Confidence Intervals and Hypothesis Testing Basic Concepts: Let s start this important chapter with a problem:

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1 Cnfidence Intervals and Hypthesis Testing Basic Cncepts: Let s start this imprtant chapter with a prblem: The Twn f Newtn sits next t tw rivers, the Sac and Vanzetti. Newtn is made up f 168 land parcels, all the same size. The twn is shwn t the right. The clrs refer t the use f the land and is shwn in the legend belw. There is nt enugh mney fr the twn t run a full census. It is an expensive prcess and many peple need t be hired. It is decided t estimate the ppulatin f the twn by sampling techniques. It is thught that a sample f 21 parcels f land will be sampled which is ne-eighth f the actual number f parcels. S an SRS f 21 samples will be dne: Using the simulatin, let s chse an SRS City Hall Sample mean: x Sample St. Dev: s: We are interested in µ, the unknwn ppulatin mean f the 168 parcels. If we can find that, we can multiply it by 168 and we will have ur ppulatin. The thught prcess is that if x =, then µ = as ur sample mean x is an unbiased estimatr f ur ppulatin mean µ. We expect µ t be apprximately the value f x. Hwever, if we tk anther sample f 21 land parcels, we wuld get a different x and thus a different estimatin f µ. Recall what the Central Limit Therem tells us abut sampling distributins: 1) x has a nrmal distributin. If we were t take many samples f 21 parcels, the result wuld be bellshaped and fllw ur % rule. 2) The mean f ur nrmal sampling distributin is the same as ur unknwn ppulatin mean µ. " 3) The standard deviatin f x fr ur 21 parcels f 21 parcels is where " is the standard 21 deviatin f all 168 land parcels. Let us suppse that the standard deviatin f ppulatin parcels is " = Then the standard deviatin f x " is then 21 = = Imprtant: Nte that this suppsitin f knwing " is cmpletely unrealistic. If we knew ", then we d knw µ and this prcess wuld be unnecessary. We are ding this t understand the prcess f hypthesis testing and cnfidence intervals. Chapter 11 will deal with the realistic situatin that we dn t knw the value f " Stu Schwartz

2 The picture t the right shws the sampling distributin f x. In repeated samples f size 21, the sample mean x wuld vary accrding t the nrmal distributin with mean equal t the unknwn µ and standard deviatin is The different values f appear alng the axis and the nrmal curve shws hw prbable values are. x these The picture t the right is anther picture f the sampling distributin. The % rules says that in 95% f all samples, x will be within tw standard deviatins f the unknwn ppulatin mean µ. That is, x will be within 14.1 f µ in 95% f samples. all S in 95% f all samples, the unknwn µ lies between µ "14.1 and µ Our sample gave x =. We say that we are 95% cnfident that the unknwn parcel mean lies between x = and x =. The grunds fr ur cnfidence lies in the fact that nly ne f the tw pssible facts are true: 1) ur true µ lies between ur x and x % f all ur samples shuld d s. 2) the true µ des nt lie between ur x and x Only 5% f ur samples shuld d s. VERY IMPORTANT: Remember, yu are basing this n ne value f x. We d nt knw whether ur sample is ne f the 95% fr which the interval x ±14.1 catches µ r ne f the unlucky 5%. The statement that we are 95% cnfident that the unknwn µ lies between and is a quick way f saying: we gt these numbers that gives crrect results 95% f the time. The interval f numbers between x ±14.1 is called a 95% cnfidence interval. It has the frm: Cnfidence Interval = Estimate ± margin f errr The estimate x is ur guess fr the unknwn parameter µ. The margin f errr ±14.1 shws hw accurate we believe ur guess is, based n the variability f the estimate. This is a 95% cnfidence interval because it catches the unknwn µ in 95% f all pssible samples. In rder t understand the prcess, we are ging t use the simulatin prgram and take 10 samples. We will then create ten 95% cnfidence intervals. Then we can extraplate thse values by multiplying by 168 t give a 95% cnfidence interval fr the twn f Newtn Stu Schwartz

3 Sample mean x x x ( x "14.1) 168( x +14.1) S a 95% cnfidence interval can be interpreted like this: Yu need t knw and understand the wrding. If the prcedure were repeated many times, 95% f the cnfidence intervals wuld catch the true mean. The prbability that ur prcedure will generate a cnfidence interval cntaining the true mean is 95%. We are 95% cnfident that ur true mean lies in ur interval. It des Nt say: 95% f the ppulatin lies in the cnfidence interval. The prbability that the true mean is in ur cnfidence interval is 95%. Again, remember, yu will never d this sampling prcess 100 times r even 10 times. Yu will d it exactly nce. Yu will be making a judgment based n that ne sample. The prcess will generate an interval that will catch the true mean µ 95% f the time. Example 1) A pll interviewed 1,500 men randmly selected frm the United States and fund that 64% watched at least ne NFL ftball game a week. The pll annunced a ± 3 percentage pints margin f errr fr 95% cnfidence in its cnclusins. a) What is the 95% cnfidence interval fr American men watching NFL games? b) Explain in crrect wrding what this says. Example 2) A gas statin lks at an SRS f 150 custmers ver a week and finds that the average number f gallns a custmer pumps is 8.4 with a margin f errr f 2.15 gallns fr 95% cnfidence. a) What is the 95% cnfidence interval fr custmer purchasing gas at this statin? b) Explain in crrect wrding what this says. Example 3) The price f a tw-liter bttle f Pepsi natinwide has a standard deviatin " f 23.4 cents. A randm sample f 50 bttles fund an average price f $1.64. a) What is the 95% cnfidence interval fr the price f a tw-liter bttle f Pepsi. b) Explain in crrect wrking what this says Stu Schwartz

