Week11-4up - STA 2023 c B.Presnell & D.Wackerly -...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: STA 2023 c B.Presnell & D.Wackerly - Lecture 17 STA 2023 c B.Presnell & D.Wackerly - Lecture 17 194 195 Thought: A committee is a group of people who, individually, can do nothing, but collectively can meet, and decide that nothing can be done. Last Time: ¡ Assignments : Point Estimator (p. 261) ¡ Interval Estimator (p. 282) ¡ Confidence Coefficient ¡ Confidence Level (p. 282) For Thursday: Exercises 7.51, 7.53, 7.59, 7.64, 7.67,  ¡ such that ¢  Wednesday: P. 306 – 308, 322 – 326    7.77, 7.79, 7.82, 7.91 (p. 282) . " #!© For Tuesday: Exercises 7.37, 7.42, 7.44–46, 7.72, ©§ ¥£ ¨¦¤¢ Today: P. 299 – 302, COMPUTER DEMONSTRATION 8.7, 8.9, 8.10, 8.13 Monday: OPTIONAL review day Tuesday: EXAM 2 STA 2023 c B.Presnell & D.Wackerly - Lecture 17 STA 2023 c B.Presnell & D.Wackerly - Lecture 17 196 197 Ex. : (#7.71, p. 314) Metalic Glass, 4 times stronger than steel, but brittle at high temps. Want to estimate @ the mean temp. at which metalic glass becomes brittle. (large sample) p. 283 36 specimens randomly selected, measured temp. © qi r&p¢ ££ xw TV R P H F  C USQIGEDA B v y €¡ true mean temp. metalic glass gets brittle is a a ‘ £ S£ ‘ s `… h d¢ £ s `… h £ Ca TV R ‘ a  ˜ ‘ C —A B s `… h £ a y C —A B s `… h £ © rq i exY h Sxont ˆh l6jihY h ƒ ¢© ƒ t sm ‡ … k s¢ © xY h Sudnt ˆh lIj¢ ƒt sm ‡ …k s a h Ca PHF QIG¡  ¡ PH Q6F  Y – ua v ts x# § ‘ ua – ua C DA B : f… gxƒ e ˜ u© ‘ s S… h w £ ”’ •“ x ¡ †¨¦£ §¥ ‡ ˆh ‡ PH Q6F  a ™ f h© d ec formula sheet ˜ ‰ a  ‚¡ …ƒ S„ confidence interval for standard errors  †§ AB CI R £h ts x# ts x# ‡ ¡ ¢b f ec b d PHF QIGgC table v ¥ f d ec b formula sheet and ££ Sw YW `X§ NOTE : the basic form of this interval is estimator and unknown, α/2 1−α AB in the upper tail of a standard normal distribution α/2 ts u# is the z value that cuts off an area of at which each became brittle : a 855% 3 1 ) 976420(&$ '% Confidence Interval for ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 17 198 199 y . ¡ Claim : Yks h `6# – confidence interval for y †§ ‰ t‡ Find a STA 2023 c B.Presnell & D.Wackerly - Lecture 17 § £ S£ this claim at the … xƒ Ca ˜ ‰ ¡ ©k SxY h `udnlk h st sm ƒ – t‡ ts x# ¡ ks Ij¢ y – ¡ t‡ – the region of “believable” values for y ‘ : Yk h `6s Ca ™ confidence interval for y Claim : interval!! . C —A B  k xY h s believable values for in the region of . 475.73 472 at the – confidence level. t‡ – 484.27 this claim at the t‡ ts x# Note : Higher confidence coefficient (.98) is confidence level – v because y – Yk `Is ¡ Y CI 484.27 t‡ 475.73 472 – ¡ †¨¦£ §¥ and because confidence level is in the interval. 484.27 STA 2023 c B.Presnell & D.Wackerly - Lecture 17 200 475.73 STA 2023 c B.Presnell & D.Wackerly - Lecture 17 y kt h uus Claim : ¡ confidence level 475.73 484.27 478 confidence level: – t‡ is in the interval. confidence level: k Is this claim at the ( so are ‘ – 484.27 487 m `£ h `xdm ƒ h lIom ˆh sts …ks ‡ 475.73 y y kts h uu# of tk lIs t‡ – At the – t‡ because ALL believable values for y – this claim at the Claim : y – 484.27 487 tks h l6# ¡ 475.73 Claim : 201 etc.). is kt uus ¡ 202 post-retirees. ¡ sheet) ¡ If  ‡ Yk S£ stay between 4 and 7 nights on a typical trip. is “large”, : (p. 300,formula is approximately normally £¢ – ua for the true proportion of post-retirement travelers who ¤¦ ¨R £ ¢ Standard error of confidence interval  and 7 nights on trips. Find a 203  ¢  ƒYƒ `x„ said that they stayed away from home between 4 © Ex.: #7.79 p. 316 Survey of STA 2023 c B.Presnell & D.Wackerly - Lecture 17 Estimation of a Proportion, STA 2023 c B.Presnell & D.Wackerly - Lecture 17 distributed. (p. 300) LOOKS FAMILIAR!!! Confidence Interval for a Proportion,  ¡ attribute. .  ¢ e dc ¡ b m   ¢ ©  PH  QIF EC ¢ ¡ . (p. 300) B £¢ ¢  ¢ ¤¦ § ¥y ¢ ¡ £¢ 204 ¡ £¢ STA 2023 c B.Presnell & D.Wackerly - Lecture 17  ƒYƒ `x„ Ex.: #7.79 p. 316 Survey of B f is an unbiased estimator for , and £w2  fdcb PH  Q6F †C is a random variable. ¢b the attribute. formula sheet table , the sample proportion with dc formula sheet standard errors ©f Estimate for : estimator f with the ¥ B and get ¢ ¢ Large Sample Take: a random sample of size attribute (“ ”) is . 855%1 ) 976S20(&$ '% Have: a popn. where the proportion with a particular STA 2023 c B.Presnell & D.Wackerly - Lecture 18 205 post-retirees. said that they stayed away from home between 4 – ua ‡ Yk S£ and 7 nights on trips. Find a confidence interval for the true proportion of post-retirement travelers who Thought: When a man is wrapped up in himself, he makes a pretty small package – (John Ruskin) stay between 4 and 7 nights on a typical trip. Assignments : § Today: P. 306 – 308, 322 – 326 ‰ ¡ †§ Y ¡ For Thursday: Exercises 7.51, 7.53, 7.59, 7.64, 7.67, B 8.7, 8.9, 8.10, 8.13 ¡ £¢  ¡ Confidence interval : Monday: OPTIONAL review day © Tuesday: EXAM 2 C or equivalently Wednesday: P. 328 – 332 k ‘ sh a C‘ Y xƒ h ‘ ¡ . Thus, the believable values, at the ¢ confidence level for the true proportion of post-retirement travelers who stay between 4 and 7 nights on a typical trip are those between and . Tuesday, 4/3/01: EXAMS RETURNED We cannot give out grades over the phone. ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 18 STA 2023 c B.Presnell & D.Wackerly - Lecture 18 206 207 Last Time: Choosing the Sample Size –a uSa Confidence Interval for a Want: ¡ f h© sample mean, . formula sheet R Yw £  PH Q6F  .025 v AB y V R … ‡h £ … ‡h £ is within of y TV R y ¥ B of the time h STA 2023 c B.Presnell & D.Wackerly - Lecture 18 209 bound I want ( ¡ If I want to estimate y to within “ ” units with (p. 307) §  ¥£ ˜  from past study or pilot study if available. , and use  or … ˜  – guess the range,  v ƒ „ R  s ™  Y x… £ ˜  Y h `Y ƒ ‘ ˜ Y h `Y ƒ ‘ P r`t h k s ©s I`t h p¢ k  ¡ ¡ ¡ … ‡h £ ƒ ˜  k 6s h £ „ … ƒ ˜ ¦¦¦ ƒ ¨¥§„ ¤ ¡ ¤ ¥ PH F Q6G – use ™ R ƒ „ Need : at least a “ballpark value for R VR and solve for R y ‡ V R … ˆh £ confidence level in prev. example) R © “” y ¢ C 95% of time. © ¢ ¥ C DA B ˜ Q6G PH F  PH Q6F   95% of the time. ƒ y R ™   ¢ © 208 Want : units of ¡ ¢y C DA B PH F Q6G That is : … ˆh £ ‡  ˜ (P. 300), LARGE of .025 TV R PH Q6F  a Want : to be within is within , how large should “ ”, the sample size be? .95 STA 2023 c B.Presnell & D.Wackerly - Lecture 18 Know: confidence.  