Question

A company sells eggs whose individual weights are normally distributed with a mean of 70 g and a standard deviation of 2 g. Suppose that these eggs are sold in packages that each contain 4 eggs that represent an SRS from the population.

What is the probability that the mean weight of 4 eggs in a package is less than 68.5 g?
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O P( x < 68.5) ≈ 0.07
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O P( x < 68.5) ≈ 0.12
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O P( x < 68.5) ≈ 0.18
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O P( x < 68.5) ≈ 0.23

O We cannot calculate this probability because the sampling distribution is not normal.

Answer 1

The probability that the mean weight of 4 eggs in a package is less than 68.5 g is approximately 0.07. This is found by calculating the standard error, obtaining the z-score, and consulting the standard normal distribution. The correct answer is option O P( x < 68.5) ≈ 0.23

Step-by-step explanation:

The question pertains to the probability distribution of sample means, which is a topic under statistics, a branch of mathematics. Specifically, it deals with the normal distribution of egg weights and the sampling distribution of the mean weight of eggs in a package.

Since we know the population mean (μ = 70 g), the population standard deviation (σ = 2 g), and the sample size (n = 4), we can find the sampling distribution of the sample mean. The standard error (SE) for the sample means is σ/ √n, which in this case is 2 g / √4, resulting in 1 g. The sampling distribution of the sample mean will also be normally distributed because the population distribution is normal.

To find the probability that the mean weight of 4 eggs in a package is less than 68.5 g, we use the standard normal distribution (z-distribution). We compute the z-score using the formula: z = (X - μ) / SE. Substituting the values gives us z = (68.5 - 70) / 1 = -1.5.

Using standard normal distribution tables or a calculator, we find the probability corresponding to z = -1.5. This value is approximately 0.0668, which we can round to 0.07. Hence, P( x < 68.5) ≈ 0.07.