week.7.homework - A null and alternative hypothesis are...

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Unformatted text preview: A null and alternative hypothesis are given. Determine whether the hypothesis test is left-tailed, right-tailed, or two—tailed. Ho: p513 Ha: p313 It the alternative hypothesis contains the not—equal—to inequality symbol, then it is a two—tailed test. If the altemative hypothesis contains the less-than inequality symbol, then it is a left—tailed test. If the alternative hypothesis contains the greater—than inequality symbol, then it is a right—tailed test. Since the alternative hypothesis, Ha, has a greater—than inequality symbol, the hypothesis test is a light—tailed test. For the statement below, write the claim as a mathematical statement. State the null and alternative hypotheses and identify 1wrhich represents the claim. A laptop manufacturer claims that the mean life of the battery for a certain model of laptap is more titan 10 hours. Rewrite the claim as a mathematical statement by translating the key words of the statement into mathematical symbols. Identify 1.vhat parameter is being tested and what its corresponding mathematical symbol is. Then identify the appropriate relation operator 5 , < ,3‘ , = , 35, or E. For example, "equals" or "equal to" in the verbal claim would correspond to the symbol " = "' . Finally identify the claim value that the test parameter is being related to. This value is often numeric. In this problem, the test parameter is the mean, which is represented by the mathematical symbol "p". The relational operator is "more than" which is expressed mathematically as " := " and the claim value is "‘l [1", since the problem statement claims the mean life of these laptop batteries lasts "more than 10 hours." Thus, the mathematical statement for this claim is p :> 1D. Find the complement of the claim. The complement represents all the possible values of p that are not contained in the range of the claim. In this problem, the complement of p :b it] is ps 1D. The null hypothesis, Ho- is a statement that contains a statement of equality, such as 5, =, or E. The null hypothesis is that the mean life of a certain type of laptop battery is not more than it] hours, or Ho: ps 10. L The altemative hypothesis, HE, is the complement of the null hypothesis. It is a statement that must be true if “D is false, and it contains a statement of inequality, such as s, at, or :=- . The alternative hypothesis is that the mean life is more than 10 hours, or Ha: 1.1}1l]. Thus, the hypothesis test to be conducted is as follows. HI l-l'l} ll] 3 The claim is the statement made about the population parameter. It can be either the null or alternative hypothesis, depending on how it is worded. Remember that the claim for this problem is p :- 1D. The claim for this problem is that the mean life is more than it] hours. This corresponds to the alternative hypothesis Ha since it is a statement containing no equality. Therefore, the alternative hypothesis is p :- 1i]. A security expert claims that less than 23% of all homeowners have a home security alarm. State Ho and He in words and in symbols. Then determine whether the hypothesis test for this claim is left—tailed, light—tailed, or two—tailed. Explain your reasoning. First state the null and the attemative hypotheses in words and in symbols. The null hypothesis, Ho- is a statement that contains a statement of equality, such as S, =, or 2. The alternative hypothesis, HE, is the complement of the null hypothesis. It is a statement that must be true if Hg is false, and it contains a statement of inequality, such as r: , a, or :=~ . To solve this problem, first identify the claim. Then express the claim symbolically by choosing the mathematical symbol corresponding to the test parameter, identitying the appropriate relational operator, and identitying the claim value in the statement. Then use the claim to express the hypotheses symbolically and to determine the appropriate relational operators for each. First, review the problem statement and identity the claim in words. The claim is that the proportion of all homeowners who own a security alarm is less than {1.23. This is found by translating the words of the problem statement into a claim statement. Next express the claim symbolically. In this problem, the test parameter is the proportion of the population, which is written symbolically as "p". The relation operator for the claim is < since the claim is that less than 23% of all homeowners have a home security alarm. Putting these symbols together, the claim expressed as a mathematical statement is p r: D23. Now find the complement of the claim above. The complement represents all the possible values of p that are not contained in the range of the claim. In this problem, the complement is p E t123. Thus, the claim is written symbolic ally as p e: D23 and the complement of the claim is written symbolically