4 Cnfidence intervals d nt have t be 95% althugh that is usually what is given in many statistical prblems. Other cnfidence intervals are 90% and 99%. 90% cnfidence intervals are less cnfident than 95% while 99% cnfidence intervals are mre cnfident. We use 95% unless therwise stated. Fr instance, in example 2 abve, we fund the average number gallns that a custmer pumps is 8.4 with a margin f errr f 2.15 fr 95% cnfidence. The fllwing table shws a pssible margin r errr and cnfidence interval fr nt nly 95% but 90% and 99% as well. Cnfidence x Margin f errr Lwer bund Upper bund 90% % % Ntice hw the margin f errr ges dwn when the cnfidence ges dwn and vice versa. If we d nt need t be super-cnfident f ur result, we d nt need a large margin f errr. But if we need t be extremely cnfident f ur result, then we need a large margin f errr. We will cme back t this relatinship. We can have any percentage cnfidence levels althugh 90%, 95%, and 99% are the mst usual. S the rather cmplicated f a cnfidence interval is as fllws: Cnfidence Interval A level C cnfidence interval fr a parameter is an interval cmputed frm sample data by a methd that has prbability C f prducing an interval cntaining the true value f the parameter. S here is the methd f finding a C cnfidence interval fr a ppulatin mean. Draw an SRS f size n frm a ppulatin having an unknwn mean µ and knwn standard deviatin ". A level C cnfidence interval fr µ is: # x ± z *% $ " & n ' ( z * # " & % $ n ( is the margin f errr. ' The value z* is a value fund at the bttm f table A at the back f yur bks. The typical values fr z* are: 90%: %: %: Example 4) Questinnaires were sent t 180 teachers in the Philadelphia area and 155 were returned. The average number f years f teaching f these 155 teachers was 13.6 years. Assume that the ppulatin standard deviatin f years teaching is 7.3 years. Find a) a 95% cnfidence interval b) a 90% cnfidence interval c) a 99% cnfidence interval d) Explain in wrds what yur 95% cnfidence interval says in the cntext f the prblem Stu Schwartz

5 Example 5) The number f French fries in a Medium cntainer f fries at McDnald s is nt always the same. Here are the number f fries fr a sample f 30 rders in different Bucks Cunty McDnalds a) We expect the distributin f fries t be nrmal. Make a stemplt r histgram f these 30 values and describe its shape. b. Suppse that McDnalds published the standard deviatin " f the number f fries in a medium cntainer t be 4.2 fries. Find a 98% cnfidence interval fr the number f fries. c. Explain in wrds what b) abve means in the cntext f the prblem. d. Wuld yu trust the cnclusin in part b if all the fries came frm ne McDnalds? Why? Hw cnfidence intervals behave Typically a persn perfrming an bservatinal study chses the cnfidence he desires and the margin f errr fllws frm this chice. We usually want high cnfidence and a small margin f errr, but we cannt have bth. There is usually a trade-ff. If we ask fr high cnfidence, we have t allw urselves a large margin f errr. Example: If I want t predict yur average in a curse with 99% cnfidence, I might say that I am 99% cnfident that yu will get a 75% with a margin f errr f 25%. That is saying that we are 99% cnfident that yu will get between 50% and 100%. Ntice that this desn t say much ther than yu will prbably pass the curse. If we want a small margin f errr, we have t ask fr a smaller cnfidence level. Example: If I want t predict yur average with a margin f errr f 2 pints, I might say that I am 50% cnfident that yu will get a 92% with a margin f errr f 2 percentage pints. That is saying that I am 50% cnfident that yu will get between a 90 and 94 in the curse. Again, the small range is impressive but with 50% cnfidence, I am nt very cnfident at all. It is a cin flip. # Since the margin f errr is z * " & % (, we find that the margin f errr gets smaller when either: $ n ' z* gets smaller (which is the same thing as saying that the cnfidence level is smaller, T btain a smaller margin f errr, yu must be willing t accept lwer cnfidence. " gets smaller. The standard deviatin " is the measure f variatin in the ppulatin. If " is small, it is easier t pin dwn µ. n gets larger. Increasing the sample size n reduces the margin f errr. We must take 4 times as many bservatins t cut the margin f errr in half Stu Schwartz

6 Smetimes we wish t establish a specified margin f errr fr a certain cnfidence level. That fixes z* and " certainly cannt change. The nly way we can achieve what we want is t change n, the sample size. Sample Size fr Desired Margin f Errr T determine the sample size n that will achieve a cnfidence interval fr a ppulatin mean with a specified margin f errr m, use the expressin belw t be less than r equal t m and slve fr n. The answer shuld be runded up t the next integer. # z *% $ " & ( ) m n ' Example 6) In the French fry prblem abve, hw many samples wuld we need t take in rder t have the margin f errr be less than 0.75 fries with 95% cnfidence? Example 7) The standard deviatin f purchases made at a Wawa stre is $2.11. We wish t predict the average price f a Wawa sale with 98% cnfidence with a margin f errr f less than 25 cents. Hw many purchases must we sample? If we find that we can accept 90% cnfidence, then hw many shuld we sample? Finally, sme warnings: # Our handy-dandy frmula fr prducing cnfidence intervals: x ± z *% $ The data must cme frm an SRS. " & ( is fraught with dangers: n ' The margin f errr in a cnfidence interval is based nly n randmness and nt because f issues f undercverage and nn respnse. The frmula is nt accurate fr prbability samples mre cmplex than an SRS. Stratified and multistage samples have their wn set f frmulas and are nt cvered in this curse. If there is any bias in yur sample, this and any methd is useless. Garbage in-garbage ut. Because x is strngly influence by extreme bservatins, utliers can have a large effect n the cnfidence interval (as we saw in hmewrk prblem 8). Outliers shuld be remved if pssible. There are prcedures t deal with utliers t generate cnfidence intervals but again, they are nt cvered in this curse. If the sample size n is small and the ppulatin is nt nrmal, the frmula will nt be accurate. It is best t have a sample size n 15 in these prcedures, especially if yur sample shws strng skew. Again, yu must knw the standard deviatin " f the ppulatin. This is unrealistic and thus we rarely use this exact prcedure in real-life. We are nly ding it t understand the prcedure s when we get t chapter 11 and are faced with prblems where we d nt knw ", yu will understand the prcess Stu Schwartz

