Week11_001 - STA 2023 c B.Presnell & D.Wackerly -...

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Unformatted text preview: STA 2023 c B.Presnell & D.Wackerly - Lecture 17 194 Thought: A committee is a group of people who, individually, can do nothing, but collectively can meet, and decide that nothing can be done. Assignments : Today: P. 299 – 302, COMPUTER DEMONSTRATION For Tuesday: Exercises 7.37, 7.42, 7.44–46, 7.72, 7.77, 7.79, 7.82, 7.91 Wednesday: P. 306 – 308, 322 – 326 For Thursday: Exercises 7.51, 7.53, 7.59, 7.64, 7.67, 8.7, 8.9, 8.10, 8.13 Monday: OPTIONAL review day Tuesday: EXAM 2 STA 2023 c B.Presnell & D.Wackerly - Lecture 17 195 Last Time: Confidence Coefficient Confidence Level (p. 282) ¨ ©§ ¤ £¡ ¢ ¨  ¦ §  §¡  such that  Interval Estimator (p. 282) ¦¥ Point Estimator (p. 261) (p. 282) . STA 2023 c B.Presnell & D.Wackerly - Lecture 17 196  §§ ©¨¡ Confidence Interval for ¦¥¤ £ ¢ ¡ (large sample) p. 283 " # !§   & '% ¥   is the z value that cuts off an area of in  $ § the upper tail of a standard normal distribution α/2 α/2 1−α §   §   § ¤ ( NOTE : the basic form of this interval is   ¡ $§ table formula sheet 53 1 40 1 ) 3 40 )  3 1 20 ) formula sheet standard errors ¦ estimator STA 2023 c B.Presnell & D.Wackerly - Lecture 17 197 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. 36 specimens randomly selected, measured temp. ( ¥£ ¤   ¦ at which each became brittle : and ¡ ¢ ¡ ¢¢ ¦  true mean temp. metalic glass gets brittle is § unknown, ¢¢ ( £ ¤ ¦ (¢ 5 5  # ¢ £ ¢¢ ¦  & £¡ £¢  ( £ ¤ 5  ©  5  5 ¡ ¡ ¦ & ( 43¤ 10(¡ ©¤£2 )£ ¦&( 5 ©¤ £2 ¥98¤  75 )£ 65(¡ : ¦ !  " § § 5  & ¢ £ # "    5  & ¢ £ ¦ (   5  ¥(   ¥ 5 ¨ ¤¢ ¥  & $ %( $ %( confidence interval for ¨ ¥  ©  CI 5  # ¢ £   and ¨  5 &( ©      ( ¥£ ¤  '   STA 2023 c B.Presnell & D.Wackerly - Lecture 17 $¤ confidence interval for § .  ¥   ¥  © ¦    ¨   ( ¥£ ¤ & $¤   Note : Higher confidence coefficient (.98)  5 ) & £ ' ¥ CI  ( ¥£ ¤ ¤¢  and ¢¢  Find a 198 interval!! 5 5 £¤ £2 © #¥981)  5(¡ )£ § $¤ – the region of “believable” values for confidence level. 475.73 484.27 § : ¦ ¥& ) confidence interval for at the . $¤   STA 2023 c B.Presnell & D.Wackerly - Lecture 17 &) '0£ 5 § this claim at the is believable values for . 5 &) '0£ § Claim : 475.73 472 because interval.  this claim at the 484.27 $¤ – confidence level in the region of § because &) '0£ – 484.27  475.73 472 $¤  Claim : 199 confidence level is in the STA 2023 c B.Presnell & D.Wackerly - Lecture 17 confidence level because ALL believable values for this claim at the is  § – 484.27 487 $¤  475.73 5 ) ¥£ ¤ Claim : § this claim at the  – 484.27 487 $¤ § 475.73 5 ) ¥£ ¤ Claim : 200 in the interval. confidence level: ) ¥£ ¤ STA 2023 c B.Presnell & D.Wackerly - Lecture 17 5 ¤) 60£  § Claim : 201 475.73 484.27 478 $¤ confidence level: ¤) 65£ 5 5  75 2&¢ &43© 6542  5£ £¤£2 ) £ ) §  – At the of ( so are etc.). is STA 2023 c B.Presnell & D.Wackerly - Lecture 17 202 © && ©  ¨ Ex.: #7.79 p. 316 Survey of Estimation of a Proportion, post-retirees. &)¢ said that they stayed away from home between 4 $ %(  and 7 nights on trips. Find a confidence interval for the true proportion of post-retirement travelers who stay between 4 and 7 nights on a typical trip. Have: a popn. where the proportion with a particular attribute (“ ”) is . ¢ and get  ¨ ¡ Take: a random sample of size with the attribute.  ¨  £ ¢ Estimate for : , the sample proportion with ¢ the attribute. is a random variable. ¢  ¤ ¥ § is an unbiased estimator for , . (p. 300) ¢ ¢ £ £ ¢ STA 2023 c B.Presnell & D.Wackerly - Lecture 17 203  ¨ ¤ ¥ : ¢ £ ¢ Standard error of sheet) ¢ ¨ is “large”, is approximately normally £ If (p. 300,formula distributed. (p. 300) LOOKS FAMILIAR!!! Confidence   ¡ § standard errors table formula sheet ¡ ¢ 1 20 ¡ and 3   1 ) 3 20 ) 3 1 40 ) formula sheet ¦ estimator  £ ¢ ¡ Interval for a Proportion, ¤¢ §§¡ ©!¥ ¤ Large Sample . ¡ ¡ ¨  ¡ ¢ 2 ¨ ¢  $ §  ¡ ¢ STA 2023 c B.Presnell & D.Wackerly - Lecture 17 © && ©  ¨ Ex.: #7.79 p. 316 Survey of 204 post-retirees. &)¢ said that they stayed away from home between 4 $ %(  and 7 nights on trips. Find a confidence interval for the true proportion of post-retirement travelers who stay between 4 and 7 nights on a typical trip. ¥    & ¥    ¨  £ Confidence interval : ¢  or equivalently )£( 5   & © 5 . 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 . STA 2023 c B.Presnell & D.Wackerly - Lecture 18 205 Thought: When a man is wrapped up in himself, he makes a pretty small package – (John Ruskin) Assignments : Today: P. 306 – 308, 322 – 326 For Thursday: Exercises 7.51, 7.53, 7.59, 7.64, 7.67, 8.7, 8.9, 8.10, 8.13 Monday: OPTIONAL review day Tuesday: EXAM 2 Wednesday: P. 328 – 332 Tuesday, 4/3/01: EXAMS RETURNED We cannot give out grades over the phone. STA 2023 c B.Presnell & D.Wackerly - Lecture 18 206 Last Time: Confidence Interval for a $ %( ( ¢ ¦¥ ¤ £¡ ¢ PARAMETER   ¡ $§ formula sheet 1 40  1 ) 3 40 ) 3 1 20 α/2 1−α §   §   § ¤ ( § (P. 283), LARGE sample: ¨   ' ¢ ¨  $ §   – Population Proportion,   § – Population mean, ¦ ) α/2 53 formula sheet ¦ standard errors table estimator (P. 300), LARGE sample: ¡ ¡ 5 ¢ ¨  $ §  ¡ ¢ STA 2023 c B.Presnell & D.Wackerly - Lecture 18 207 Choosing the Sample Size Want: § Estimate the population mean, , by using the  sample mean, . © $  with units of the true value of § Estimate to be within confidence.  Question : If the population standard deviation is ¨ , how large should “ ”, the sample size be? &¢  .025 .025 .95   5 ¢ § § § of "# # "#   5 ¢   75 ¢ is within   ¤ § That is : of the time STA 2023 c B.Presnell & D.Wackerly - Lecture 18  ¢  95% of time.   ©  © # © ¨  ¨  ¨ &  5 ¢   75    ¨ 5 ) 0£ ¢  © units of 95% of the time.  of §  75 Want : is within # Want : to be within § Know: 208  5 £  ¦ &¤ ) ¡ 5 £&¤ ) © 5 &  && © ¨ 5 &  ©&& ¨ ¨ ¢   5 ¢ STA 2023 c B.Presnell & D.Wackerly - Lecture 18 in prev. example) § to within “ ” units with ¥ (p. 307) ¤¢ ¨  $ § ¨ Need : at least a “ballpark value for from past study or pilot study if available. , and use or ¡ – guess the range, ¡ ¦ – use ' and solve for  confidence level   If I want to estimate bound I want ( © “” 209 £ ' ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 18  § Ex. Want to estimate 210 average cost of freshman year to within $500 with prob. .95. Know – cost between have to be?  ¨ $4800 and $13,000. How large does ¥  & ¥    !   § ' ( '¥( &  £ '   ¡  ¨ ( '( &  ¨ 5  ¦ © ( ¤ ¡  5  © ( ¤ ¨   ( ' (  ¨  ¨   75 ¢  75 ¢ STA 2023 c B.Presnell & D.Wackerly - Lecture 18 211 Tests of Hypotheses: Section 8.1 Ex. : A manufacturer of light bulbs would like to claim that the average lifetime of our bulbs exceeds 1325 hours.  £  ¨ Put a random sample of bulbs on test, record time to burnout for each. ©© && ¦ ( ©¢   Data : Can the claim be made? (VALIDLY) Choose between two HYPOTHESES. 5  &©¢ (NULL hypothesis) &©¢ (ALTERNATIVE hypothesis)  §   § § ¢ ¢ ¡ £ We do not know the true value of true mean lifelength of the manufacturer’s bulbs. ( FIXED, but UNKNOWN ) Decision is to be made using sample data. STA 2023 c B.Presnell & D.Wackerly - Lecture 18 212 Sample DOES NOT contain ALL of the information about the POPULATION. Using sample data to make decision – always the possibility of making an ERROR. (See Table 8.1, p. 325) Reality Decision Ho true Ha true Accept Ho Correct Type II error Reject Ho Type I error Correct Consider the lightbulb example Type I Error – Claim when really Type II Error – Claim when really STA 2023 c B.Presnell & D.Wackerly - Lecture 18 213  (p. 323) 5 ¡ Type I Error  ¥ 3 40 ) 1 alpha  (p. 325) 5 ¡ Type II Error  ¢ 3 40 ) 1 beta ¥ and are PROBABILITIES ¢ Note: ( numbers between 0 and 1 ) ¨ is a fixed sample size. ¥ Suppose Ideally, want BOTH How about both very small? Type I Error we  ¡ ¡ ¥ ” OFTEN. ¡ – Thus, “accept  must decide to “reject . – If we want to REDUCE to be ¢ and ” – Leading to an OFTEN. in ¡ Type II Error   ¢ STA 2023 c B.Presnell & D.Wackerly - Lecture 18 ¨ For fixed 214 . 2 2 ¢ move in ¢ ¥ ¥ ¥ ¢ ¡ and directions. Our strategy ¥ Choose small enough to be “CONVINCING” – If we decide that the ALTERNATIVE IS CORRECT The ONLY type of error possible is a ¢ ERROR. The PROBABILITY of making a TYPE I error ¥ – By making . SMALL we our confidence in our conclusion to ACCECT £ ¢ is . STA 2023 c B.Presnell & D.Wackerly - Lecture 18 215 The hypothesis of MAIN INTEREST is the ALTERNATIVE HYPOTHESIS, or the Research Hypothesis. – What we are “trying to prove” in an objective, fair manner £ – Denote by . (p. 322) The “other” hypothesis is called the NULL HYPOTHESIS ¡ – Denote by . (p. 322) In out set-ups The NULL hypothesis is always that a PARAMETER EQUALS the value used in specifying the ALTERNATIVE or “RESEARCH” hypothesis. &©¢ (ALTERNATIVE hypothesis) (1) &©¢ (NULL hypothesis) (2) §   § ¢ ¢ £ ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 18 216  ¡ Type I error  ¥ £ when true ¡  ¡ accepting  ¥  ¥   ¥ when ¡ saying   SIGNIFICANCE LEVEL of the test, LEVEL of ¥ the test. ¥ for the ’s that we pick. ¢ The test that we will discuss have the SMALLEST ’s ¡ saying what we would like to say when we should not ...
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