Let <em>X</em> be the random variable representing the weight of a randomly selected widget. You're given that the mean and standard deviation of <em>X</em> (which is normally distributed) are 41 oz and 11 oz, respectively.
Then
Pr[<em>X</em> > 19] = Pr[(<em>X</em> - 41)/11 > (19 - 41)/11] = Pr[<em>Z</em> > -2]
where <em>Z</em> follows the standard normal distribution with mean 0 and s.d. 1.
I assume you're familiar with the 68-95-99.7 rule, the important part of which says that approximately 95% of any normal distribution lies within 2 standard deviations of the mean. Mathematically, this is to say
Pr[-2<em>σ</em> < <em>X</em> < 2<em>σ</em>] ≈ 0.95
where <em>σ</em> is the s.d. of <em>X</em>, or in terms of <em>Z</em>,
Pr[-2 < <em>Z</em> < 2] ≈ 0.95
This means that roughly 5% of the distribution falls outside this range:
Pr[(<em>Z</em> < -2) or (<em>Z</em> > 2)] = 1 - Pr[-2 < <em>Z</em> < 2] ≈ 0.05
and because the distribution is symmetric about its mean, the probability of falling within either tail of the distribution is half of this, or roughly 2.5%
Pr[<em>Z</em> < -2] ≈ 0.05/2 ≈ 0.025
Then the probability of the complement is
Pr[<em>Z</em> > -2] = 1 - Pr[<em>Z</em> < -2] ≈ 1 - 0.025 ≈ 0.975
so that Pr[<em>X</em> > 19] ≈ 97.5%.
