Hypothesis Test Statistics and Confidence Intervals 1 - Confidence Interval Hypothesis Test Value (Statistic) NULL Hypothesis: Use the statement containing the condition of Point Estimate Maximum Error E equality either directly or implied, as the Null Hypothesis Ho. (TI-84) Single Population (TI-84) One Sample for mean ( is known) (ZInterval) (Z-Test) Use the Normal z -Table x x z z 2 n for the critical value n One Sample for mean ( is unknown) (TInterval) (T-Test) df = n - 1 s x Use the t-distribution Table xt 2 t n for the critical value t sn One Sample for Proportion p (1-PropZInt) (1-PropZTest) pqˆˆ Use the Normal -Table ppˆ pˆ z z 2 n for the critical value pq/ n (TI-84) Dual Population (TI-84)
Dependent Paired for d (TInterval) (T-Test) s df = n - 1 d Use the t-distribution Table d d d t 2 t Use Ho : 0 n for the critical value t d snd
Two Independent Samples for 1 - 2 ( 1, 2 are known) (2-SampZInt) (2-SampZTest) 22 ()()xx 1 2 1 2 12 z Use Ho : ()xx z Use the Normal -Table 2 2 120 1 2 2 2 nn12 for the critical value 1 nn12
Two Independent Samples for 1 - 2 ( 1, 2 are unknown) (2-SampTInt) (2-SampTTest) 22 ()()xx ss12 1 2 1 2 ()x x t df = smaller of n 1 – 1 or n 2 – 1 t 1 2 2 nn ss22 12Use the t-distribution Table for the critical value t 12 Use Ho : 120 Use “NOT POOLED” on the calculator. nn12 Two Independent Samples for Proportions p 1 - p 2 (2-PropZInt) (2-PropZTest)
Use the Normal - Table ()()pˆˆ1 p 2 p 1 p 2 pˆ q ˆ pˆ q ˆ z 1 1 2 2 for the critical value pq pq Use Ho : ()ppˆˆ z pp12 0 1 2 2 nn 12 nn12 x x n pˆˆ n p where, pp1 2 or 1 1 2 2 qp1 n1 n 2 n 1 n 2
Xx X1 X 2 x 1 x 2 x 1 x 2 n 1 pˆˆ 1 n 2 p 2 1-Prop: ppˆ Dual Prop: p1 p 2 pˆˆ 1 p 2 p or p Nn N1 N 2 n 1 n 2 n 1 n 2 n 1 n 2 q11 p qˆˆ p q11 p 1 q 2 1 p 2 qˆ 1 1 pˆ 1 q ˆ 2 1 pˆ 2 q 1 p Sample Size Determination for Mean for Proportion p 22 2 22 zz z22pq z (.25) n 22 (round up) nnor use 2 22 EE EE (if p, q unknown) Handout created by Professor Jahn on 3/01/00 and updated by Ms. Neginsky on 1/26/18 Areas, Confidence Intervals Student’s t – Table and Hypothesis Tests Formula Sheet For means, use when is NOT known ( s is given ) Hypothesis Test Also used for other applications. Steps : H0 statement Tail Types, Critical Z Signs for 1) Set up H0 & H1 of Hypothesis Tests (Probability Density Function) Student's T-distribution H0: Use the “=” or equality The tail type is determined by H1 Degrees Confidence indicate equality, etc. 50% 80% 90% 95% 98% 99% of Intervals H0 H1 Type Critical Value Sign H1: Use the “”, “>”, “<” Freedom One Tail, 0.25 0.10 0.05 0.025 0.01 0.005 or indicate inequality, < left 1-tail negative (-) df Two Tails, 0.50 0.20 0.10 0.05 0.02 0.01 dependence, etc.