sample: units of the true value of Question : If the population standard deviation is (P. 283), LARGE sample: – Population Proportion, ‡ b ¡ α/2 1−α – Population mean, with – x‘ d ec α/2 Estimate to be within ƒ f table , by using the B f ec b d PHF QIGgC ¢b d ec formula sheet Estimate the population mean, standard errors y ¡ ©§ ¥£ ¨¦¤¢ estimator y £ PARAMETER £ … ‡h £ … ˆh £ ‡ B STA 2023 c B.Presnell & D.Wackerly - Lecture 18 Tests of Hypotheses: Section 8.1 average cost of freshman Ex. : A manufacturer of light bulbs would like to claim year to within $500 with prob. .95. Know – cost between that the average lifetime of our bulbs exceeds 1325 have to be? hours.   $4800 and $13,000. How large does 211 § ¡ bulbs on test, record time to burnout for each. a v B ƒ£ ‡ &i ™ Can the claim be made? (VALIDLY) (NULL hypothesis) ‘ (ALTERNATIVE hypothesis)  gy Yƒ x&£ ™ Y ‰ R true mean ¤ ˜ ˜  ¡ ©… uxƒ h t ¢ a Type I Error Type II Error ¡ f d§c GeGb alpha  ¡ Using sample data to make decision – always the ¡ fc d b possibility of making an ERROR. (See Table 8.1, beta p. 325) Reality Note: and § ¡ about the POPULATION.  … ‡h £ Sample DOES NOT contain ALL of the information § ¦ ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 18 © £  212 213 (p. 323) (p. 325) h aa S`‘ Y … ˆh £ ‡ STA 2023 c B.Presnell & D.Wackerly - Lecture 18 Decision is to be made using sample data. § ¨ a‘ `ua P rSƒ h t … a UNKNOWN ) © £ ˜  ¤  lifelength of the manufacturer’s bulbs. ( FIXED, but h  s  We do not know the true value of y Yƒ£ x&w a‘ `xa £¦¢ ¥ £¡ ¤¢ y ” P “ S ’ R ¡ Choose between two HYPOTHESES. ‘ w ¡ Data : ƒƒ h SSY © †§ Y Put a random sample of s ‡# y Ex. Want to estimate 210 STA 2023 c B.Presnell & D.Wackerly - Lecture 18 are PROBABILITIES ( numbers between 0 and 1 ) Ha true Correct Type II error Reject Ho Type I error Correct Suppose is a fixed sample size. ¡ Ideally, want BOTH ¡ How about both very small? and § Consider the lightbulb example – Claim when really – If we want to REDUCE must decide to “reject Type II Error ” . Type I Error we OFTEN. OFTEN. – Leading to an § ¦  when really ” in © – Claim ¡  – Thus, “accept Type II Error §¡ ¨ †§ ¡ Type I Error to be © Accept Ho  Ho true  Decision ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 18 STA 2023 c B.Presnell & D.Wackerly - Lecture 18 214 ¡ The hypothesis of MAIN INTEREST is the . ALTERNATIVE HYPOTHESIS, or the Research   Hypothesis.  manner move in directions. – Denote by ¡ § ¡ ¢ ¡  § m  § ¡ ¡ HYPOTHESIS Our strategy – Denote by small enough to be “CONVINCING” § ¡ Choose . (p. 322) The “other” hypothesis is called the NULL ¡  m and – What we are “trying to prove” in an objective, fair ¥  For fixed 215 . (p. 322) – If we decide that the ALTERNATIVE IS In out set-ups CORRECT ¡ PARAMETER EQUALS the value used in ERROR. The PROBABILITY of making a TYPE I error specifying the ALTERNATIVE or “RESEARCH” is hypothesis. ¡  when true © © ¥ accepting 216 § ¦ § ¦ †§ ¡ ¡ †§ ¡ § ¦ ¡ when § ¦ SIGNIFICANCE LEVEL of the test, LEVEL of †§ ¡ the test. ’s that we pick.  The test that we will discuss have the SMALLEST ’s © †§ saying © †§ saying what we would like to say when we should not y § STA 2023 c B.Presnell & D.Wackerly - Lecture 18 Y£ xƒ w . (ALTERNATIVE hypothesis) (1) ‘ ¥ £ confidence in our conclusion to ACCECT Type I error £¡ £¥ our  †y SMALL we (NULL hypothesis) Y xƒ £ . – By making § for the The NULL hypothesis is always that a ‘ £ The ONLY type of error possible is a (2) ...
View Full Document

This note was uploaded on 12/15/2011 for the course STA 2023 taught by Professor Ripol during the Spring '08 term at University of Florida.

Ask a homework question - tutors are online