Question

The weight of a sack of potatoes is normally distributed with a mean of 240 ounces and a standard deviation of 6 ounces. Round your answers below to 2 decimal places. (a) What is the probability (in percent) that a sack has more than 252 ounces of potatoes? % (b) What is the probability (in percent) that a sack has more than 222 ounces of potatoes? % (c) What is the probability (in percent) that a sack has between 234 and 246 ounces of potatoes? % (d) Where are the first and third quartiles located? Q1 is at ounces and Q3 is at ounces.

Answer 1

To solve these questions, we'll use the properties of the normal distribution and the z-score formula.

(a) Probability that a sack has more than 252 ounces of potatoes:

Step 1: Calculate the z-score for 252 ounces.
z = (x - μ) / σ
where x = 252 ounces, μ = 240 ounces (mean), and σ = 6 ounces (standard deviation).

z = (252 - 240) / 6
z = 2

Step 2: Find the probability of the z-score using a standard normal distribution table or calculator. The probability of having a z-score greater than 2 is approximately 2.28%.

(b) Probability that a sack has more than 222 ounces of potatoes:

Step 1: Calculate the z-score for 222 ounces.
z = (x - μ) / σ
where x = 222 ounces, μ = 240 ounces (mean), and σ = 6 ounces (standard deviation).

z = (222 - 240) / 6
z = -3

Step 2: Find the probability of the z-score using a standard normal distribution table or calculator. The probability of having a z-score greater than -3 (which is the same as being more than 3 standard deviations below the mean) is almost 0%. We can approximate it to 0%.

(c) Probability that a sack has between 234 and 246 ounces of potatoes:

Step 1: Calculate the z-scores for 234 ounces and 246 ounces.
z1 = (234 - 240) / 6
z1 = -1

z2 = (246 - 240) / 6
z2 = 1

Step 2: Find the probabilities for both z-scores separately.

Probability (z