Question

The number of chocolate chips in a bag of chocolate chip cookies is approximately normally distributed with mean 1261 and a standard deviation of 118. ​(a) Determine the 30th 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 98​% of bags. ​(c) What is the interquartile range of the number of chocolate chips in a bag of chocolate chip​ cookies?

Answer 1

a) The 30th percentile for the number of chocolate chips in a bag is approximately 1149.6.

b) The number of chocolate chips in a bag that make up the middle 98​% of bags is between 935.4 and 1586.6.

c) The interquartile range for the number of chocolate chips in a bag of chocolate chip cookies is 213.6.

(a) Determine the 30th percentile for the number of chocolate chips in a bag.

The 30th percentile is the value below which 30% of the bags fall, and above which 70% of the bags fall. We can find the 30th percentile using the following formula:

30th percentile = mean + (0.30) * (standard deviation)

Plugging in the values for mean and standard deviation, we get:

30th percentile = 1261 + (0.30) * (118) = 1149.6

Therefore, the 30th percentile for the number of chocolate chips in a bag is approximately 1149.6.

(b) Determine the number of chocolate chips in a bag that make up the middle 98​% of bags.

The middle 98​% of bags excludes 1​% of bags on each tail of the distribution. These tails are equal, so each tail contains 0.5​% of the bags. We can find the values that separate these tails from the middle 98​% of bags using the following formula:

Value = mean + (z) * (standard deviation)

where z is the z-score corresponding to the desired percentile. For 0.5​%, the z-score is approximately -2.33. Plugging in the values for mean, standard deviation, and z-score, we get:

Value = 1261 + (-2.33) * (118) = 935.4

Therefore, the number of chocolate chips in a bag that make up the middle 98​% of bags is between 935.4 and 1586.6.

(c) What is the interquartile range of the number of chocolate chips in a bag of chocolate chip​ cookies?

The interquartile range (IQR) is the difference between the 75th percentile and the 25th percentile. We can find the 75th percentile using the same formula we used for the 30th percentile, but with a z-score of 0.67:

75th percentile = 1261 + (0.67) * (118) = 1367.8

The 25th percentile is simply the 100th percentile - the 75th percentile, which is:

25th percentile = 100th percentile - 75th percentile = mean - (0.67) * (standard deviation) = 1154.2

Therefore, the interquartile range for the number of chocolate chips in a bag of chocolate chip cookies is 1367.8 - 1154.2 = 213.6.

Answer 2

Answer:

(A) 1199.168

(B) 1503.372

(C) 159.17728

Explanation:

(A) To determine the 30th percentile for the number of chocolate chips in the bag, we find the z-score for the 30th percentile.

Found using a z-table or z-calculator, the z-score for the 30th percentile is -0.524

The formula for finding X (the number of items in a given percentile) is:

X = M + Z(S.D.)

Where M is the mean, Z is the specific z-score of the sought percentile and S.D. is the standard deviation.

So for the 30th percentile,

X = 1261 + (-0.524)(118)

X = 1261 - 61.832 = 1199.168

(B) The number of chocolate chips that make up the middle 98% of chips in the bag is

X = 1261 + (2.054)(118)

X = 1261 + 242.372 = 1503.372

(C) For normal distributions, Interquartile range is Q3 - Q1, that is; 3rd quartile minus 1st quartile.

This is within 1.34896 standard deviations of the mean.

IQR = (1.34896)(118)

IQR = 159.17728