as p202} Use this information to identify the null and alternative hypotheses. The null hypothesis, Hg, is a statement that contains a statement of equality, such as S, =, or E. The alternative hypothesis, HE, is the complement of the null hypothesis. It is a statement that must be true if Ht] is false, and it contains a statement of inequality, such as c. , i, or := . Use this information to identity the null hypothesis and the altemative hypothesis and then express the two hypotheses symbolically. The null hypothesis is expressed symbolically as HG: p s [123 and the alternative hypothesis is expressed symbolically as Ha: p a: I123. To express these symbolic statements verbally, refer baclc to the context of the claim in the problem statement and translate the symbolic notation into words. In this problem, the attemative hypothesis is expressed verbally as, r‘lhe proportion of all homeowners who own a security alarm is less than [l2 3" and the null hypothesis is expressed verbally as, "the proportion of all homeowners who own a security alarm is at least D23." To determine whether the hypothesis test is left—tailed, right—tailed, or two—tailed, consider the alternative hypothesis of the test. If the alternative hypothesis, HE, contains the less-than inequality symbol [:1], the hypothesis test is a left—tailed test. Similarty, it it contains the greater-than inequality symbol [:a }, it is a right-tailed test, and if it contains the not-equal-to symbol {of ]I, it is a two-tailed test. Since the alternative hypothesis H E,: p s: [1.23 contains the less-than inequality symbol, the test is left—tailed. Determine whether the claim stated below represents the null hypothesis or the alternative hypothesis. It a hypothesis test is performed, how should you interpret a decision that (a) rejects the null hypothesis or [b] fails to reject the null hypothesis? A scientist claims that the mean incubation period for the eggs of a species of bird is at most :11 days. Does the claim represent the null hypothesis or the altematiye hypothesis? Sincetheclaim contains astatementofequality,itrepresentsthe null hypothesis. (a) Hortr should you interpret a decision hat rejects the null hypothesis? There is sutlicient evidence to reject the claim that the mean incubation period for the eggs ofa species of bird is at most 41 days. lb) How should you interpret a decision that fails to reject the null hypothesis? There is insutlicient evidence to reject the claim that the mean incubation period for the eggs of a species of bird is at most 41 days. A security expert claims that less than 14% of all homeowners have a home security alamr. State HD and Hg in words and in symbols. Then detemrine whether the hypothesis test for this claim is left-tailed, right-tailed, or two-tailed. Explain your reasoning. State the null hypothesis in words and in symbols. Choose the correct answer below. A The null hypothesis expressed in words is, "the proportion of all homeowners who owh a home security alamr is at most 11.14." The null hypothesis is expressed symbolically as, "HG: ps i}. 14." 3. The null hypothesis expressed in words is, "the proportion of all homeowners who owh a home security atom is not l1.14.' The null hypothesis is expressed symbolically as "H“: p #l1.14." IVE. The null hypothesis expressed in words is, "the proportion of all homeowners who owh a home security alamr is at least l1.14." The null hypothesis is expressed symbolically as, "Hg: p a £1.14." Iii The null hypothesis expressed in words is, "the proportion of all homeowners who owh a home security alamr is less than 11.14." The null hypothesis is expressed symbolically as "H”: p «1 11.14." State the alternative hypothesis in words and in symbols. Choose the correct answer below. We The alternative hypothesis expressed in words is, "the proportion of all homeowners who own a home security alarm is less than l1.14." The alternative hypothesis is expressed symbolically as, "Ha: p< 11.14." -. 3. The alternative hypothesis expressed in words is, "the proportion of all homeowners who own a home security alarm is more than 0.14." The alternative hypothesis is expressed symbolically as, "Ha: p :- l1. 1 4." -. III. The alternative hypothesis expressed in words is, ”the proportion of all homeowners who own a home security alarm is 0.14." The alternative hypothesis is expressed symbolically as, "H : p= l1.14.' -. 1 The alternative hypothesis expressed in words is, "the proportion of all homeowners who own a home security alarm is not equal to 11.14." The altemative hypothesis is expressed symbolically as, "Ha: p#l1.14." The hypothesistest is lelt-tailed because the alternative hypothesis contains (. Determine whether to reject er fail to reject Hr] at the level of significance of a} ct = DUE and b} c: = (1139. HD1p=11Et Hg: 3) De you reject or fall to reject Hfl at the [1.415 level of significance? j.r #1113, and F: [141759. 