How can i calculate probabilities regarding candle factories? (details inside)

hello, i'm having trouble with this question. would appreciate your help with it:

in a package of candles there are 45 candles, with random sizes(lengths) and no dependency between the lengths.
in a certain factory, candles are produced so that the distribution of the length (in cm) (of each) is normal with the parameters 13 and 0.1^2

1)what is the probability that in a random package there will be at least 30 candles whose length is within the range of 12.82 and 13.06 cm?

2)what is the length of the candle that 92% of the candles are shorter than him?

1 Answer
May 27, 2018

1) 0.6903

2) 13.141 cm

Explanation:

Let X be the length of a single candle. Then X" ~ N"(13, 0.1^2).

1)

We first find the probability of selecting a single candle that's between 12.82 and 13.06 cm.

"P"(12.82 < X < 13.06)

= "P"((12.82-13)/0.1 < Z < (13.06-13)/0.1)

= "P"(–1.8 < Z < 0.6)

= "P"(Z < 0.6) - "P"(Z < –1.8)

= 0.7257 - 0.0359

=0.6898

This is the probability of "success" for a single candle.

Let Y be the number of candles out of 45 that are between 12.82 and 13.06 cm. Then Y" ~ Bin(45, 0.6898)."

We now seek "P"(Y >= 30). Usually that would mean calculating

"P"(Y "=" 30) + "P"(Y "=" 31) + ... + "P"(Y "=" 45)

This would take a long time. However, since n=45 is fairly large and p=0.6898 is fairly close to 0.5, we can use the normal approximation for Y.

Y " "stackrel "approx." ~ "N"(np, npq) " "="N"(31.041, 9.6289)

Using a continuity correction, we get

stackrel"Binomial" overbrace("P"(Y >= 30)) ~~ stackrel"Normal" overbrace("P"(Y >= 29.5))

color(white)("P"(Y >= 30)) = 1- "P"(Y < 29.5)

color(white)("P"(Y >= 30)) = 1- "P"(Z < (29.5-31.041)/sqrt9.6289)

color(white)("P"(Y >= 30)) = 1- "P"(Z < –0.4966)

color(white)("P"(Y >= 30)) = 1- 0.3097" " (from software)

color(white)("P"(Y >= 30)) = 0.6903

The actual Binomial probability is "P"(Y <= 30) ~~ 0.6955 so our Normal approximation is pretty good.

2)

We seek x such that "P"(X < x) = 0.92. This is the same as

"P"(Z < (x - mu)/sigma) = 0.92

Through table lookup, we get

(x - mu)/sigma = z ~~ 1.41

:. (x - 13)/0.1 ~~ 1.41

"       "x - 13 ~~0.141

"                "x ~~13.141

So 92% of the candles will be shorter than 13.141 cm.