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

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?

1 Answer
May 31, 2018

The standard deviation is sigma = 0.2001.

Explanation:

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