On the sixth day of Statistics, my teacher gave to me, six geese-a-laying. The eggs they were a-laying were normally distributed and had a mean of 7cm and standard deviation of 2cm. What is the probability that if you randomly select one egg from each of the six geese, the mean of all six will be less than 6 cm?

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I know we must use z-score but how

Define table laying

To find the probability that the mean of all six eggs will be less than 6 cm, we can use the concept of the Central Limit Theorem. According to the Central Limit Theorem, the distribution of sample means from any population approaches a normal distribution as the sample size increases.

Here's how we can compute the probability step by step:

Step 1: Calculate the standard deviation of the sampling distribution of the mean.
The standard deviation of the sampling distribution of the mean (also known as the standard error) is equal to the standard deviation of the population divided by the square root of the sample size. In this case, the standard deviation of the population is 2 cm, and since we are randomly selecting one egg from each of the six geese, the sample size is 6.

Standard deviation of the sampling distribution of the mean = standard deviation of the population / square root of the sample size
= 2 cm / sqrt(6) cm

Step 2: Calculate the z-score for the desired mean.
The z-score represents the number of standard deviations the desired mean is away from the population mean. The formula for calculating the z-score is:

z = (x - μ) / σ

Where:
- x is the desired mean (6 cm in this case)
- μ is the population mean (7 cm in this case)
- σ is the standard deviation of the sampling distribution of the mean

Plugging in the values, we have:
z = (6 cm - 7 cm) / (2 cm / sqrt(6) cm)

Step 3: Calculate the probability using the z-score.
Once we have the z-score, we can find the probability using a standard normal distribution table or calculator. The probability can be interpreted as the area under the normal curve to the left of the z-score.

So, the probability can be calculated as:
P(X < 6 cm) = P(Z < z-score)