Question

. Suppose the weight of Chipotle burritos follows a normal distribution with mean of 450 grams, and variance of 100 grams2 . Define a random variable to be the weight of a randomly chosen burrito. (a) What is the probability that a Chipotle burrito weighs less than 445 grams? (3 points) (b) 20% of Chipotle burritos weigh more than what weig

Answer 1

Complete Question

Suppose the weight of Chipotle burritos follows a normal distribution with mean of 450 grams, and variance of 100 grams2 . Define a random variable to be the weight of a randomly chosen burrito.

(a) What is the probability that a Chipotle burrito weighs less than 445 grams? (3 points)

(b) 20% of Chipotle burritos weigh more than what weight

Answer:

a

`P(X < 445 )= 0.3085`

b

`k = 458.42`

Explanation:

From question we are told that

The population mean is
`\mu = 450 \ g`

The variance is
`var = 100 \ g^2`

The consider weight is
`x = 445 \ g`

The standard deviation is mathematically represented as

`\sigma = √(var)`

substituting values

`\sigma = √( 100)`

`\sigma = 10`

Given that weight of Chipotle burritos follows a normal distribution the the probability that a Chipotle burrito weighs less than x grams is mathematically represented as

`P(X < x ) = P ( (X - \mu )/(\sigma ) < (x - \mu )/(\sigma ) )`

Where
`(X - \mu )/(\sigma )` is equal to z (the standardized values of the random number X )

So

`P(X < x ) = P (Z < (x - \mu )/(\sigma ) )`

substituting values

`P(X < 445 ) = P (Z < (445 - 450 )/(10) )`

`P(X < 445 ) = P (Z <-0.5 )`

Now from the normal distribution table the value for
`P (Z <-0.5 )` is

`P(X < 445 ) = P (Z <-0.5 ) = 0.3085`

=>
`P(X < 445 )= 0.3085`

Let the probability of the Chipotle burritos weighting more that k be 20% so

`P(X > k ) = P ( (X - \mu )/(\sigma ) > (k - \mu )/(\sigma ) ) = 0.2`

=>
`P (Z> (k - \mu )/(\sigma ) ) = 0.2`

=>
`P (Z> (k - 450)/(10 ) ) = 0.2`

From the normal distribution table the value of z for
`P (Z> (k - \mu )/(\sigma ) ) = 0.2` is

`z = 0.8416`

=>
`(k - 450)/(10 ) = 0.8416`

=>
`k = 458.42`