7 Using the Calculatr The TI-84 has built-in tests fr inferences (which is what taking cnfidence intervals is). They are fund in the STAT TESTS menu. We want #7:ZInterval. Which gives a screen similar t the ne t the right: Let s d # 3 frm the examples abve using the calculatr: The price f a tw-liter bttle f Pepsi natinwide has a standard deviatin " f 23.4 cents. A randm sample f 50 bttles fund an average price f $1.64. In this case, we are given summary statistics, nt the actual data itself. S In the ZInterval menu, yu use Stats. There are 4 inputs, ",x,n, and C - Level, and we knw all f them. Simply input them, remembering t put the cnfidence level in decimal frm: Then press Calculate. The calculatr gives yu the interval in up t 4-decimal place accuracy. Yur answers might nt cmpletely match the nes dne by frmula because in thse, yur values f z* was nt cmpletely accurate. Thus the calculatr s answers will always be mre accurate than the nes dne by frmula and it is recmmended that yu use the calculatr. Hwever, it is vital that yu understand the prcess f cnfidence intervals as yu will be tested n it in the AP exam. Als nte that althugh the calculatr desn t directly give the margin f errr, yu culd easily find it by either subtracting x frm the upper bund r subtracting the lwer bund frm x. If yu are given raw data, the calculatr can find cnfidence intervals n thse as well. Let s repeat prblem # 5 abve. The number f French fries in a Medium cntainer f fries at McDnald s is nt always the same. Here are the number f fries fr a sample f 30 rders in different Bucks Cunty McDnalds. Suppse that McDnalds published the standard deviatin " f the number f fries in a medium cntainer t be 4.2 fries. Find a 98% cnfidence interval fr the number f fries When we did this prblem, we had t calculate ur value f x and it was prudent t examine the data fr utliers. We can, f curse, d this by inputting ur data int a list and generating a histgram. When we chse ZInterval frm ur STAT TESTS menu, we nw chse Data. Yu still input ", yur chice f Lists, and the cnfidence level. The upper and lwer bund are given alng with x and Sx, the sample standard deviatin. Again, yu can easily find the margin f errr by either subtracting x frm the upper bund r subtracting the lwer bund frm x. It is strngly recmmended that yu red hmewrk prblems using the calculatr Stu Schwartz

8 Hypthesis Testing: A cmpany puts ut a diet drink that suppsedly will help peple lse weight immediately. Ten different peple use the prduct in a carefully cntrlled experiment fr a week and the fllwing are their weight lsses: Mst peple lst weight. But the lsses are small and tw peple (the negative numbers) actually gained weight. Des this data present gd evidence that the diet drink actually makes peple lse weight? Fr this prblem, let s assume that " = 2.5 (althugh, nrmally we wuld nt knw that infrmatin). Nte that the average weight lss is x = 1.6. That s nt a large lss and ten different peple participating in the experiment wuld mst prbably have gtten a different result. S, a significance test asks the questin: Des the x = 1.6 represent a real weight lss caused by the diet drink (if lurking variables are cntrlled)? Or Culd the x = 1.6 simply have happened by chance? T start the prblem, we need t identify the parameter f interest. In this case the parameter f interest is µ which represents the average weight lss fr the first week if the entire ppulatin were using the diet drink. As we said befre, µ exists but there is n way t actually find it. We then make what we call is ur null hypthesis. The Null (meaning nne) hypthesis says that that the diet drink des nt wrk, that any weight lss was by chance. We write the null hypthesis like this: H 0 : µ = 0 We then make what we call is ur alternative hypthesis: We believe that the diet drink actually wrks and that peple will lse weight with it. S we believe that µ (ur parameter f interest) is psitive: We write the alternative hypthesis like this: H a : µ > 0 The reasning fr a hypthesis test is this: We start with the idea that the null hypthesis is true that the diet drink des nt cause peple t lse weight. But we have this x = 1.6 (10 peple lst an average f 1.6 punds ver a week in a carefully cntrlled experiment). It culd have happened by chance. The questin is: is that x = 1.6 surprisingly large assuming that the null hypthesis H 0 is true. By analgy: if smene is n trial fr murdering his wife, the premise is that the persn is nt guilty. The prsecutin then tries t pke hles in that argument. Prsecutr: If yu did nt cmmit the murder, why was yur fingerprints n the knife? Accused: I bught the knife s my fingerprints wuld be n it. Prsecutr: If yu didn t cmmit the murder, why did yu recently take ut a millin dllar insurance plicy? Accused: It is just cincidence. Prsecutr: On the knife, we fund traces f yur bld as well as yur wife s, cnfirmed by a DNA test. If yu didn t cmmit the murder, what is yur bld ding n the knife? Accused Lawyer: DNA tests can be wrng, can t they? Prsecutr: DNA tests are wrng 1 in 10 millin times. Accused Lawyer: Then this is that ne time. Prsecutr: (sarcastically) Quite a string f cincidences, aren t they? Stu Schwartz

