normal distribution: part 2

Ø Solving!for!Unknown!Values!in!the!Normal!Model

Ø Sum!of!Normal!Random!Variables

ØNormal!Approximation!for!the!Binomial

Normal Distribution: Part 2

Lecture!13Sections!7.2,!7.4,!7.5

Review: Empirical Rule

• Empirical Rule: approximation!of!percentage!of!observations!that!lie!within!a!certain!number!of!standard!deviations!from!the!mean!in!data!that!is!normal!(symmetric!and!unimodal)

• In!a!normal dataset!or!normal distribution,!approximately:• 68%!of!observations!are!within!1!standard!deviation!of!the!mean

• 95%!of!observations!are!within!2!standard!deviations!of!the!mean

• 99.7%!of!observations!are!within!3!standard!deviations!of!the!mean

Example: Empirical Rule

• Scenario: IQ!scores!have!a!normal!distribution!with!mean!100!and!standard!deviation!15

• Question: What!is!the!IQ!of!a!person!whose!IQ!is!higher!than!approximately!97.5%!of!all!people?

• Answer: ________• Below!_____ includes!everything!but!the!______________

Example: Empirical Rule


• Question: What!is!the!IQ!of!a!person!whose!IQ!is!higher!than!approximately!90%!of!all!people?

• Answer: ____________________________________________________• 115!has!________!to!the!left;!130!has!________!to!the!left

• Best!estimate:!__________________________

Review: Normal Probabilities

• Known: Value!of!observation!! from!normally!distributed!random!variable!with!mean!" and!standard!deviation!#

• Unknown: Area!(probability)!to!the!left!of!!

• Strategy:

• Standardize!using!$ =%&'

(

• Find!probability!using!standard!normal!table

Normal Percentiles

• Known: Area!to!the!left!(or!right)!of!desired!observation!from!normally!distributed!random!variable

• Unknown: Either�• Value!of!the!observation

• Mean!of!the!distribution

• Standard!deviation!of!the!distribution

• Strategy:• Find!Z-score!corresponding!to!desired!percentile• Locate!the!closest!percentile/probability!inside the!table

• Work!to!the!left!and!up!to!the!top!row!to!solve!for!the!Z-score

• Solve!for!the!unknown!value!in!$ =%&'

(

Example: Unknown Observation



• Picture: • Sub-Question 1: What!Z-score!will!give!us!90%!of!the!area!in!the!lower!tail?• Find!the!___________________________!_________________________• Solve!for!______________• 90th percentile:!___________________

_______

_______

_______

_______

Example: Unknown Observation



• Picture: • Sub-Question 2: What!specific!value!is!this!many!standard!deviations!above!the!mean?

• Set!up!__________:!__________________

• Solve!for!__________________________:!_____________________________________

Example: Unknown Mean

• Scenario: Tire!manufacturer!promises!that!its!tires!will!last!60,000!miles!or!patrons!get!a!refund.!!Assume!# = 7000miles.

• Question: If!mileages!are!normal,!what!must!the!mean!mileage!be!such!that!no!more!than!3%!of!tires!last!less!than!60,000!miles?

• Picture: • Sub-Question 1: What!Z-score!will!give!us!3%!of!the!area!in!the!lower!tail?• Find!the!probability!______ inside!the!table• Solve!for!_____________• 3rd percentile:!____________________

_______

_______

_______

_______

Example: Unknown Mean

• Scenario: Tire!manufacturer!promises!that!its!tires!will!last!60,000!miles!or!patrons!get!a!refund.!!Assume!# = 7000miles.

• Question: If!mileages!are!normal,!what!must!the!mean!mileage!be!such!that!no!more!than!3%!of!tires!last!less!than!60,000!miles?

• Picture: • Sub-Question 2: What!should!the!mean!tires!length!be!if!3%!of!the!area!is!in!the!lower!tail?

• Set!up!Z-score:!____________________

• Solve!for!missing!mean:!!!!!!!!!!!" = ________________________________

= ____________

Example: Unknown Standard Deviation

• Scenario:More!consistent!speeds!on!the!highway!tend!to!reduce!accidents.!!The!speed!limit!on!PA!highways!is!70!mph.!!Suppose!7%!of!cars!travel!faster!than!80!mph!and!speeds!are!normal.

• Question: What!is!the!standard!deviation!of!the!speeds!of!cars?

