Good day everyone! In this video lesson, we will discuss about converting a normal random variable to a standard normal variable and vice versa. So ang gagamitin natin dito is yung z-score wherein the z-score is measures the distance between an observation and the mean measured in units of standard deviation.
So, the areas under the normal curve are given in terms of Z values or scores. So meron tayong formula na gagamitin and that is z is equal to, okay, so dapat isang equals lang yan, tanggalin natin yung isang equals. Okay, so ito yung gagamitin natin formula kung saan yung z, that is the standard normal score or the z score. And x is any data value in a normal distribution. Okay, this symbol, the mu or that is the mean.
And this symbol is the standard deviation. So, bakit maimportante yung z-score? Kasi minsan class, in our previous video lesson, diniscuss natin yung normal curve Merong mga raw score na may be composed of large values So, hindi natin siya ipapakita doon sa baseline ng normal curve Lalo na kapag masyadong malaki na So, the large values cannot be accommodated at the baseline of the normal curve So, kaya kailangan na natin siyang i-convert. Kailangan natin i-convert para makita natin doon sa baseline ng normal curve.
So, sa pamamagitan ng z-score. Okay. So, if the z-score is positive, so, ibig sabihin the positive score is above the mean. If the z-score is 0, the score is the same as the mean. And if the z-score is negative, the score is below the mean.
So tatandaan natin sa normal curve natin, yung pinakagitna yun yung mean natin. Okay, for example number 1, determine the Z value for each of the following X values for a normal distribution with a mean of 16 and the standard deviation of 3. For letter A, let's say our x value is 12. So, kunin natin yung z value nito. So, using the formula. So, 12 minus 16. Bakit 16? Okay, so based on the given.
Ayan yung given natin So, substitute lang natin sa formula 12 minus 16 Divide 3 The answer is negative 1.33 Since negative sya Ibig sabihin Yung mean natin dito is Below the mean Okay, yung 12 natin is below the mean Next If our x is 8 So, substitute again on the given formula So So 8 minus 16, divide 3, so the answer is negative 2.67. So, same with letter A. So, yung 8 natin ay below the mean.
Another, if we have x is equal to 22, so, using the formula, substitute lang natin, 22 minus 16, divide 3, okay, using your calculator, the answer is 2. Since positive yung z value natin, So, ibig sabihin, ang 22 is above the mean. Another, if we have x is equal to 25. So, 25 minus 16, the answer is 3. Kasi 25 minus 16, that is 9. 9 divide 3, the answer is 3. And this is letter D na. Letter D na ito. So, palitan natin letter D.
Okay. So, ibig sabihin, yung x is equal to 25. above the mean kasi positive siya. Another example, we have a given the mean of 50 and the standard deviation of 4. So, I think yung symbol natin dito parang iba.
So, palitan lang natin. Okay, given the mean of 50 and the standard deviation of 4 of a population of reading scores. So, find the Z value that corresponds to a score of 58. So, kukunin natin yung Z value corresponds to 58. So, gamit lang din yung Z score formula.
So, X minus the mean over the standard deviation. So, what are yung mga given natin? So, yung mean natin 50, standard deviation is 4, tapos yung X natin 58. In-substitute lang natin. So, dito, pag pinakita natin sa normal curve, o gumuhit tayo yung normal curve natin, so, lagi natin itapat sa 0, yung sa gitna, yung given na mean.
Okay? Since yung given natin is 50, so, dito dapat siya nasa gitna. Okay? Dapat nasa gitna siya.
And then, yung standard deviation natin is 4. So, ibig sabihin, mag-add ka ng 4 papuntang right side at mag-subtract ka ng 4 papuntang left side. So kagaya nito. So yung average natin is 50 so dapat na sa gitna siya.