9 S, again, ur questin is: is the diet drink desn t wrk (ur null hypthesis), hw is it that we gt an average weight lss f 1.6 fr 10 peple? T answer the questin, we start with the premise that the sampling distributin f x is nrmal with mean µ = 0 (ur null hypthesis) and standard deviatin " n = 2.5 =.791 (this cmes frm the Central Limit Therem). 10 As yu can see frm the diagram, ur value f x =1.6 is slightly mre than tw standard deviatins abve the mean. That is far ut t the right! If the null hypthesis is true, then it gt there simply by chance. If the null hypthesis is nt true (and the alternative hypthesis is), then x =1.6 gt there nt by chance but simply because the diet drink causes peple t lse weight. The further ut x is in the psitive directin, the mre cnvinced we are that the ppulatin mean µ is nt zer but psitive. We measure the strength f the evidence against H 0 by the prbability under the nrmal curve t the right f the bserved x. This is the shaded area in the diagram. We call this the P-value. The P-value is the prbability f a result as far ut as the result we gt. The lwer this prbability, the mre surprising ur result and the strnger the evidence against the null hypthesis. T find the P-Value, yu need t find the z-scre fr ur value f x =1.6. This is called the test statistic. Recall frm chapter 2 that z = x " µ. Hwever, since we are using a sample f 10 peple, ur frmula changes # t: z = x " µ. # n Ding the calculatin, we get z = Mving t yur table A, we find the value f the shaded sectin in the chart as Hwever, ur P-Value in the diagram abve is the shaded regin t the right f x =1.6 which is thus =.022. What this says is the prbability that ur value f x =1.6 r greater assuming the null hypthesis is true is 2.2%. It is certainly a surprising result if the null hypthesis is true Stu Schwartz

10 What cnstitutes surprising? We usually use a standard f 5%. Anything belw that is surprising and thus statistically significant. We call this standard f 5% the Greek letter alpha ("). It can be any value but " =.05 is usually the standard unless we want a tugher standard, in which case " might be.01 r.01. In this case ur P-Value is.022 which is less than.05. Since this is statistically significant, we decide that we have enugh evidence t reject the null hypthesis H 0 and thus accept the alternative hypthesis H a. We then write ur cnclusins in English and in the cntext f the prblem: YOU SAY: There is gd evidence t suggest that the diet drink causes weight lss in the first week. Here is what yu DON T SAY: This prves that the diet drink wrks. We have shwn that the diet drink causes weight lss. (the underlined wrds are red-flags) This establishes that the diet drink causes an average weight f 1.6 lbs. This whle prcess is called hypthesis testing and this specific test is called a z-test. Yu will learn many ther types f hypthesis tests befre we are dne. The z-test lks like this in general: T test the hypthesis H 0 : µ = µ 0 based n an SRS f size n frm a ppulatin with unknwn mean µ and knwn standard deviatin ", cmpute the z-test statistic: x " µ in terms f a variable x having the standard nrmal distributin fr a test f H 0 against: # n H a : µ > µ 0 ( right tailed test): H a : µ < µ 0 ( left tailed test) H a : µ " µ 0 ( tw - tailed test) Here is the general prcedure fr a z-test: 1) Decide n yur parameter f interest µ and write what it represents in English. 2) Yur chice f test is a z-test. We knw it s a z-test because we knw ". 3) Write dwn given infrmatin: In this case it is n, x, and ". Yu might have t calculate x. 4) All tests have cnditins: Let s nt wrry abut that yet ther than the fact ur data cmes frm an SRS. 5) Write the null hypthesis in bth English and variables. The null hypthesis is always in the frm H 0 : µ = µ 0 6) Write the alternative hypthesis in bth English and variables. The null hypthesis is always in the frm: H a : µ > µ 0 ( right tailed),h a : µ < µ 0 ( left tailed),h a : µ " µ 0 ( tw - tailed) 7) Calculate yur test-statistic z by the frmula abve. 8) Remember that z is the number f standard deviatins abve r belw µ, shade in area n the nrmal curve. Als establish ". If nt given in the prblem, use " =.05. 9) Find the p-value by using the Table A chart. If yu are using the psitive z-chart, subtract yur result frm 1. If yu are ding a tw-tailed test, yu must duble the p-value. Decide if the p-value is less than ". 10) If p-value <", it is a significant result. Reject H 0 and accept H a. If p-value >", it is nt a significant result. We fail t reject H 0. (we dn t accept H 0, just fail t reject it). (If a persn is fund nt guilty, we dn t say he is inncent, just that there is nt enugh evidence t cnvict) 11) Make yur cnclusins in the cntext f the prblem using prper English. Use the wrding mentined abve. Use wrds like: the evidence suggests, the evidence strngly suggests, nt enugh evidence Stu Schwartz

11 T help yu with this strange prcedure, I have prvided a chart fr the prblems we will tackle. It will guide yu as t what yu need t d fr each prblem. 1. The head f the physics department has fund that ver the years, students in the final exam average 70% with a standard deviatin f 6.3%. One f his instructrs seems t have a greater number f failures in his classes s the department head decides t d a hypthesis test. He takes a randm sample f 20 final exams frm that teacher and finds that the average scre is 67.3%. Is there evidence at the 5% level that these students have dne wrse than the general ppulatin f physics students? 1. Parameter f Interest: 2. Chice f test: 3. Given infrmatin: n = x = " = 4. Check f cnditins: Statement f necessary cnditins: Verificatin f satisfactin: 5. Null Hypthesis: H 0 : (English) 6. Alternative Hypthesis: H 0 : (symbls) H a : (English) H a : (symbls) 7. Test Statistic: Frmula: Value: 8. Test: Level f Significance " = Sketch f sampling distributin assuming that H 0 is true. Identify the lcatin f the test statistic in the sketch and shade the apprpriate regin fr the p-value 9. p-value: Exact p-value: Recnciliatin with Critical Value f Rejectin: 10. Recmmended Decisin: Regarding significance: Regarding H 0 : 11. Interpretatin: (English) Can we claim that the reasn this teacher s students haven t dne as well is because f the teacher? Wh r why nt? Stu Schwartz

