Week12 - STA 2023 c D.Wackerly - Lecture 20 273 Thought:...

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Unformatted text preview: STA 2023 c D.Wackerly - Lecture 20 273 Thought: People will accept your ideas much more readily if you tell them that Benjamin Franklin said it first. Assignments Today : P. 343 – 345 (p-values), P. 374 – 383 (Sec. 9.1, “Large Sample”) For Thursday: Exer. 8.49(c), 8.54(c), 8.106(c), 107, 8.117, 9.1, 9.13, 9.19, 9.22, 9.24, 9.25 Monday : P. 378 – 383 (rest of Sec. 9.1) For Tuesday: Exer. 9.7, 9.16, 9.17, 9.95, 9.105, 9.114 STA 2023 c D.Wackerly - Lecture 20 274 Last Time : Tests and Confidence Intervals for a Population Mean, based on SMALL samples. ¡ Assumption : POPULATION approx. NORMALLY dist. : ¨ ¦ ©§¥ ¤ ¢ £ Small Sample (p. 292):  Hypothesis Tests   &%#!¥ $" calc. value of       versus  RR p-value 0 ) ¥1 3 $" 42%# ¥ 0 5 ¥1 3 2%#!¥ $" ¦¥  "  )¥ ) OR ¦¥ 6  5¥ 5 OR @ 9 or (tail area) ' (¥ ¡ Confidence Interval for ¨ 8¦ ¥ 6 ¨ ©¦ ¥ )¥ 5¥  7 ¡ STA 2023 c D.Wackerly - Lecture 20 275 ¡ Test statistic :   ¡ ¢ £ ¥  ) depends on ¨ ©¦ ¥ ¦¥ ¡ (new) ¢ 6 (and (looks just like !!) (like before) AND #d.f. ¡ Table does not allow us to get exact p-values. ¡ In Everglades example, lower tail test, d.f.  £ ¤£ , ) to §¥ ¨¦' £ 9 9  §¥ ¨¦' £ 6 ¥ p-value Look at table, and  are £ ©£ Closest values in table ( with d.f. ¡ ¡ STA 2023 c D.Wackerly - Lecture 20 3 ¡¥ £ ¦' £ 3 ) ¥1 § ¡ §' £ 6 0 ¡ ¡ ¢ ¥ ' £ 3 5 ¥1 ¡ From the table, From the table, 276 0 5 ¥1 6 0 ¡ Thus, in this case, best we can say is that Can’t be any more precise using these tables!  § ¨£ ' Ho .  § ¢©'  § ¡ ©'  ¥§ ©' Ho .  §§ ©©' ¢¤  ¥ ¦£ ' Ho . Ho . ¢¤ ¢¤ ¢¤ ¢¤ ¢¤ ¡ STA 2023 c D.Wackerly - Lecture 20 277 § £ ' ¥ ¡    1 is  §§ ¢¨£ ' 3 ' £ ¡¥ 3 if d.f. ¥ § ¡¥ ' £ ) ¥1 0 3 ) ¥1 £ ' § ¡ ¡ ¡ 0 ¡ ' £ ) ¥1 0 ¡ So p-value in table for d.f.  Closest value to ¥ 0 p-value = 3 ¡ d.f. , Ex. : New case, upper tail test, STA 2023 c D.Wackerly - Lecture 20 278 6 9 ¥ , § £'   Ex. : New case, lower tail test, d.f. p-value = § ¨£ ' 9 6 5 ¥1 3 0  § ¡ £§ ' 9 6  ¡ ¡ So then § ¡ ©' £  If d.f. are . . 3 ¡ and  in table for d.f.  ¡ Closest values to and 5 ¥1 0 STA 2023 c D.Wackerly - Lecture 20 279 9 6 ¥ , § ©©' § ¨£  Ex. : New case, two-tailed test, d.f. ¡ p-value =  § ©©' § ©©' are 9 ¡ ¡ ¥' 6 9 5 ¥1 0 3 and 5 5 tail area p-value 0 ¡ 9 . 3 5 ¥1 6 § ¨£ ¢¡ £ ' § 5 5 6  ¡ ¡ ¡ And § ¨£ 3 So then 9 0 and If d.f. ) ¥1 in table for d.f. § ¤ ' Closest values to 3 tail area . . 5 ¥1 0 STA 2023 c D.Wackerly - Lecture 20 280 Exercise 10.52 Pouring temperature of molten iron. Target setting is 2550 degrees. Actual mean pouring  ¢ §£ measurements   2543 2541 2544 2620 2562 2553 2552 . 2560 2559 £ ©' § temperature different than target? Use 2553 § £¥ ¥ 9    ¡  ¡ Want : two tailed test  § ¢¥ ¥ 7 9  " Minitab? ¡ Basic Statistics Stat 1-Sample ¥ ¡ Punch in data values Select variable and click Radio button “Test mean” Select Alternative, Type in Null Value, OK ¡ ¡ T-Test of the Mean Test of mu = 2550.00 Variable C1 N 10 vs Mean 2558.70 mu not = 2550.00 StDev 22.75 SE Mean 7.19 T 1.21 P 0.26 STA 2023 c D.Wackerly - Lecture 20 281 ¡ Conclusion at .01 level? ¡ Is ? ¡ claim that the mean pour temperature differs from 2550 at the .01 level of significance. Minitab CI? Basic Statistics 1-Sample ¥ ¡ Stat Select variable and click Radio button “Confidence ¡ interval” Type in Confidence Level OK ¡ T Confidence Intervals Test of mu = 2550.00 vs Variable C1 N 10 Mean 2558.70 mu not = 2550.00 StDev 22.75 SE Mean 7.19 99% CI (2535.32, 2582.08) STA 2023 c D.Wackerly - Lecture 