How can i find t (candle factory statistics question - a bit difficult, need it to prepare for an exam)?

Question

in some candle factory, the length disturbance (cm) of each candle is normal with the parameters 15 and #t^2#
. knowing that the probability that the shortest candle in a random package is longer than 14.6 cm (out of 45 candles) is 0.354206. how can i find t?

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Answer 1 · 2018-05-31T06:16:45

The standard deviation is #sigma = 0.2001#.

Let #X_i# be the length of any candle in the box. Then #X_i" ~ N"(15, sigma^2)", for " i=1,...,45.#

Let the CDF of each #X_i# be denoted #F_X(x)="P"(X_i < x)#.

The CDF of the minimum of all #X"'s"# is then

#"P"[min(X_i) < x]=1-[1-F_X(x)]^45#.

From the given information, we know #"P"(min(X_i) > 14.6)=0.354206,# which means

#"P"[min(X_i) < 14.6]=1-"P"[min(X_i) > 14.6]#

Substituting in known values gives

#1-[1-F_X(14.6)]^45=1-0.354206#

#=>" "[1-F_X(14.6)]^45=" "0.354206#

#=>" "1-F_X(14.6)" "=" "root45(0.354206)#

#=>" "1-F_X(14.6)=0.9772#

#=>" "F_X(14.6)=0.0228#

#=>" P"(X_i < 14.6)=0.0228#

#=>"P"(Z < (14.6-15)/sigma)=0.0228#

#=> (–0.4)/sigma = –1.999# (from table lookup)

#=> sigma = (–0.4)/(–1.999)= 0.2001 ~~ 0.2#