12 2. A light bulb cmpany prclaims that its new flurescent bulbs will average 225 hurs f light. Cnsumer Reprts decides t test at the 2% level t see if there is evidence that the claim is nt true. They test 12 light bulbs and keep them n until they burn ut. Assume a ppulatin standard deviatin f 8. The results are: Hurs Parameter f Interest: 2. Chice f test: 3. Given infrmatin: n = x = " = 4. Check f cnditins: Statement f necessary cnditins: Verificatin f satisfactin: 5. Null Hypthesis: H 0 : (English) 6. Alternative Hypthesis: H 0 : (symbls) H a : (English) H a : (symbls) 7. Test Statistic: Frmula: Value: 8. Test: Level f Significance " = Sketch f sampling distributin assuming that H 0 is true. Identify the lcatin f the test statistic in the sketch and shade the apprpriate regin fr the p-value 9. p-value: Exact p-value: Recnciliatin with Critical Value f Rejectin: 10. Recmmended Decisin: Regarding significance: Regarding H 0 : 11. Interpretatin: (English) Althugh we are nt tld specifically that an SRS f bulbs was taken, we g thrugh with the test. This ccurs in the fllwing chapters: when cnditins are nt met, we take nte f it and prceed with the test anyway. Remember that I am giving yu this chart as a cnvenience. After this chapter, yu must prvide it. This is what yu wuld be expected t write fr this prblem n a test and the AP test: Parameter : µ : the average lifetime f the cmpany's lightbulb (hrs) z " test H 0 : µ = 225 p " value =.077 >.02 x = ,n =12,# = 8 H a : µ $ 225 fail t reject H 0 SRS is assumed z = "1.77 N evidence that the cmpany's bulb desn't average 225 hrs Stu Schwartz

13 Using the Calculatr fr Hypthesis Testing Yes, the Ti-84 can run a hypthesis test. This is less impressive than it sunds. It will nt create yur hyptheses fr yu, decide n significance, and make a decisin as t yur hyptheses. But it can d the calculatins f yur test-statistic and the p-value. When we get t mre cmplicated prblems, that will be a huge savings as yu will nt have t d the number crunching. Let s g back t the physics prblem: The head f the physics department has fund that ver the years, students in the final exam average 70% with a standard deviatin f 6.3%. One f his instructrs seems t have a greater number f failures in his classes s the department head decides t d a hypthesis test. He takes a randm sample f 20 final exams frm that teacher and finds that the average scre is 67.3%. Is there evidence at the 5% level that these students have dne wrse than the general ppulatin f physics students? T use the calculatr, we start the same way we did fr cnfidence intervals: STAT TESTS. We want 1:Z-Test. Once in that screen, yu need t tell the calculatr that yu are using Summary Statistics (Stats) and then type in yur µ 0,", x, and n values. Since this is a left tailed test, yu want t chse µ < µ 0. Once yu d this, yu have a chice f Calculate r Draw. Pressing Calculate will give yu tw imprtant pieces f infrmatin that yu will reprt, the z-statistic as well as the p- value. Pressing Draw will give yu the same infrmatin but in a graphical way. If yu use this ptin, be sure that there are n equatins being graphed in Y =. It is up t yu t, write hyptheses, decide n significance and write cnclusins. If yu are given the actual data rather than summary statistics, the calculatr can handle it as well. Let s d prblem # 2 n the previus page: A light bulb cmpany prclaims that its new flurescent bulbs will average 225 hurs f light. Cnsumer Reprts decides t test at the 2% level t see if there is evidence that the claim is nt true. They test 12 light bulbs and keep them n until they burn ut. Assume a ppulatin standard deviatin f 8. The results are as fllws First input yur data int L1 (r any ther list). Press STAT TESTS and again, chse 1:Z-Test. But nw, press Data (as yu have data and nt summary statistics). As befre, yu type in µ 0 and " but d nt need x and n as they will be fund by lking at the list f data (which yu specify as well). Finally, chse whether it is a tw, left, r right tailed test. (this is a tw-tailed test). As befre, chse Calculate r Draw and yu will get yur z- statistic and p-value. x and Sx (the sample standard deviatin) are displayed as well. These are generated ff yur list. As befre, it is up t yu t, write hyptheses, decide n significance and write cnclusins Stu Schwartz

14 Type I and Type II Errrs When we perfrm an hypthesis test, we must make a decisin whether t reject r fail t reject the null hypthesis. Based n prbability thery and the value f ", we will mst likely be right. But smetimes we are wrng. When we make a wrng decisin, we make either a Type I r Type II errr. In general, hypthesis testing has us making a decisin abut the ppulatin based n a sample. The truth abut the ppulatin exists but it is either difficult r impssible t find. In ur hypthesis tests s far, that decisin is abut the ppulatin mean µ but it desn t have t be. The fllwing diagram sums up the fur situatins that can crp up based n the truth f ur cnjecture and the decisin we make based n the sample. Truth abut the ppulatin H true H a true Decisin based Reject H Type I errr Crrect Decisin n sample Accept H (r fail t reject) Crrect Decisin Type II errr A Type I errr is made when the null hypthesis H is actually true but the alternative hypthesis is chsen. A Type II errr is made when the alternative hypthesis is actually true but the cnservative step f accepting the null hypthesis is actually made. Example 1: I have gd reasn t believe that this student cheated in an exam. Shuld I punish him? H : the student did nt cheat H a : the student cheated Truth abut the student (that nly the student knws) H true (student did nt cheat) H a true (student cheated) Decisin based Reject H Type I errr cme t the cnclusin that the student cheated when he did nt Crrect Decisin n evidence Accept H Crrect Decisin Type II errr cme t the cnclusin that the student did nt cheat when he really did. Ramificatins f these errrs: Type I errr punishing a persn wh is truly inncent. Student is n lnger trusted, lses privileges, etc. Type II errr student gets away with cheating and perhaps thinks he always can. Later in life, this can lead t larger crimes. As a teacher, I have t guard against Type I errrs and try nt t let them happen. My philsphy is that if I make a Type II errr, then the student will learn later in life abut what happens when he cheats. It is his prblem, nt mine Stu Schwartz

