Question

The weights of ice cream cartons are normally distributed with a mean weight of 8 ounces and a standard deviation of 0.4 ounce. a) What is the probability that a randomly selected carton has a weight greater than 8.13 ounces? b) A sample of 16 cartons is randomly selected. What is the probability that their mean weight is greater than 8.13 ounces?

Answer 1

Answer:

a) 0.373 probability that a randomly selected carton has a weight greater than 8.13 ounces

b) 0.097 probability that mean weight of 16 randomly selected carton is greater than 8.13 ounces

Step-by-step explanation:

We are given the following information in the question:

Mean, μ =  8 ounces

Standard Deviation, σ = 0.4 ounce

We are given that the distribution of weights of ice cream cartons is a bell shaped distribution that is a normal distribution.

Formula:

$z_{score} = \displaystyle\frac{x-\mu}{\sigma}$

a) P(weight greater than 8.13 ounces)

P(x > 8.13)

$P( x > 8.13) = P( z > \displaystyle\frac{8.13 - 8}{0.4}) = P(z > 0.325)$

$= 1 - P(z \leq 0.325)$

Calculation the value from standard normal z table, we have,  

$P(x > 610) = 1 - 0.627 = 0.373 = 37.3\%$

b) P(weight of 16 randomly selected cartons is greater than 8.13 ounces)

P(x > 8.13)

$P( x > 8.13) = P( z > \displaystyle\frac{8.13 - 8}{\frac{0.4}{\sqrt{16}}}) = P(z > 1.3)$

$= 1 - P(z \leq 1.3)$

Calculation the value from standard normal z table, we have,  

$P(x > 8.13) = 1 - 0.903 = 0.097 = 9.7\%$

Answer 2

The probability that a randomly selected carton has a weight greater than 8.13 ounces is 0.3745.

What is a Z-table?

A z-table also known as the standard normal distribution table, helps us to know the percentage of values that are below (or to the left of the Distribution) a z-score in the standard normal distribution.

As it is given that the mean weight is 8 ounces, while the standard deviation is 0.4 ounces. And it is given that the weights of ice cream cartons are normally distributed.

A.) The probability that a randomly selected carton has a weight greater than 8.13 ounces,

$P(X > 8.13)= 1-P(z < \dfrac{8.13-8}{0.4})$

                     $=1-0.6255\\\\=0.3745$

Hence, the probability that a randomly selected carton has a weight greater than 8.13 ounces is 0.3745.

B.) When a sample of 16 cartons is randomly selected then the probability that their mean weight is greater than 8.13 ounces, can be written as,

$P(X > 8.13)= 1-P(z < \dfrac{8.13-8}{\frac{0.4}{\sqrt{0.16}}})$

                     $= 1-P(z < 1.3)\\\\=1-0.9032\\\\=0.0968$

Hence, the probability that their mean weight is greater than 8.13 ounces is 0.0968.

Learn more about Z-table:

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