 2.479 2.779 3.435 3.707 1.415 1.895 2.365 2.998 3499 4,785 5.408 26 1.315 1.706 1.314 1.703 2.052 2.473 2.771 3.421 3.690 1.397 1.860 2.306 2.896 3.355 4.501 5.041 1.313 1.701 2.048 2.467 2.763 3.408 3.674 1.383 1.833 2.262 2.821 3.250 4.297 4.781 1.311 1.699 2.045 2.462 2.756 3.396 3.659 1.372 1.812 2.228 2.764 3.169 4.144 4.587 1.310 1.697 2.042 2.457 2.750 3.38 3.646 11 1.363 1.796 2.201 2.718 3.106 4.025 4.437 40 1,303 1.684 2.021 2.423 2.704 3.307 3.551 12 1.356 1.782 2.179 2.681 3.055 3.930 4.318 60 1.296 1.671 2.000 2.390 2.660 3.232 3.460 1.350 1.771 2.160 2.650 3.012 3.852 4.221 120 1.289 1.658 1.980 2.358 2.617 3.160 3.373 1.345 1.761 2.145 2.624 2.977 3.787 4.140 1.282 1.645 1.960 2.326 2.576 3.090 3.291 1.341 1.753 2.131 2.602 2.947 3.733 4.073 To find a value in the Table of t-distribution, there is a need to adjust the sample size n by converting it to degrees of freedom df. df = n - 1, where n = sample size Use the t-distribution table to find the critical value. Df is the first column and a is the first row then find their intersection. 1. a = 0.05, df = 8 Critical value: 1.860 2. a = 0.025, df = 12 Critical value: 2.179 3. a = 0.01, df = 20 Critical value: 1.325Example 1: Suppose you do a study of acupuncture to determine how effective it is in relieving pain and you suspect that the data you collected do not represent the target population. The population mean is 7.54. You measure sensory rates for 15 subjects with the results given. Use the sample data to construct a 95% confidence level. 8.6 9.4 7.9 6.8 8.3 7.3 9.2 9.6 8.7 4 10.3 5.4 8.1 5.5 6.9 Step 1: Find the sample mean and sample standard deviation. Sample mean x = = 8.2267 n Sample standard deviation = 1.6722 Step 2: find the degrees of freedom (df = n - 1) df = 14 Step 3: Find the critical value. Confidence level is 95%. (1- a) 100% = 95% 1 - a = 0.95 a = 0.05 Use the t - distribution table and move to the right until the column headed 0.05 with df = 14. Hence, the critical value is 1.761. Step 4: Compute the test statistic t. t = X - H 8.2267-7.54 1.6722 = 1.5904 Vn V15 rty Test statistic Critical value 1.590 1.761 The value of the test statistic or the computed t-value is less than the critical value 1.761. therefore, the student is wrong in suspecting that the data are not representative of the target population. Identifying Percentiles Using the t-Distribution Table Example 1: The graph of the distribution below has a df = 6 a. If the shaded area is 0.025, what is the area to the left of t? P = (1 - @) 100% = (1 - 0.025) 100% = (0.975) 100% P = 97.5% P = 0.975 area to the left of t b. What does t represent? Hence, t represents the 97.5th percentile c. Find the value of t. To find the value of t, look under the column headed df. Move to the right until the column headed for 0.025 is reached. The result is 2.447