20 282 Ex. Investigation between “handedness” and motor competence in preschool children. Scores on motor skills tests. £   41 97.5 17.5 Right-handed (2) 41 98.1 19.2 § Find a ¢ Left-handed (1) confidence interval for the true difference in mean motor skills scores for left and right handed preschoolers. ¡ Samples from two populations ¡ Objective is to compare two means ¡ HOW???? STA 2023 c D.Wackerly - Lecture 20 283 Large Sample Inferences about Differences Between Two Population Means, Independent Samples (Section 9.1) ¡ Independent samples from two pop’s. Pop. 1 Pop. 2       ¢ £  ¢£   . 6 ¢£ 6 – Use the difference in sample means, ¢ £  Want: Estimate for ¡ Sample variance ¡ Sample mean  Sample size  Pop. variance  Pop. mean (p. 375)  [formula sheet, P. 375]  [formula sheet, P. 375]    6 ¡  ¡   ¨ ¦¤ ©¢ £ §¥¢ £ ¨ ¢£ ¦ ¤ ¢£ ¡ ¡ STA 2023 c D.Wackerly - Lecture 20 284 ¡ ¡ ¢ £ 6 ¢£   “large”, then ¢£  and 6 If ¢ £ If both populations normal, then approx normal, normal. regardless of shapes of sampled populations. (P. 375)  large.   formula sheet      ¡ ¡ ’s     ¨ 8¦ ¡ ¢ ¤ 3£ 6 ¢£ 1 ’s, use them, otherwise use  ¤  ¡ Know       table standard errors  ¡  formula sheet ¨ 1 ©¦ estimator '3  and © ¦ ¥¤ £¢ ¡¡ Both © ¨§ (p. 376) ¦ Large Sample CI for STA 2023 c D.Wackerly - Lecture 20 285 Ex. Investigation between “handedness” and motor competence in preschool children. Scores on motor skills tests.  £  41 97.5 17.5 Right-handed (2) 41 98.1 19.2 § Find a ¢ Left-handed (1) confidence interval for the true difference in mean motor skills scores for left and right handed preschoolers.  ¡   ¡ ¨ ©¦  ¡ ¢ 90% CI is        ¨ 8¦ ¡ ¢ ¤ 3£  ¤ ¥ ¡' ¡ ¤ 6 ¢£ 1 STA 2023 c D.Wackerly - Lecture 20  6 6 equals ? 6  equals  by §  is than ' ©¥ 6 ¡ What does it mean if 9 is ¥ ¡ What does it mean if ? by  than  is equals ¥ ©' ¡ § What does it mean if 286 ? ¡ How can we interpret the 90% confidence interval, , for the difference in mean motor skills 3 ¥ §' ¡ ¥ 9 ' ©¥ 6 1 scores for left and right handed preschoolers? (the true mean dexterity score for left-handers) larger than  ¡ could be as much as (the true mean dexterity score for right-handers) . (the true mean dexterity score for left-handers) smaller than  ¡ could be as much as (the true mean dexterity score for right-handers) . or anything in between. ¡ STA 2023 c D.Wackerly - Lecture 20 287 ¡ Claim : no difference in mean dexterity scores for left and right handers. This claim § rejected at the be confidence level. ¡ Is it plausible that the mean dexterity score for left handed preschoolers is larger than the mean for RH preschoolers by 5 points? ¡ Is it plausible that the mean dexterity score for left handed preschoolers is smaller than the mean for RH preschoolers by 5 points? Is it plausible that the mean dexterity score for left handed preschoolers is larger than the mean for RH preschoolers by 7 points? ¡ STA 2023 c D.Wackerly - Lecture 20 288 Consider testing (p. 376) calc. value of  ¡ p-value ¡ 3 &%# $" ¡ 3 &%# $"     6  RR ¡  2%# $"  versus a fixed particular value of difference ¡ 0 1 ¦ 0 1 ) ) ¡ )  ¡  " 6 OR 5 ¡ ¦ 6 ¡ 5 5  ¡ 6 OR @ ¨ ©¦ or (tail area) 9 ¨ 8¦ 6 ¡ ) 5 ¡ ¡  7  6 ¡ TEST STATISTIC ¤ ¢¤ NULL HYPOTHESIS 6£     6    £ £ ¡  ¡ Formula Sheet ¡ standard error  6 hypothesized value ¡ ¢¡ estimator     6£  6 ¡  ¡ £  ¡ ¤ ¡ ...
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This note was uploaded on 07/28/2011 for the course STA 2023 taught by Professor Ripol during the Fall '08 term at University of Florida.

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