Question

Problem 5: [8 points) The amount of fill (weights of contents) put into a glass jar of spaghetti sauce is normally distributed with mean j = 850 g and a standard deviation o=8 g (a) Find the probability that one jar selected at random contains between 848g and 855g.[1] (b) Find the probability that a randomly selected jar contains less than 855g. [1] (e) Let Y equal the number out of 24 whose amount of fill is less then 855 g Describe the distribution of Y and find the probability of (Y < 2). [3] (d) Describe the distribution of X, the mean amount of fill for the 24 randomly selected jars and find P(848 < X < 855) [3]

Answer 1

Let X denotes the amount of spaghetti sauce in a randomly selected jar.

Here,

X ~ Normal(850, 8²)

a)

P(848.00 < X < 855.00) = P(848.00 < X < 855.00) = P(X < 855.000) – P(X < 848.00) X - 850.00 855.00 – 850.00 X – 850.00 848.00 – 850.00 = P(848.00 < X < 855.00) = P(- PC 8.00 8.00 8.00 8.00 855.00 – 850.00) - P(Z = 848.00 – 850.00 = P(848.00 < X < 855.00) = P(Z < 8.00 8.00 → P(848.00 < X < 855.00) = P(Z < 0.62) – P(Z < -0.25) = P(848.00 < X < 855.00) = 0.734014 – 0.401294 = 0.332721 ~ 0.3327

b)

c) Here,

X - 850.00 P(X < 855.00) = P(X < 855.00) = P( 8.00 855.00 – 850.00 8.00 = P(X < 855.00) = P(Z < 855.00 – 850.00 8.00 = P(X < 855.00) = P(Z < 0.625000) + P(X < 855.00) = 0.734014 ~ 0.7340

Y ~ Binomial(n = 24, p = 0.734)

The probability mass function of X is

$P(Y = y) = \binom{24}{y}*0.734^y*(1-0.734)^{24-y},y =0,1,2,...,24$

Now,

$P(Y \leq 2) =\sum_{y = 0}^{2} \binom{24}{y}*0.734^y*(1-0.734)^{24-y} \approx 0$

d)

Let X denotes the sample mean for a random sample of size n =24 Here, X - Normal(850, -- Normal(850, 1.632) Now, 8.002 24 ) or X X – 850 - PC 1.63 848.00 - 850 1.63 X – 850 855.00 – 850 P(848.00 < X < 855.00) = P(848.00 < Å < 855.00) = P(X < 855.00) – P(X < 848.00) = P( 1.63 1.63 855.00 – 850 848.00 – 850 = P(848.00 < X < 855.00) = P(Z < -) - P(Z < 1.63 1.63 = P(848.00 < X < 855.00) = P(Z < 3.061862) – P(Z < -1.224745) + P(848.00 < < 855.00) = 0.998900 – 0.110336 = 0.888564 ~0.8886