So since yung standard deviation is 4, so yun ang gagamitin natin. So 50 plus 4 that is 54 plus 4, 58 plus 4, 62. Mula naman dito, papuntang left side, magma-minus tayo ng 4. 50 minus 4, 46 minus 4, 42 minus 4. 38. So, ang kukunin natin ngayon, yung z value corresponds to 58. So, dito pa lang, kitang kita mo na no, that yung 58 corresponds, yung row score na 58 natin corresponds sa z value na 2. So, ang gawin na lang natin, ipakita natin using the formula. Okay, substitute lang 58 minus 50 over 4, that is 8, divide 4, that is 2. So, ibig sabihin, okay, ang z value na corresponds kay 58 is 2. Thus, the z value that corresponds to the raw score 58 is 2 in a population distribution.
So, the score of 58 is above average kasi positive, no? So, ito yung mean natin. So, nandito siya. So, above average. Next, locate the z value that corresponds to PE.
score of 39 given the mean value of 45 and standard deviation of 6. Okay? So, using our given, and then, dito makikita na natin, no? So, yung mean natin is 45, and then the standard deviation of 6, so mag-add lang din tayo ng 6 papuntang right side. 45 plus 6, 51, plus 6, 57, plus 6, that is 63. Dito sa left side, minus 6. 45 minus 6, 39. Minus 6, 33. Minus 6, 27. Okay, so dito makikita na natin ano yung corresponding Z value ni 39. Ito yung 39 natin, that is negative 1. So using the formula, makikita rin natin yan. Okay, so using the formula, so substitute lang natin.
39 minus 45, divide 6. So, negative 6 divided by 6, the answer is negative 1. So, kagaya na nakikita natin dito, makikita nyo agad na si 39, yung corresponding Z value ni 39 is negative 1. So, therefore, the Z value that corresponds to the row score 39 is negative 1 in a population distribution. The score of 39 is below average kasi nasa baba siya ng mean natin. Another example, a consumer group.
tested a sample of 100 light bulbs, it found that the mean light expectancy of the bulbs was 842 hours. With a standard deviation of 90, one particular light bulb from Dora Bright Company had a z-score of 1.2. What was the lifespan of this light bulb? Okay, sa problem natin, class, given yung z value natin, which is 1.2. So, Ang mga given natin, meron tayong given as a value that is 1.2.
And then yung mean natin dito is 842. And the standard deviation is 90. Okay. Okay, since kaganitong problem class, given yung z-score natin, so ang gagamitin natin yung formula, so, Ibig sabihin hinahanap natin dito yung x, yung value ng x. So, para makuha yung value ng x, so, z times standard deviation, or the z-score times the standard deviation, plus the value of your mean. Okay, so gamit yung formula na yan, substitute lang natin.
So, 1.2 times 90 plus 842, x is equal to 950. Okay, so ibig sabihin class, So, ang tanong na what was the lifespan on this light bulb? So, the answer is, the light bulb had a lifespan of 950 hours. Another example, for a continuous random variable that has a normal distribution with mean of 20 and standard deviation of 4, find the area under the normal curve from x is equal to 20. And x is equal to 27. So, paano gagawin ito, class?
Meron tayong dalawang value ng x. So, first, kunin muna natin yung z-score ng dalawang value ng x natin. So, kunin natin muna yung x is equal to 20. So, using the given, so 20 minus 20. So, yung mean natin is 20. Divide 4, which is the standard deviation.
20 minus 20, divide 4. The answer is 0. So... If x is 20, yung z value natin is 0. Sa 27 naman, sa substitute lang natin, that is 27 minus 20, divide 4, and the answer is 1.75. Ang pinapakuha sa atin dito is yung area under the normal curve.
So, yung area, since between 0 and 1.75, so gamit yung ano nyo, pag tinignan nyo yan dyan sa Z table, Okay, pwede nyo i-download yun. Pag tinignan natin yung Z table nyo, so yung normal curve, o yung area under the normal curve, between 0 and 1.75, so tingnan nyo dun sa 1.75, the answer is 0.4608. So therefore, the area under the normal curve between x is equal to 20 and x is equal to 27 is 0.4608. Another example, the scores of a group of students in standardized tests are normally distributed with mean of 60 and standard deviation of 8. So, let's answer the following questions. First, how many percent of the students got below 72?
