THE NORMAL DISTRIBUTION AND Z-SCORES COMMON CORE ALGEBRA II

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1 Name: Date: THE NORMAL DISTRIBUTION AND Z-SCORES COMMON CORE ALGEBRA II The normal dstrbuton can be used n ncrements other than half-standard devatons. In fact, we can use ether our calculators or tables to determne probabltes (or proportons) for almost any data value wthn a normally dstrbuted populaton, as long as we know the populaton mean,, and the populaton standard devaton,. But, frst, we wll ntroduce a concept known as a data value s z-score. Exercse #1: Boy s heghts n seventh grade are normally dstrbuted wth a mean heght of 62 nches and a standard devaton of 3.2 nches. Fnd z-scores, rounded to the nearest hundredth, for each of the followng heghts. Show the calculaton that leads to your answer. (a) x 66 nches (b) x 57 nches (c) x 70 nches THE Z-SCORE OF A DATA VALUE x For a data pont x, ts z-score s calculated by: z. It calculates how far from the mean, n terms of standard devatons, a data pont les. It can be postve f the data pont les above the mean or negatve f the data pont les below the mean. Z-scores gve us a way to compare how far a data pont s away from ts mean n terms of standard devatons. We should be able to compute a z-score for a data value and go n the opposte drecton. Exercse #2: Jeremah took a standardzed test where the mean score was a 560 and the standard devaton was 45. If Jeremah s score resulted n a z-value of 1.84, then what was Jeremah s score to the nearest whole number? Wth z-scores, we can then determne the probablty that a subject pcked from a normally dstrbuted populaton would have a characterstc n a certan range. Z-score tables come n many dfferent varetes. The one that comes wth ths lesson shows only the rght hand sde, so symmetry wll have to be used to determne probabltes. Exercse #3: The lengths of full grown sockeye salmon are normally dstrbuted wth a mean of 29.2 nches and a standard devaton of 2.4 nches. (a) Fnd z-scores for sockeye salmon whose lengths are 25 nches to 32 nches. Round to the nearest hundredth. (b) Use the z-score table to determne the proporton of the sockeye salmon populaton, to the nearest percent, that les between 25 nches and 32 nches. Illustrate your work graphcally.

2 Exercse #4: If the scores on a standardzed test are normally dstrbuted wth a mean of 560 and a standard devaton of 75. Answer the followng questons by usng z-scores and the normal dstrbuton table. (a) Fnd the probablty that a test pcked at random would have a score larger than 720. Round to the nearest tenth of a percent. (b) Fnd the probablty that a completed test pcked at random would have a score less than 500. Round to the nearest tenth of a percent. (c) Fnd the probablty that a completed test pcked at random would have a score between 500 and 600. (d) Fnd the probablty that a completed test pcked at random would have a score between 600 and 700. Ths process s sometmes used to determne a partcular data pont s percentle, whch s the percent of the populaton less than the data pont. Exercse #5: The average weght of full grown beef cows s 1470 pounds wth a standard devaton of 230 pounds. If the weghts are normally dstrbuted, what s the percentle rank of a cow that weghs 1,750 pounds? (1) 89 th (3) 49 th (2) 76 th (4) 35 th Your graphng calculator can also fnd these proportons or percent values. Each calculator s nputs and language wll be slghtly dfferent, although many wll do much of the work for you, even allowng you to not thnk about the z-scores. Exercse #6: Gven that the volume of soda n a 12 ounce bottle from a factory vares normally wth a mean of 12.2 ounces and a standard devaton of 0.6 ounces, use your calculator to determne the probablty that a bottle chosen at random would have a volume: (a) Greater than 13 ounces. (b) Less than 11 ounces (c) Between 11.5 and 12.5 ounces

3 Name: Date: THE NORMAL DISTRIBUTION AND Z-SCORES COMMON CORE ALGEBRA II HOMEWORK FLUENCY 1. A populaton has a mean of 24.8 and a standard devaton of 4.2. For each of the followng data values, calculate the z-value to the nearest hundredth. You do not need to read the Normal table. (a) x 30 (b) x 35 (c) x 19 (d) x 15.4 (e) x 24.8 (f) x A populaton has a mean of and a standard devaton of If a data pont has a z-value of 1.87 then whch of the followng s the value of the data pont? (1) 28.8 (3) (2) 86.7 (4) APPLICATIONS Get practce wth both the Normal Dstrbuton Table and your calculator when dong the followng problems. 3. A recent study found that the mean amount spent by ndvduals on a musc servce webste was normally dstrbuted wth a mean of $384 wth a standard devaton of $48. Whch of the followng gves the proporton of the ndvduals that spend more than $400? (1) 0.43 (3) 0.12 (2) 0.74 (4) The hold tme experenced by people callng a government agency was found to be normally dstrbuted wth a mean of 12.4 mnutes and a standard devaton of 4.3 mnutes. Whch percent below represents the percent of calls answered n less than 5 mnutes? (1) 4.3% (3) 6.8% (2) 5.3% (4) 12.9%

4 5. The natonal average prce per gallon for gasolne s normally dstrbuted wth a mean (currently) of $2.34 per gallon wth a standard devaton of $0.26 per gallon. Whch of the followng represents the proporton of the gas prces that le between $2.00 and $3.00? (1) 56% (3) 84% (2) 72% (4) 90% 6. If the average teacher salary n the Unted States s $45,753 and salares are normally dstrbuted wth a standard devaton of $7890, would a salary of $40,000 per year be n the lowest quntle of teacher salary? (Do a quck Internet search on the term quntle f you don't know what t means). 7. The average rent for a one bedroom apartment (n the Wnter of 2015) n New York Cty s a whoppng $2801 per month wth a standard devaton of $920. (a) If rents are normally dstrbuted, what percent of the apartments wll be less than $2,500 per month? (b) If rents are normally dstrbuted, how realstc s t to beleve you wll be able to rent a onebedroom n New York Cty for less than $1,500 per month? Justfy your answer. (c) A one-bedroom on the Upper East Sde wth a doorman and vews of Central Park was lsted at $5,000 per month. How rare s ths? Assume the rents are normally dstrbuted. (d) Do you thnk the rents are normally dstrbuted? Keep n mnd the normal dstrbuton s symmetrc about ts mean (looks the same on both sdes). If t sn't symmetrc, what does t look lke? 8. A natonal math competton advances to the second round only the top 5% of all partcpants based on scores from a frst round exam. Ther scores are normally dstrbuted wth a mean of 76.2 and a standard devaton of What score, to the nearest whole number, would be necessary to make t to the second round? To start, look at the table and see f you can determne the z-value that corresponds to the top 5%.

5 STANDARD NORMAL DISTRIBUTION BASED ON Z-VALUE z Area/proporton gven n table. 0 z

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