Question

4.5 Hen eggs: The distribution of the number of eggs laid by a certain species of hen during their breeding period is 35 eggs with a standard deviation of 18.2. Suppose a group of researchers randomly samples 45 hens of this species, counts the number of eggs laid during their breeding period, and records the sample mean. They repeat this 1,000 times, and build a distribution of sample means. What is this distribution called? Would you expect the shape of this distribution to be symmetric, right skewed, or left skewed? Explain your reasoning. Calculate the variability of this distribution and state the appropriate term used to refer to this value. Suppose the researchers' budget is reduced and they are only able to collect random samples of 10 hens. The sample mean of the number of eggs is recorded, and we repeat this 1.000 times, and build a new distribution of sample means. How will the variability of this new distribution compare to the variability of the original distribution?

Answer 1

Answer:

This distribution is called the sampling distribution of the mean and it is the distribution of the sample means taken from the same population.

We would expect this distribution to be symmetric because the distribution of sample means will always approach a normal distribution as the sample size increases. (Sample size greater than 30 and the approach increases if the variable has a normal distribution itself).

The variability of this distribution is the standard distribution of the sample means and it is called the standard error of the mean (σₓ).

σₓ = σ/√n

Where σ is the standard deviation of the population and n is the sample size. As we can see from the formula, the bigger the sample size, the smaller the standard error.

σₓ = 18.2/√45 = 2.7

If the researcher reduces the sample size to 10, then the variability will increase. If n decreases, the standard error σₓ is going to increase.

Answer 2

The variability of this new distribution compare to the variability of the original distribution is the,  If we reduces the sample size to 10, then the variability is  increase. when n decreases, the standard error σₓ is  increase.

We have given that standard deviation =18.2

sample size n=45

What is the meaning of distribution?

This distribution is called the sampling distribution of the mean and it is the distribution of the sample means taken from the same population.

Therefore,We have to  expect this distribution to be symmetric because the distribution of sample means it is always approach a normal distribution when the sample size increases.

The variability of this distribution is the standard distribution of the sample means and it is called the standard error of the mean (σₓ).

And it is calculated by the formula.

σₓ = σ/√n

Where the  σ is the standard deviation of the population and n is the sample size. From the formula, the bigger the sample size, the smaller the standard error and it is given by,

σₓ = 18.2/√45 = 2.7

Therefore we can say that,If we reduces the sample size to 10, then the variability is  increase. when  n decreases, the standard error σₓ is  increase.

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