1 1.000 3.078 6.314 12.706 31.821 63.656 (Label the Claim) > right 1-tail positive (+) 2 0.816 1.886 2.920 4.303 6.965 9.925 2) Determine the both 2-tail pos & neg () 3 0.765 1.638 2.353 3.182 4.541 5.841 critical number(s) 4 0.741 1.533 2.132 2.776 3.747 4.604 When is known, refer to the ( Note: All Confidence Intervals have 2-tails) 5 0.727 1.476 2.015 2.571 3.365 4.032 Z-table. Also, use the Z-table for proportions. When is 6 0.718 1.440 1.943 2.447 3.143 3.707 Z-Table For means, use when is known NOT known (and s is given), (See “Choosing z or t” flowchart, below, for details). 7 0.711 1.415 1.895 2.365 2.998 3.499 use the t-table. For others Also used for other applications. 8 0.706 1.397 1.860 2.306 2.896 3.355 (Chi-square, regression, F- Also used for other applications. dist, ANOVA, etc.) refer to the Also, use for proportions (when normal - use then q 5 check) 9 0.703 1.383 1.833 2.262 2.821 3.250 proper table or method. Conf Int (All) 10 0.700 1.372 1.812 2.228 2.764 3.169 3) Draw a curve 11 0.697 1.363 1.796 2.201 2.718 3.106 and plot the Sig Lev Conf Lev ±z for 2-Tail z for 1-Tail 12 0.695 1.356 1.782 2.179 2.681 3.055 critical number(s). 0.10 90% 1.645 1.28 13 0.694 1.350 1.771 2.160 2.650 3.012 Label the “Fail to Reject H0 0.09 91% 1.695 1.34 14 0.692 1.345 1.761 2.145 2.624 2.977 Zone” and, label and shade 15 0.691 1.341 1.753 2.131 2.602 2.947 the “Reject H0 Zone” 0.08 92% 1.75 1.405 16 0.690 1.337 1.746 2.120 2.583 2.921 4) Determine the 0.07 93% 1.81 1.476 test statistic - see 0.06 94% 1.88 1.555 17 0.689 1.333 1.740 2.110 2.567 2.898 the applicable formula sheet. 18 0.688 1.330 1.734 2.101 2.552 2.878 and plot it. 0.05 95% 1.96 1.645 19 0.688 1.328 1.729 2.093 2.539 2.861 5) Reject H0 or Fail 0.045 95.5% 2.005 1.695 20 0.687 1.325 1.725 2.086 2.528 2.845 0 to Reject H0. 0.04 96% 2.054 1.75 21 0.686 1.323 1.721 2.080 2.518 2.831 1) Reject if the test stat 0.035 96.5% 2.11 1.81 22 0.686 1.321 1.717 2.074 2.508 2.819 is in the Critical Zone or 0.03 97% 2.17 1.88 23 0.685 1.319 1.714 2.069 2.500 2.807 2) If using p-values, 0.025 97.5% 2.24 1.96 24 0.685 1.318 1.711 2.064 2.492 2.797 Reject if p-value or Fail to reject if 0.02 98% 2.326 2.054 25 0.684 1.316 1.708 2.060 2.485 2.787 < p-value.) 26 0.684 1.315 1.706 2.056 2.479 2.779 0.015 98.5% 2.43 2.17 6) Write the final 27 0.684 1.314 1.703 2.052 2.473 2.771 Write the final 0.01 99% 2.576 2.326 28 0.683 1.313 1.701 2.048 2.467 2.763 conclusion (see the flowchart below) 0.005 99.5% 2.81 2.576 29 0.683 1.311 1.699 2.045 2.462 2.756 Z's for Confidence Intervals 30 0.683 1.310 1.697 2.042 2.457 2.750 Confidence Interval Steps: Discrete Probability 31 0.682 1.309 1.696 2.040 2.453 2.744 & 1 and 2 tail Hypothesis Test Critical Points (values) 32 0.682 1.309 1.694 2.037 2.449 2.738 1) Find the critical z or t Distributions
34 0.682 1.307 1.691 2.032 2.441 2.728 When is known, refer to the Z-table. x P() x 36 0.681 1.306 1.688 2.028 2.434 2.719 Also, use the Z-table for proportions. 38 0.681 1.304 1.686 2.024 2.429 2.712 22 40 0.681 1.303 1.684 2.021 2.423 2.704 When is NOT known (and s is given), x P() x use the Student's t-table. For others, 45 use closest, 0.680 1.301 1.679 2.014 2.412 2.690 refer to the proper table or method. 50 or largest 0.679 1.299 1.676 2.009 2.403 2.678 55 if exactly 0.679 1.297 1.673 2.004 2.396 2.668 Binomials 2) Calculate the Maximum Error 60 in-between 0.679 1.296 1.671 2.000 2.390 2.660 (see formullas – reverse page) np 65 0.678 1.295 1.669 1.997 2.385 2.654 3) The Interval is: 70 0.678 1.294 1.667 1.994 2.381 2.648 75 0.678 1.293 1.665 1.992 2.377 2.643 Point Est. Max. Error n p q 80 0.678 1.292 1.664 1.990 2.374 2.639
90 0.677 1.291 1.662 1.987 2.368 2.632 100 0.677 1.290 1.660 1.984 2.364 2.626 200 0.676 1.286 1.653 1.972 2.345 2.601 Hypothesis Tests: Determining 300 0.675 1.284 1.650 1.968 2.339 2.592 the Final Conclusion 400 0.675 1.284 1.649 1.966 2.336 2.588 500 0.675 1.283 1.648 1.965 2.334 2.586 750 0.675 1.283 1.647 1.963 2.331 2.582 1000 0.675 1.282 1.646 1.962 2.330 2.581 2000 0.675 1.282 1.646 1.961 2.328 2.578 100000 Large 0.674 1.282 1.645 1.960 2.326 2.576
Hypothesis Testing using p-Values: Left tail: Area to the left of the test statistic
Right tail: Area to the right of the test statistic
Two tail: If Test Stat to left of center: Twice the area to the left of the test statistic If Test Stat to right of center: Twice the area to the right of the test statistic
(To find the areas, use the “Strategies to Find Areas”)
Reject Ho if, p ≤ α or Page 1 of 2 Last Update/Print: 4/25/18 4:48 PM snd_tdist_tab_triola9.doc Fail to Reject Ho if, p > α