Question

The number of chocolate chips in a bag of chocolate chip cookies is approximately normally distributed with mean 1263 and a standard deviation of 117. ​(a) Determine the 29th percentile for the number of chocolate chips in a bag.​(b) Determine the number of chocolate chips in a bag that make up the middle 95​% of bags.​(c) What is the interquartile range of the number of chocolate chips in a bag of chocolate chip​ cookies?

Answer 1

Answer:

(a) The 29th percentile for the number of chocolate chips in a bag is 1198.65.

(b) The number of chocolate chips in a bag that make up the middle 95​% of bags are [1146, 1380].

(c) The inter-quartile range of the number of chocolate chips in a bag of chocolate chip​ cookies is 157.83.

Step-by-step explanation:

Let the random variable X represent the number of chocolate chips in a bag of chocolate chip cookies.

The random variable X is normally distributed with mean, μ = 1263 and a standard deviation, σ = 117.

(a)

Compute the 29th percentile for the number of chocolate chips in a bag as follows:

P (X < x) = 0.29

⇒ P (Z < z) = 0.29

The value of z for the above probability is, z = -0.55.

Compute the value of x as follows:

$z=\frac{x-\mu}{\sogma}\\-0.55=\frac{x-1263}{117}\\x=1263-(117\times 0.55)\\x=1198.65$

Thus, the 29th percentile for the number of chocolate chips in a bag is 1198.65.

(b)

According to the Empirical rule 95% of the normally distributed data lies within 2 standard deviations of the mean.

P (μ - σ < X < μ + σ) = 0.95

P (1263 - 117 < X < 1263 + 117) = 0.95

P (1146 < X < 1380) = 0.95

Thus, the number of chocolate chips in a bag that make up the middle 95​% of bags are [1146, 1380].

(c)

The inter-quartile range of the normal distribution is:

IQR = 1.349 σ

Compute the inter-quartile range of the number of chocolate chips in a bag of chocolate chip​ cookies as follows:

IQR = 1.349 σ

      = 1.349 × 117

      = 157.833

Thus, the inter-quartile range of the number of chocolate chips in a bag of chocolate chip​ cookies is 157.83.