• Picture: • Sub-Question 1: What!Z-score!will!give!us!7%!of!the!area!in!the!upper!tail?• 7%!in!upper!tail!means!_________!percentile• ________!percentile:!________________

_______

_______

Example: Unknown Standard Deviation

• Scenario:More!consistent!speeds!on!the!highway!tend!to!reduce!accidents.!!The!speed!limit!on!PA!highways!is!70!mph.!!Suppose!7%!of!cars!travel!faster!than!80!mph!and!speeds!are!normal.

• Question: What!is!the!standard!deviation!of!the!speeds!of!cars?

• Picture: • Sub-Question 2: With!mean!70,!what!standard!deviation!yields!7%!of!the!area!in!the!upper!tail?

• Set!up!Z-score:!____________________

• Solve!for!standard!deviation:!

# = ________________________________

Using Excel

• One!function!will!find!the!value!of!missing!observation:• =NORM.INV([PROBABILITY], [MEAN], [STANDARD DEVIATION])• Used!before!standardizing

• One!function!will!find!the!Z-score!corresponding!to!a!percentile:• =NORM.S.DIST([PROBABILITY])

• Used!if!the!probability!is!known!but!either!mean!or!standard!deviation!is!missing!so!more!work!needs!to!be!done

Sum and Difference of Normal Random Variables

• Suppose!) and!* are!two!different!normal!distributions!with!means!"+ and!", and!standard!deviations!#+ and!#,.!!Then:

• The!distribution!of!the!sum!) - * is!normal!with:• Mean:!"+., = "+ - ",

• Standard!Deviation:!#+., = #+/ - #,

/

• The!distribution!of!the!difference!) 1 * is!normal!with:• Mean:!"+., = "+ 1 ",

• Standard!Deviation:!#+., = #+/ - #,

/

Example: Sum of Normal Random Variables

• Scenario: Sales!at!a!coffee!shop!on!weekdays!average!$1500!with!a!standard!deviation!of!$120!while!sales!on!weekends!average!$1200!with!a!standard!deviation!of!$150.

• Question: What!are!the!mean!and!standard!deviation!of!the!total!sales!over!a!weekend?

• Answer: ) = ______________;!* = _____________• Mean: "+., = ___________________________________________

• Standard Deviation: #+., = _________________________________________________

Example: Sum of Normal Random Variables


• Question: What!is!the!probability!that!sales!are!less!than!$2000!over!a!weekend?

• Answer:

____________________!= ___________________________

= _____________________

= ____________

Example: Difference of Normal Random Variables


• Question: What!are!the!mean!and!standard!deviation!of!the!difference!in!sales!between!a!weekday!and!weekend!day?

• Answer: ) = ______________;!* = _____________• Mean: "+&, = ___________________________________________

• Standard Deviation: #+&, = _________________________________________________

Example: Difference of Normal Random Variables


• Question: What!is!the!probability!that!sales!on!a!randomly!selected!weekday!will!exceed!those!on!a!random!weekend!day?

• Answer:

____________________!= ___________________________

= _____________________

= _____________________

= _____________________

= ____________

Motivation: Normal Approximation for the Binomial

• Scenario: 8%!of!males!in!the!general!population!are!color!blind.

• Question: In!a!random!sample!of!500!males,!what!is!the!probability!that!35!or!fewer!males!in!the!sample!are!color!blind?

• Problem: Calculating!this!probability!without!software!would!require!____________________________________• 2 ) 3 45 = ________________________________________________________________

• Solution: Use!the!__________________!to!_______________!the!probability

Normal Approximation for the Binomial

• If!68 9 :0 and!6; 9 :0,!then!the!binomial!model!can!be!approximated!using!a!normal!model!with:• Mean: " = 68

• Standard Deviation: # = 68;

• Note:!The!normal!model!will!not!be!appropriate!if!the!sample!size!is!_______________________________.

Example: Normal Approximation for the Binomial

• Scenario: 8%!of!males!in!the!general!population!are!color!blind.

• Question: In!a!random!sample!of!500!males,!what!is!the!probability!that!35!or!fewer!males!in!the!sample!are!color!blind?

• Answer:• Mean: " = ___________________

• Standard Deviation: # = ___________________________

• Approximate Probability:

____________________!= ___________________________

= _____________________

= ____________

• Exact Probability: ____________

normal distribution: part 2

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