A cake company produces cakes with approximately normally distributed with a mean of 300g and a standard deviation of 3g.Find the proportion of cakes that are under 298g. [1 mark]0.10560.40130.25230.1587There is a 3% reject rate for cakes defined as under-weight products. What is the minimum acceptable weight? [1 mark]
Question
A cake company produces cakes with approximately normally distributed with a mean of 300g and a standard deviation of 3g.Find the proportion of cakes that are under 298g. [1 mark]0.10560.40130.25230.1587There is a 3% reject rate for cakes defined as under-weight products. What is the minimum acceptable weight? [1 mark]
Solution
To find the proportion of cakes that are under 298g, we can use the standard normal distribution.
Step 1: Calculate the z-score The z-score is calculated using the formula: z = (x - μ) / σ, where x is the value we want to find the proportion for, μ is the mean, and σ is the standard deviation.
In this case, x = 298g, μ = 300g, and σ = 3g.
z = (298 - 300) / 3 z = -2 / 3 z ≈ -0.6667
Step 2: Find the proportion using the z-score We can use a standard normal distribution table or a calculator to find the proportion associated with the z-score.
Looking up the z-score of -0.6667 in the table, we find that the proportion is approximately 0.2523.
Therefore, the proportion of cakes that are under 298g is approximately 0.2523.
To find the minimum acceptable weight with a 3% reject rate for under-weight products, we need to find the z-score associated with a 3% proportion.
Step 1: Find the z-score Using a standard normal distribution table or a calculator, we can find the z-score associated with a 3% proportion.
Looking up the proportion of 0.03 in the table, we find that the z-score is approximately -1.8808.
Step 2: Calculate the minimum acceptable weight We can use the z-score formula to find the minimum acceptable weight.
z = (x - μ) / σ
In this case, we want to find x, the minimum acceptable weight. μ is still 300g, and σ is still 3g.
-1.8808 = (x - 300) / 3
Solving for x, we get:
-1.8808 * 3 = x - 300 -5.6424 = x - 300 x = 300 - 5.6424 x ≈ 294.3576
Therefore, the minimum acceptable weight is approximately 294.3576g.
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