Adrian has a bag full of pebbles that all look about the same. He weighs some of the pebbles and finds that their weights are normally distributed, with a mean of 2.6 grams and a standard deviation of 0.4 grams.
What percentage of the pebbles weigh more than 2.1 grams? Round to the nearest whole percent.



























[tex] \square [/tex] % of the pebbles weigh more than 2.1 grams.

Calculate the z-score: z = 0.4 2.1 − 2.6 ​ = − 1.25 .

Find the corresponding probability from the z-table: P ( x ≤ 2.1 ) = 0.11507 .
Calculate the probability of pebbles weighing more than 2.1 grams: 2.1) = 1 - 0.11507 = 0.88493"> P ( x > 2.1 ) = 1 − 0.11507 = 0.88493 .
Round to the nearest whole percent: 88 ​

Explanation

Understand the problem and provided data We are given that the weights of pebbles are normally distributed with a mean ( μ ) of 2.6 grams and a standard deviation ( σ ) of 0.4 grams. We want to find the percentage of pebbles that weigh more than 2.1 grams.

Calculate the z-score First, we need to calculate the z-score for 2.1 grams. The z-score formula is: z = σ x − μ ​ where x is the value we are interested in (2.1 grams), μ is the mean (2.6 grams), and σ is the standard deviation (0.4 grams).

Compute the z-score Plugging in the values, we get: z = 0.4 2.1 − 2.6 ​ = 0.4 − 0.5 ​ = − 1.25

Find the corresponding probability from the z-table Now we need to find the area to the right of z = − 1.25 in the standard normal distribution. The z-table usually gives the area to the left of a given z-score. From the provided z-table, we can see that the value corresponding to z = − 1.2 is 0.11507. We can approximate the value for z = − 1.25 by taking the average of the values for z = − 1.2 and z = − 1.3 . However, since we don't have the exact value for z = − 1.25 in the table, we will use the closest value we can find. Looking at the table, the value for z = − 1.2 is 0.11507. The value for z = − 1.3 is 0.09680. We can estimate the value for z = − 1.25 by taking the average of these two values. However, we can also look for the value closest to -1.25 in the table. The closest value in the table is -1.2, which corresponds to 0.11507.

Calculate the probability of pebbles weighing more than 2.1 grams The value from the z-table (0.11507) represents the probability that a pebble weighs less than or equal to 2.1 grams. To find the probability that a pebble weighs more than 2.1 grams, we subtract this value from 1: 2.1) = 1 - P(x \leq 2.1) = 1 - 0.11507 = 0.88493"> P ( x > 2.1 ) = 1 − P ( x ≤ 2.1 ) = 1 − 0.11507 = 0.88493

Convert the probability to a percentage To express this probability as a percentage, we multiply by 100: 0.88493 × 100 = 88.493%

Round to the nearest whole percent Finally, we round the percentage to the nearest whole percent: 88.493% ≈ 88%


Examples
In quality control, understanding the distribution of product weights is crucial. For instance, if a cereal company wants to ensure that at least 90% of its cereal boxes contain more than a certain weight, they can use normal distribution calculations to determine the appropriate fill level. This helps them avoid underfilling boxes, which could lead to customer dissatisfaction, or overfilling, which increases costs. By analyzing the mean and standard deviation of their filling process, they can calculate the percentage of boxes that meet their weight requirements.

Answered by GinnyAnswer | 2025-07-04