Use technology or a z-distribution table to find the indicated area.
The weights of grapefruits in a bin are normally distributed with a mean of 261 grams and a standard deviation of 9.4 grams.
Approximately 20% of the grapefruits weigh less than which amount?
239 g
247 g
253 g
261 g
To find the approximate weight that corresponds to 20% of the grapefruits weighing less, we need to use the z-distribution table.
1. Firstly, we need to convert the weight values to z-scores. The formula for calculating the z-score is:
z = (x - μ) / σ
where:
- x is the weight value
- μ is the mean of the distribution (261 grams in this case)
- σ is the standard deviation of the distribution (9.4 grams in this case)
2. Once we have the z-scores, we can use the z-distribution table to find the corresponding area (probability) for each z-score.
3. We want to find the weight value that corresponds to an area of approximately 20%. Looking at the z-distribution table, we need to find the z-score that corresponds to an area of 0.20 (20%).
4. Using the table or an online z-distribution calculator, we find that the z-score corresponding to an area of 0.20 is approximately -0.84.
5. Now, we can solve for the weight value corresponding to this z-score:
z = (x - μ) / σ
-0.84 = (x - 261) / 9.4
Rearranging the equation and solving for x:
-7.896 = x - 261
x = 253.104
So, approximately 20% of the grapefruits weigh less than 253.1 grams. Therefore, the answer is 253 g.
To find the weight of grapefruits that corresponds to the 20th percentile (or approximately 20% of the grapefruits weigh less), we can use the standard normal distribution table, also known as the Z-table.
First, we need to find the z-value that corresponds to the 20th percentile. This can be calculated using the formula:
z = (x - μ) / σ
Where:
x = the desired weight (unknown)
μ = mean weight of the grapefruits = 261 grams
σ = standard deviation = 9.4 grams
Now, rearranging the equation, we have:
x = (z * σ) + μ
To find the corresponding z value for the 20th percentile, we subtract 0.20 from 1 (since the table gives the area to the left of the z value):
z = 1 - 0.20 = 0.80
Now, we need to find the corresponding value from the z-table.
Using the z-table, the closest value to 0.80 is 0.7995. This value is found at the intersection of the row labeled 0.8 and the column labeled 0.09 (since 0.80 corresponds to z = 0.80, which is equivalent to 0.8 standard deviations from the mean).
Thus, the z value is approximately 0.7995.
Now, we can substitute the values into the equation:
x = (0.7995 * 9.4) + 261
x = 7.5953 + 261
x ≈ 268.5953
So, approximately 20% of the grapefruits weigh less than 268.5953 grams.
Out of the given options, the closest weight to 268.5953 grams is 261 grams. Therefore, the correct answer is 261 g.
from a z-score table
the lower 20% is below ≈.842 σ