'lrMhnn unll nee-Ilrnn H19 nrlll hunn‘ll'rraeic is: In ID. -'_1 DJJ-ullra. WI: '5 hamihneic tncf it: H19 nrnh-Jhili'hr nf nhtnininn .3 llllI—rll juu uuuulllh- IIILr Ilull IIIIJUIII'LHJIJ I-uJ lulu-u, LI I turns..- 1." u IIJIFUIII'Lr-d'ld LLr-IJI. I._| IIILI- PIUWUIIIII UI UIJLLIIIIIIIH u sample statistic with a value as extreme or mere extrerne than the one determined from the sample data. If the P-value is less than or equal tn the level of significance, then reject HD. If the P—value is greater than the level of significance, then fail to reject HEI- Fail to reject HI] because 5.5?55 :- 5.55. b) Db you reject or fail to reject H4] at the 5.55 level of significance? If the P-value is less than or equal tn the level of significance, then reject HD. If the P—value is greater than the level of significance, then fail to reject HEI- Reject Hfll because 5.5?55 c 5.55. The P—value for a hypothesis test is shown. Use the P-value to decide whether to reject l'le when the level of significance is [a] ct: 5.51, {b} u = 5.55, and {c} at = 5.15. P = 5.5555 - .5. Fail to reject HD because the P—value, 5.5555, is less than u = 5.51. - 3. Reject Hfl. because the P—value, 5.5555, is less than a =5.51. w Reject l'le because the P—value, 5.5555, is greater than cc: 5.51. 1-“ :1. Fail to reject HD because the P-value, 5.5555, is greater than u = 5.51. {b} Do you reject er tail to reject HD at the 5.55 level of significance? {:2- fii. Reject Hfll because the P—value, 5.5555, is greater than t: = 5.55. is! 3. Fail to reject “I: because the P—value, 5.5555, is greater than u = 5.55. {1.3- L. Fail to reject HI:| because the P-value, 5.5555, is less than u = 5.55. I'. Reject Hfll because the P—value, 5.5555, is less than at =5.55. {c} Db 1,rbu reject or fail to reject Ht} at the 5.15 level of significance? J" :- Reject Hfll because the P—value, 5.5555, is less than or = 5.15. 3 Fail to reject HI:| because the P—value, 5.5555, is less than u = 5.15. I; Reject Ho because the P—value, 5.5555, is greater than or = 5.15. Fail to reject HI:| because the P-value, 5.5555, is greater than u = 5.15. Determine whether to reject or fail to reject H” at the level of significance of a} u = 5.51 and b} u = 5.55. HD: p =1I15,Ha: p #145, and P: 55155. a] Do you reject or fail to reject Ho at the 5.51 level of significance”? H Reject Ho because F' a 5.51. 3 Reject Ho because P c 5.51 . I; Fail to reject HI] because P c 5.51. I“ II Fail to reject HD because P :b 5.51. b} Do you reject or fail to reject H9 at the 5.55 level of significance? ; Reject Hfll because P :e 5.55. 3 Fail to reject HI:| because P r: 5.55. 2-9" l: Reject HD because P e nos. Fail to reject Ho because P :b 5.55. Use technology to help you test the claim about the population mean, p, at the given level of significance, cc, using the given sample statistics. Assume the population is nom‘rally distributed. Claim: '1} 1325; or: 5.55; o=215.?2. Sample statistics: 5:13:1115, n= 355 First identity the null and alternative hypotheses. Recall that the null hypothesis Ho is a statistical hypothesis that contains a statement of eg uality. The II I' I .I ' II ' .I I I [II II I .I ' II' I I I r I I I'. l—'|I aIIemauve nypomesls l'la Is Ine complement or me nuu nypotnesls. It Is a statement or stnct Inequality. tltner hypothesis—the null hypothesis or the alternative hypothesis—may represent the original claim. The problem statement states that the claim is p 3132(1. Its complement is p s 1320. Because the claim contains the statement of inequality, it becomes the alternative hypothesis. The hypotheses are given below. HD: [451321] Ha: ps132flfclaim} The standardized test statistic, 2, can be found by using the formula given below, 1.vhere J_( is the sample mean, p is the hypothesized mean value, 13 is the population standard deviation, and n is the sample size. i—u Substitute the hypothesized p = 1320 and o = 215.22, and the given sample statistics it = 1342.19 and n = 311D into the formula to find the test statistic. i—u E 134?.19 -132l] = 215.22 Substitute. «ll'Sflfl H 2.1 E Simplify. The P-value is the probability of getting a value of the test statistic that is at least as extreme as the one representing the sample data, assuming that the null hypothesis is tnie. For a left—tailed test, the P—value is the area to the left of the test statistic. For a rig ht—tailed test, the P—value is the area to the right of the test statistic. For a hero—tailed test, the P—value is twice the area in the tail of the test statistic. Determine if the test is left-tailed, right-tailed, or two-tailed. A hypothesis test is right—tailed if the alternative hypothesis contains the greater—than inequality symbol {3* }, left—tailed if it contains the less—than inequality symbol (a: }, or vao-tailed if it contains the not-eguaI—to symbol f at ]. Since the alternative hypothesis is Ha: p :b 132E], this is a right—tailed test. Because this is a right-tailed test, the P—value is equal to the area 3| to the right of the standardized