15 When we reduce the prbability f a type I errr, we autmatically increase the prbability f a Type II errr and vice versa. In the previus prblem, if I want t reduce the chance f a type I errr (punishing a nncheater), I will need t let students g unless I am % sure f the fact that the student cheated. In that way, I will nly catch cheaters, but nw students wh did cheat but I cannt ttally prve will get away with it (a Type II errr). And if I want t reduce the prbability f a Type II errr (letting a cheater ff), I will punish anyne fr whm I have any suspicin f cheating. That will be sure f punishing a cheater but in the prcess, a number f students wh didn t cheat but may appear slightly suspicius will get punished as well (a Type I errr). Example 2: We sampled 100 pretzels that cme ff the assembly line f 100,000 and find that in general, they are t salty. Shuld we pull the entire batch? H : the pretzels are nt t salty. H a : the pretzels are t salty. Truth abut the Pretzels H true (pretzels are nt t salty) H a true (pretzels are t salty) Decisin based Reject H Type I errr Cme t the cnclusin that the pretzels are t salty when in fact they are fine. Crrect Decisin n histry Accept H Crrect Decisin Type II errr cme t the Ramificatins: cnclusin that the pretzels are nt t salty when in fact they are. Type I Errr: The cmpany is hurt. They end up thrwing away pretzels that are perfectly fine, lsing a lt f mney. Type II errr: The cnsumer may be hurt. They end up purchasing salty pretzels that they may nt expect and culd very well thrw them ut. The Type II errr may very well be wrse here. Althugh the Type I errr hurts the cmpany financially (thrwing ut 10,000 pretzels is expensive), the type II errr affects the cnsumers and if peple dn t like the salty pretzels, they may very well nt purchase thse pretzels again even thugh future batches may be fine. They als may nt purchase any ther f the cmpany s prducts. Terms Assciated with Type Errrs The prbability f a Type I errr is simply yur value f ". That is the prbability that we will reject the null hypthesis (accept the alternative) when the null hypthesis is actually true. The prbability f a Type II errr is mre cmplicated and is nt par f the curriculum. The Pwer f a test is 1 the prbability f a Type II errr. If the prbability f a Type II errr is.04, then the pwer f a test I =.96. If we increase ", then we reduce the prbability f a type I errr which then increases the prbability f a Type II errr which increases the pwer f a test Stu Schwartz

16 Example 3: Describe in wrds what H and H a are. Cnsider the decisin that yu have t make based upn yur cnjectures. Then make a chart similar t the nes I created and fill in wrds what the Type I and Type II errrs mean. Finally, describe the ramificatins f making these errrs within the cntext f the prblem and describe which f the 2 errrs are wrse (in yur pinin). My histry teacher hasn t quizzed us in awhile and very well may give us a quiz tmrrw. Shuld I study? H 0 : H a : Truth abut the quiz tmrrw Yur Reject H H true Type I errr cme t the cnclusin that H a true Crrect Decisin decisin Accept H Crrect Decisin Type II errr cme t the cnclusin that Ramificatins f a Type I errr: Ramificatins f a Type II errr: Which is wrse and why (pinin). Finally, we end the chapter with a discussin f making sense f statistical significance. When designing an experiment, we have t chse a level f significance ("). Usually it is 5% but can be lwer r higher. That decisin is based n hw much yu want t reduce the prbability f a Type I errr. In a murder trial, we want t really reduce the prbability f a type I errr (an inncent man put in jail r wrse) s we reduce ". The term is: beynd a reasnable dubt. Hwever, if the persn is n trial fr a lesser crime like pirating music, the level f " is increased. In a civil jury trial, nly 9 f 12 vtes are needed. In any hypthesis test, " represents the demarcatin line between significant and nt significant. In real life, it isn t that easy. If we are testing a bunch f student averages n whether they are greater than 80. The by sample had x = 84.5 and the p-value was.051 while the girl sample had x = 84.4 and the p-value was.049. The girl data is significant statistically while the by data in n significant statistically, but in reality there is n practical difference between the by and girl data. They are bth clse t being statistically significant with the girls slightly mre than the bys Stu Schwartz

17 Hmewrk Prblems: 1. The lifetime f African Cichlids (trpical fish) is unknwn but a sample f 75 cichlids were taken and its mean lifetime was 5.2 years with a margin f errr f 420 days fr 95% cnfidence. a) What is the 95% cnfidence interval fr the lifetime f African Cichlids? b) Explain in crrect wrding what this says. 2. A pll f 2,000 Americans was taken and the findings shwed that 52% percent preferred Obama ver McCain in an electin with a margin f errr f 3.5% pints fr 95% cnfidence. a) What is the 95% cnfidence interval fr the percentage f peple vting fr Obama? b) Explain in crrect wrding what this says. c) What des this say abut predicting the winner f the upcming electin? 3. Every week, a website displays the average price f a galln f regular gasline natinwide. This is based n a randm pll f 80 gas statins. Suppse the average price was $2.94 and histrically the standard deviatin f a galln f gas was 8.3 cents. a) What is the 95% cnfidence interval fr the price f a galln f gasline natinwide? b) Explain in crrect wrding what this says. 4. The lifetime f a AA battery varies but histrically, the standard deviatin f the lifetime f a battery is 2.9 hurs. 100 randm batteries were tested playing a CD player at the same vlume and the average battery lifetime was 65.6 hurs. a) What is the 95% cnfidence interval fr lifetime f a AA battery? b) Explain in crrect wrding what this says. 5. A ppular ink cmpany advertises that a cartridge f black ink will last between and pages with 95% cnfidence. a) What is the margin f errr? b) Explain in crrect wrding what this says. 6. An Internet pll fund that 64% f the adults plled said that they wanted the use f marijuana legalized. The pll prclaimed itself t have 95% cnfidence ± 2.5 percentage pints. a) Make a cnfidence statement abut the percent f all adults wh favr legalizing marijuana. b) Express reasns why there is dubt as t the accuracy f the pll Stu Schwartz

