Question

Measurement error that is normally distributed with a mean of 0 and a standard deviation of 0.5 gram is added to the true weight of a sample. Then the measurement is rounded to the nearest gram. Suppose that the true weight of a sample is 166.0 grams.
(a) What is the probability that the rounded result is 167 grams?
(b) What is the probability that the rounded result is 167 grams or more?

Answer 1

Answer:

(a)$0.15731$

(b)0.02275

Step-by-step explanation:

We are given that

Mean=0

Standard deviation=0.5 g

True weight of a sample=166 g

Let X denote the normal random variable  with mean =166+0=166

(a)

P(166.5<X<167.5)

=$P(\frac{166.5-166}{0.5}<\frac{X-\mu}{\sigma}<\frac{167.5-166}{0.5})$

=$P(1<Z<3)$

=$P(Z<3)-P(Z<1)$

$=0.99865-0.84134$

$=0.15731$

(b)

$P(X>167)=P(Z>\frac{167-166}{0.5})$

$=P(Z>2)$

$=1-P(Z<2)$

$=1-0.97725$

$=0.02275$