The weight of potatoes produced by one particular farm are found to be approximately
normally distributed, with the mean weight of 147 g and a standard deviation of 23g.
i. What proportion of the potatoes produced by this farm will weigh less than 160 g?
ii. Of all the potatoes produced on the farm, 5% are considered too light. What is the
maximum weight that will be considered too light?
i. Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability related to that Z score.
ii. Reverse that process for this problem.
To find the answers to both questions, we need to use the concept of the standard normal distribution (also known as the Z-distribution). We can convert the given data into a standard normal distribution by subtracting the mean and dividing by the standard deviation.
Let's start by calculating the Z-score for the weight of 160g in question i.
Step 1: Calculate the Z-score using the formula:
Z = (X - μ) / σ
where:
X is the value we want to find the proportion for (in this case, 160g),
μ is the mean weight of the potatoes (147g), and
σ is the standard deviation (23g).
Plugging in the values, we get:
Z = (160 - 147) / 23
Z = 0.565
Step 2: Look up the corresponding area/proportion using a standard normal distribution table or calculator.
The Z-score of 0.565 corresponds to an area of approximately 0.7157. This means that approximately 71.57% (or 0.7157) of the potatoes produced by this farm will weigh less than 160g.
Moving on to the second question, we need to find the weight that corresponds to the 5th percentile of the standard normal distribution.
Step 1: Find the Z-score that corresponds to the 5th percentile.
To do this, we need to find the Z-score value for which the area to the left is 0.05 (since the 5th percentile is equivalent to an area of 0.05 to the left).
Using a standard normal distribution table or calculator, we find that the Z-score corresponding to the 5th percentile is approximately -1.645.
Step 2: Convert the Z-score back to the original weight (X) using the formula:
X = Z * σ + μ
Plugging in the values, we get:
X = -1.645 * 23 + 147
X ≈ 107.51
Therefore, the maximum weight that will be considered too light is approximately 107.51g.