18 7. A study f the career paths f htel general managers sent questinnaires t an SRS f 175 htels belnging t majr U.S. htel chains. There were 123 respnses. The average time these 123 managers spent with their cmpany was years. Give the fllwing cnfidence intervals fr the mean number f years f majrchain htels have spent with their current cmpany. (Take it as knwn that the standard deviatin f time spent with their cmpany fr all general managers is 2.9 years.) a) 90% b) 95% c) 99% d) explain in wrds what b) abve says. 8. A large university des a test f its incming freshmen t determine their reading speed (number f lines read in a minute). Here are the number f lines per minute that a sample f 40 freshmen can read: Assume that the ppulatin reading speed " is a. We expect that the distributin f reading speed be clse t nrmal. Make a stemplt r histgram f the distributin f these 40 scres and describe its shape. b. There is clearly ne utlier. Further study shws that this persn tk a speed-reading curse. Make a decisin abut whether r nt t include the utlier and give a 96% cnfidence interval fr the mean reading speed fr the freshmen class in that university. c. Hw wuld b) change if yu made the ppsite decisin abut the utlier? 9. Here are measurements (in millimeters) f a critical dimensin n a sample f mtrcycle engine crankshafts. The data came frm a prductin prcess that is knwn t have standard deviatin " =.072 mm. The prcess means is suppsed t be µ = 225 mm but can drift away frm this target during time f prductin a. We expect that the distributin f these values t be clse t nrmal. Make a stemplt r histgram f the distributin f these 15 values and describe its shape. b. Give a 95% cnfidence interval fr the prcess mean at the time these crankshafts were prduced and describe yur findings in wrds Stu Schwartz

19 10. A bld pressure test is nt perfectly precise and the BP will vary frm day t day. Suppse that histrically, S. Tlik s bld pressure standard deviatin varies nrmally with " = 4.8. a. On March 1, his BP was taken nce and the result is 125. Give a 95% cnfidence interval fr his mean BP. b. On March 2, his BP was taken twice and the result was x = 125. Give a 95% cnfidence interval fr his mean BP. c. On March 5, his BP was taken 5 times and the result was x = 125. Give a 95% cnfidence interval fr his mean BP. d. On March 20, his BP was taken 20 times and the result was x = 125. Give a 95% cnfidence interval fr his mean BP. e) Explain what is happening t the margin f errr thrughut the mnth f March and why it is happening in the cntext f the prblem. 11. The standard deviatin f the lifetime f a certain brand f tire is 2,825 miles. The tire cmpany wishes t estimate the average lifetime f that tire with 95% cnfidence with a margin f errr f 1,000 miles. Hw many tires shuld it sample? 12. A refrigeratr cmpany is testing certain types f refrigeratrs. A setting f 3 n the refrigeratr shuld set the refrigeratr s internal temperature t 37 F. The histry f this refrigeratr shws that the standard deviatin f the temperatures n this setting is 1.2 F. The cmpany wishes t be very accurate s it decides t publish a 99% cnfidence interval with a margin f errr f 0.3 F. Hw many refrigeratrs shuld it test? Stu Schwartz

20 Sample Quiz: Suppse yu administer a certain aptitude test t a randm sample f 16 students in yur schl and that the average scre is 84. We want t determine the mean µ f the ppulatin f all students in the schl. Assume a standard deviatin f! = 7. 5 fr the test. We wish t calculate a 90% cnfidence interval f the mean. 1. What is z*? 2. What is the margin f errr? Shw hw yu gt it. 3. What is the 80% cnfidence interval fr the mean scre µ f the whle schl (2 decimal pl). 4. Draw the sample distributin shwing the key numbers n the sketch. 5. Write a sentence that explains the significance f the cnfidence interval. 6. If yu needed t d the prblem abve but with a 99% cnfidence interval, what wuld be the margin f errr nw? Shw hw yu gt it. 7. What is the 99% cnfidence interval? (2 decimal places) 8. What sample size wuld be need t have a margin f errr f at mst 4 pints with 95% cnfidence. Shw wrk Stu Schwartz

21 Hypthesis Testing: 1. A small cup f French fries in the cafeteria is suppsed t hld 55 fries when full. A student believes that the cafeteria wrker wh fills the cups is shrtchanging students. S he decides t d a hypthesis test. He takes a sample f 10 cups and cunts the number f fries. They are as fllws: Is there evidence at the 5% level that the cafeteria wrker is averaging less than 55 fries per cup? Use a ppulatin standard deviatin f 4 fries. 1. Parameter f Interest: 2. Chice f test: 3. Given infrmatin: n = x = " = 4. Check f cnditins: Statement f necessary cnditins: Verificatin f satisfactin: 5. Null Hypthesis: H 0 : (English) 6. Alternative Hypthesis: H 0 : (symbls) H a : (English) H a : (symbls) 7. Test Statistic: Frmula: Value: 8. Test: Level f Significance " = Sketch f sampling distributin assuming that H 0 is true. Identify the lcatin f the test statistic in the sketch and shade the apprpriate regin fr the p-value 9. p-value: Exact p-value: Recnciliatin with Critical Value f Rejectin: 10. Recmmended Decisin: Regarding significance: Regarding H 0 : 11. Interpretatin: (English) Stu Schwartz

