# Confidence Intervals QMET103

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1 Cofidece Itervals QMET103 Library, Teachig ad Learig

4 2 If the oly iformatio give is mea ad sample stadard deviatio orvariace, ukow a t score is used istead of a Z score x t 1 Use - 1. From the sample, calculate x. (This may be give to you.) s 2. Calculate the degrees of freedom. For a oe sample mea, this is Look up a t score from the t- table (*see below). Note level of cofidece required ad use the correct degrees of freedom (df). 5. Calculate the stadard error of the sample statistic. s se. For a oe sample mea, mea Example: For a set of data, x, s,ad, se ( mea) degrees of freedom fid a 95% ad 95% upper tail = So from table: t score : df t.100 t.050 t.025 t.010 t.005 t.001 t Hece, the = That is, t.025 = =( , ) That is, we ca be 95% cofidet that the true populatio mea lies betwee approximately ad Note that this iterval is oly slightly greater tha the oe calculated previously usig populatio s.d ad a Z score. 4

6 p 1 p Use p Z score 1. From the sample, calculate p or ote if give. 2. Look up a Z score from the stadard table (*see below). Note the level of cofidece required. 3. Calculate the stadard error of the sample statistic. For a proportio, the s.e. is p1 p Example: I a Rugby World Cup, a radom sample of supporters were asked, Which coutry do you thik will wi the 2003 Rugby World Cup? The results are summarised: Coutry Number of supporters who thik their coutry will wi Australia 116 Eglad 13 Frace 25 New Zealad 140 Wester Samoa 50 South Africa 47 Wales 24 Udecided 65 Total 480 Calculate a 90% cofidece Iterval for the proportio who had ot yet decided. Solutio: ot decided p ; 90% cofidece Z = ( We ca be 90% cofidet betwee 11% ad 16% of the populatio were udecided. (If you chage the fractios to decimals, there will be a slight roudig error, but this will usually ot be greatly sigificat.) Practice Questios 1. Samples of size are take from populatios with a probability of success p. Use the values of ad p, the sample size ad proportio, give below, to fid cofidece itervals for the populatio proportio with the levels of cofidece idicated. p Cofidece Level a % b % c %. ) ,

8 7. x 4. 85, s d.f. 19, p t , That is, we ca be 95%cofidet that the true acceleratio time is betwee 4.16 ad 5.54 secods. 8. x ,s (after eterig data i calculator). That is we ca be 99% cofidet that the true voltage i the power packs is betwee ad volts. Aswers (proportio) a , b. C , p 0.34, 95% Z c , , 7 df 6, p t , , , p % Z , , ,

9 To fid a z-score (Usig Z table with upper half shadig oly) Probability Table etry is probability at or below z Z etc To locate the appropriate Z-score For example, to fid the correct Z-score for a 98% cofidece iterval: Chage the %age to a decimal 98% = 0.98 Halve this decimal 0.98/2 = 0.49 Fid this value i the table Closest value =.4901 Use the correspodig Z-score Z-score = 2.33 To fid a z-score (Usig Z table with left had tail shaded) Probability Table etry is probability at or below z Z etc To locate the appropriate Z-score For example, to fid the correct Z-score for a 98% cofidece iterval: Chage the %age to a decimal 98% = 0.98 Halve this decimal 0.98/2 = 0.49 Add = 0.99 Fid this value i the table Use the correspodig Z-score Closest value = Z-score = 2.33

10 10

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