Okay, dun sa previous video lesson natin, meron na tayong introduction about dun sa different kung paano kukunin yung area. Magmasa-subtract ba tayo from 0.5 or mag-a-add ng 0.5? So, pwede nyo balikan yung video lesson natin, no?
Yung pagkuhan ng area. So, dito, meron tayong given problem and then sagutan natin kung ilang students o ilang percent ng estudyante ang nakakuha ng score na below 72. Okay? So, using the z-score formula, so, substitute lang natin. So, yung x natin dito is 72. Ito yun.
Minus 60. Minus 60, that is our mean. And divide 8. Standard deviation natin is 8. So, 72 minus 60, divide 8, that is 12 over 8, is equal to 1.5. So, ngayon, ang kukuni natin dito is yung area ng 1.5. So, tingin ulit kayo sa z-table nyo. Tingnan nyo yung Z table nyo.
Hanapin nyo yung 1.5. Okay. So, kapag re-know natin yung ano natin, no?
I-sketch natin yung normal curve natin. Ito yung 1.5 natin dito. Since below 7, ah, below daw, okay, below 1.5, kasi yan, yan, nang-o.
Tingnan nyo rin itong nakalagay dito. Makikita natin na papunta siya doon at yung half ng normal curve ay nasakop niya. Diba ang paliwanag ko sa inyo kapag nasakop o ng graph nyo Yung isang kalahati, ibig sabihin yung kalahati nyan, automatic 0.5 0.5 na agad to Plus yung area ni 1.5 So, ibig sabihin i-add natin Yung area ni 1.5 plus 0.5 Okay? So, that is Ang area ni 1.5, pag tinignan nyo yan That is, no Ah Inadd na pala natin yung area ni 1.5 plus 0.5, the answer is 0.9332.
Okay? So, therefore, about 93.32% of the group gets below 72. Again, paano natin nakuha yung 0.9332? Kunin nyo muna yung area ni 1.5.
Okay? Pagkakuha ng area ni 1.5, Add natin kay 0.5. Okay, add kay 0.5. And the answer is 0.9332.
Another question, what part of the group scored between 58 and 76? So, meron tayong dalawa, 58 and 76. So, kukuni natin yung z-score or z-value nitong dalawa, nung 58 and 76. So, first, yung 58 muna. So, 58 minus 60, divide 8. The answer is negative 0.25.
Yung isa naman, so pag kinuha natin yung area ng 0.25, since wala naman tayong negative dun sa area, okay, that is 0.0987. Yung isa, 76 minus 60 divide 8, the answer is 2. At yung area dun sa Z table natin ng 2 is 0.4772. So, kapag ganito class, paano natin kukunin yung area kapag meron tayong dalawa tapos different ang sign? Anong ginagawa dyan?
So, ito yan, no? Okay, this is negative 0.25 at ito ay positive 2. So, paano gagawin dyan? Yes, tama po kayo.
I-add natin yung area nila kapag different sign between different sign, yung dalawang say value. Add natin yung area nila. So, therefore, 0.0987 plus 0.4772, the answer is 0.5759.
So, therefore, about 57.59% of the group got a score between 58 and 76. Another, if there were 250 students who took the test, About how many students scored higher than 64? So, kukunin natin kung ilang estudyante ang nakakuha ng score higher than 64. So, ganun din, kunin natin yung Z value. 64 minus 60 divided 8, the answer is 0.5. At yung area ni 0.5 is 0.1915. Since ang nakalagay dito is higher than.
Tapos, higher than, tapos positive, anong gagawin natin? Yes, tama po. Ang gagawin natin, ima-minus natin yung area ni 0.5 kay 0.5. So, therefore, 0.5 minus 0.1915, the answer is 0.3085.
Again, paano natin nakuha yung 0.3085? Yung area nito, Mininus natin kay 0.5. So, therefore, since there were 250 students who took the test, about 250 times 0.3085 equals 77.13 or 77 students got a score higher than 64. Thank you for watching this video. I hope you learned something. Don't forget to like, subscribe, and hit the bell button para updated kayo for more video tutorial.
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