A particular fruit's weights are normally distributed, with a mean of 594 grams and a standard deviation of 16 grams.
If you pick a fruit at random, then 18% of the time, their mean weight will be greater than how many grams?
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To find the weight at which the mean weight will be greater 18% of the time, we will need to use the Z-score formula.
The Z-score formula is: Z = (X - μ) / σ
Where:
- Z is the Z-score
- X is the raw score
- μ is the mean
- σ is the standard deviation
In this case, the mean is 594 grams (μ), and the standard deviation is 16 grams (σ).
We need to find X, so we rearrange the formula:
X = Z * σ + μ
To find the Z-score corresponding to 18% of the area under the normal distribution curve, we can use a Z-table or a calculator. Looking up the closest Z-score to 0.18, we find it to be approximately Z = 0.91.
Now we can substitute the values into the formula:
X = 0.91 * 16 + 594
X ≈ 14.56 + 594
X ≈ 608.56
Therefore, if you pick a fruit at random, then 18% of the time, their mean weight will be greater than approximately 608.56 grams.
To find the weight in grams for which the mean weight will be greater 18% of the time, we need to use the properties of the normal distribution.
Step 1: Convert the percentage to a z-score.
The percentage indicates the area under the curve to the right of a particular z-score. We want to find the z-score that corresponds to the area to the left of it being 82% (100% - 18%).
The z-score can be found using a standard normal distribution table or a calculator. In this case, we want the z-score for the left-tailed area of 82%, which corresponds to a z-score of approximately 0.91.
Step 2: Find the weight value corresponding to the z-score.
To determine the weight value, we need to use the formula for converting a z-score to a raw score:
z = (X - μ) / σ
Where:
X is the raw score (weight in grams)
μ is the population mean (mean weight of the fruit)
σ is the population standard deviation (standard deviation of the fruit's weights)
Rearranging the formula to solve for X:
X = z * σ + μ
Substituting the values:
X = 0.91 * 16 + 594
X = 14.56 + 594
X ≈ 608.56
Therefore, if you pick a fruit at random, then 18% of the time, their mean weight will be greater than approximately 608.56 grams.
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability (.18) and its Z score.
Z = (score-mean)/SD