test statistic, z = 2.13. Use technology to find this area, rounding to three decimal places. The area to the right of: = 2.18 is {11115. Thus, the P—value is P = [11115. It is given that that the significance level is u = ELIE. Reject Ho ifthe P—value is less than or equal to :1. Otherwise, fail to reject Ho- Since the claim is the alternative hypothesis, if Hfll is rejected, then there is enough evidence to support the claim. If Hg. is not rejected, then there is not enough evidence to support the claim. JI- i] 2.153 4 J -J'n. 5 Use technology to help you test the claim about the population mean, p, atthe given level of significance, at, using the given sample statistics. Assu me the population is nomially distributed. Claim: p :1220; or = DDT; o = 210.3? Sample statistics: t=1243.53, n = BUD Identify the null and allemative hypotheses. Choose the correct answer below. :ps1220 a. HD Zp>1220 H, :psizsn o. Hfl Ip51220 H, :p2124153 r. Hfl 2p<124153 H, Calculate the standardized test statistic. :psiztsss :psizisss :pélZZfl :pa122n :ps124353 :p5124153 The standardized test statistic is 1.94 . (Round to two decimal places as needed.) Determine the P—value. P = [1025 {Round to three decimal places as needed.) Determine the outcome and conclusion of the test. Reject Hflfitthe its“: significance |etre|,there is enough evidence to support the claim. A random sample of 76 eighth grade students' scores on a national mathematics assessment test has a mean score of 289. This test result prompts a state school administrator to declare that the mean score for the state's eighth graders on this exam is more than 285. Assume that the population standard deviation is 36. At a = 0.05) is there enough evidence to support the administratods claim? Complete parts (a) through (e). (a) Write the claim mathematically and identity H0 and Ha. Choose the correct answer below. a. HUIp<285 B. H02p2235 (claim) {I Hfl:p=285 (claim) Ha: p2285 (claim) Ha: p<285 Ha: pflflS II. anp5285(claim) E Hfl2p=235 3F. HDIp5235 Ha: p> 235 Ha: p> 285 (claim) Ha: p>285 (claim) (b) Find the standardized test statistic z, and its conesponding area. 2 = 0.9? (Round to two decimal places as needed.) (c) Find the P-iralue P-‘ralue = 0.166 (Round to three decimal places as needed.) (d) Decide whether to reject or fail to reject the null hypothesis. .29, Fail to reject Hp Reject Hfl (e) Interpret your decision in the context of the original claim. 1H AL. I'M .:__IE.__.. I....I lL... I. _.l _-...-L _..IJ._.. A. ...__._| .IL. _J_.I_I.l__1.J. .|_I_IL.|1L. _..-- ..._. L. AL. .1_1_|'. .:_L1L _.-J... ._ IL. ..._-. I. _... AL__ "Il'll' ‘I‘ 'l' .""" I‘III‘TI“| llll‘ |'.| .I . . ' ‘. “0‘. ‘H‘ ‘l I.|‘Il"‘. “.103, The lengths of time {in years} it took a random sample of 32 former smokers to quit smoking permanently are listed. Assume the population standard deviation is ?.1 years. At a = 9.19, is there enough evidence to reject the claim that the mean time it takes smokers to quit smoking permanently is 12 years? Complete parts {a} through idl- 21.9 22.9 9.1 23.3 22.4 19.1 9.3 14.]Ir lg 19.9 11.? 29.9 19.3 14.9 19.9 21.9 13.]Ir 19.4 9.9 19.3 9.9 12.3 19.4 29.3 11.9 1?.9 19.? 9.9 19.9 14.9 12.9 19.9 19.]Ir {a} Identify the null hypothesis and alternative hypothesis. In To find the null and altematiye hypotheses, start by writing the claim mathematically. The claim, "the mean time it takes smokers to quit smoking permanently is 12 years," expressed mathematically is p =12. The other hypothesis is the complement of the claim. The complement of p = 12 is p # 12. The null hypothesis, Ho- is a statement that contains a statement of equality, such as S, =, or E. The altematiye hypothesis, H3, is the complement of the null hypothesis. It is a statement that must be true if He is false, and it contains a statement of inequality, such as c , #, or :>_ The hypotheses Ho and HE can be stated as shown below. Ml Ho: p: 12 {claim} Ha: p3512 {b} Identify the standardized test statistic. The standardized test statistic z is given by the following formula, where x is the sample mean, p is the population mean, :5 is the population standard deviation, and n is the sample size. i—p Z—ofqlllfi 1While either the formula or technology can be used to find the standardized test statistic, for the purposes of this explanation, use technology. 2 $2.39 {c} Find the P—vralue. In order to find the P—value, first determine if the hypothesis test is left-tailed, rig ht—tailed, or hero—tailed. The test is hero—tailed. While either the Standard Normal Table or technologgr can be used to find the P—VEIIUE, for this exercise, use technology. Since the test is hero-tailed, the P-value is equal to two times the area in the tail of the standardized test statistic 2 = 2.39. Use technology to find the P—value for z = 2.39. Pellflli' The lengths of time (in years} it took a random sample of 32 former smokers to quit smoking pemianent...
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