22 2. In a New Yrk deli that makes very large sandwiches, a typical sandwich is suppsed t have 16 unces f meat in it (1 pund). The sandwich maker never weighs the meat and just puts it n by eye. The manager f the restaurant is cncerned that the sandwich guy is putting t much meat in the sandwich. He bases this n the fact that in a randm sample f 15 sandwiches, the average amunt f meat was 1.05 punds. Is there evidence at the 5% level that the sandwich maker is putting t much meat n the sandwiches? Use a ppulatin standard deviatin f 0.14 punds. 1. Parameter f Interest: 2. Chice f test: 3. Given infrmatin: n = x = " = 4. Check f cnditins: Statement f necessary cnditins: Verificatin f satisfactin: 5. Null Hypthesis: H 0 : (English) 6. Alternative Hypthesis: H 0 : (symbls) H a : (English) H a : (symbls) 7. Test Statistic: Frmula: Value: 8. Test: Level f Significance " = Sketch f sampling distributin assuming that H 0 is true. Identify the lcatin f the test statistic in the sketch and shade the apprpriate regin fr the p-value 9. p-value: Exact p-value: Recnciliatin with Critical Value f Rejectin: 10. Recmmended Decisin: Regarding significance: Regarding H 0 : 11. Interpretatin: (English) Stu Schwartz

23 3. An autmatic sda dispenser is suppsed t fill a cup up t a line. That line represents 32 unces f sda. There is cncern whether the sda dispenser is wrking prperly, whether it is actually giving the amunt f sda it is suppsed t. The machine is rated t have a standard deviatin in sda dispensed as unces. An SRS f 20 sdas were dispensed ver a perid f time and the results are as fllws: (Be careful) Is there evidence at the 5% level that the sda machine is nt averaging 32 unces f sda? 1. Parameter f Interest: 2. Chice f test: 3. Given infrmatin: n = x = " = 4. Check f cnditins: Statement f necessary cnditins: Verificatin f satisfactin: 5. Null Hypthesis: H 0 : (English) 6. Alternative Hypthesis: H 0 : (symbls) H a : (English) H a : (symbls) 7. Test Statistic: Frmula: Value: 8. Test: Level f Significance " = Sketch f sampling distributin assuming that H 0 is true. Identify the lcatin f the test statistic in the sketch and shade the apprpriate regin fr the p-value 9. p-value: Exact p-value: Recnciliatin with Critical Value f Rejectin: 10. Recmmended Decisin: Regarding significance: Regarding H 0 : 11. Interpretatin: (English) Is this the same questin as asking whether r nt the sda machine is wrking as it is suppsed t? Stu Schwartz

24 4. A certain brand f car is rated t get 30 miles per galln n the highway. A researcher believes the claim is faulty that it is t high. His cmpany drives a randm sample f 35 cars in identical highway cnditins fr a ttal f 500 miles and checks the fuel ecnmy. He finds that the average rating fr the cars is miles per galln. Using a ppulatin standard deviatin f 1.1 mpg, determine if there is evidence at the 2% level that the cmpany s claim f 30 mpg is t high. Als, find a 98% cnfidence interval fr the average mileage rating f the car and interpret the answer. 1. Parameter f Interest: 2. Chice f test: 3. Given infrmatin: n = x = " = 4. Check f cnditins: Statement f necessary cnditins: Verificatin f satisfactin: 5. Null Hypthesis: H 0 : (English) 6. Alternative Hypthesis: H 0 : (symbls) H a : (English) H a : (symbls) 7. Test Statistic: Frmula: Value: 8. Test: Level f Significance " = Sketch f sampling distributin assuming that H 0 is true. Identify the lcatin f the test statistic in the sketch and shade the apprpriate regin fr the p-value 9. p-value: Exact p-value: Recnciliatin with Critical Value f Rejectin: 10. Recmmended Decisin: Regarding significance: Regarding H 0 : 11. Interpretatin: (English) Stu Schwartz

25 Sample Quiz: Fd-Rite Supermarkets claim that a typical rder f $100 at that market will l save at least $5 n a cmpeting market. They take the rders f 12 randm shppers and purchase the same items in the cmpeting market and check ut the savings by purchasing in Fd-Rite. Fllw are thse savings Are yu cnvinced that claim f Fd-Rite is justified? Carry ut an apprpriate significance test. Use a ppulatin standard deviatin f Parameter f Interest: 2. Chice f test: 3. Given infrmatin: n = x = " = 4. Check f cnditins: Statement f necessary cnditins: Verificatin f satisfactin: 5. Null Hypthesis: H 0 : (English) 6. Alternative Hypthesis: H 0 : (symbls) H a : (English) H a : (symbls) 7. Test Statistic: Frmula: Value: 8. Test: Level f Significance " = Sketch f sampling distributin assuming that H 0 is true. Identify the lcatin f the test statistic in the sketch and shade the apprpriate regin fr the p-value 9. p-value: Exact p-value: Recnciliatin with Critical Value f Rejectin: 10. Recmmended Decisin: Regarding significance: Regarding H 0 : 11. Interpretatin: (English) Stu Schwartz

26 Assignment On Type I and type II errrs (t be handed in): Fr each statement belw, yu are t describe in wrds what H and H a are. Cnsider the decisin that yu have t make based n H and H a. Then make a chart similar t the nes I created and fill in wrds what the Type I and Type II errrs mean in the cntext f the prblem. Finally, describe the ramificatins f making these errrs within the cntext f the prblem and describe which f the 2 errrs are wrse (in yur pinin) and why. D any 4 yu wish. Prblems are t be written neatly n ntebk paper r wrd-prcessed. Yu are nt graded n yur pinin, just yur ability t describe the Type I and Type II errrs and their ramificatins. 1. We are fairly sure than this new drug can eliminate headaches. Let s put it n the market. 2. The teacher will never check hmewrk tmrrw. Maybe I shuldn t d it 3. In a big stre like Walmart, they will never catch me if I shplift. Shuld I 4. Saddam Hussein has weapns f mass destructin. Shuld we attack him? 5. Shuld I try and learn t ski? Whenever I try smething new, I end up hurting myself. 6. We in airprt security are cncerned with terrrism. We shuld check everyne carefully wh is flying. 7. There might be a bmb in the schl because smene called in a threat. We shuld evacuate. 8. I am nt sure if Heaven and Hell exist. Shuld that affect hw I live my life? Stu Schwartz