Question

1. Explain how a sampling distribution is created. What does the mean (center) of the sampling distribution tell us? What is the standard deviation of the sampling distribution called and what does it tell us?
2. Explain the central limit theorem. Why is it important for a sampling distribution of sample means and sample proportions to be normal? What requirement ensures that a sampling distribution for sample means is normal? What requirement ensures that a sampling distribution for sample proportion is normal?
3. What is margin of error? What is a confidence interval? How do we use the sample statistic and the margin of error to calculate the confidence interval?
4. Explain how to use StatKey to calculate a critical value Z-score and a critical value T-score. Explain how to use the critical value and the standard error to calculate margin of error.
5. As confidence levels decrease, what happens to the margin of error and the confidence interval?
6. As the sample size decreases, what happens to the margin of error and the confidence interval?
7. Explain how to interpret two-population confidence intervals that are either (+ , +), ( - , - ) or ( - , + ).

Answer 1

• To create a sampling distribution a researcher must:

(1) select a random sample of a specific size (N) from a population,

(2) calculate the chosen statistic for this sample (e.g. mean),

(3) plot this statistic on a frequency distribution, and

(4) repeat these steps an infinite number of times.

The mean of a distribution is the average of data set of the distribution.

The standard deviation of a sampling distribution is called the standard error. Standard deviation tells you how spread out the data is. It is a measure of how far each observed value is from the mean.

• The central limit theorem states that if you have a population with mean μ and standard deviation σ and take sufficiently large random samples from the population with replacement , then the distribution of the sample means will be approximately normally distributed. It is important for a sampling distribution of sample means and sample proportions to be normal to avoid a skewed distribution which will ensure it's normality.

Requirement ensures that a sampling distribution for sample means and proportion is normal:

1. If sample distribution is normally distributed than mean is normal or else the sample size should be large, mostly N=30 is considered as magic number, above which normality is achieved for the sample means.
2. The shape of the distribution of proportion will be approximately normal as long as the sample size n is large enough. The convention is to require both np and n(1 – p) to be at least 10, where p is the population proportion.

• A margin of error tells you how many percentage points your results will differ from the real population value,i.e., the amount of random sampling error in the results of a survey.

A confidence Interval is a range of values we are fairly sure our true value lies in.

The confidence interval is the estimate ± the margin of error.(this is how it's used)

• In StatKey to find the critical z- value and t-value following steps should be followed:

1. In the home page, go to normal distribution(for z-value) and t-distribution(for t value) under theoritical distribution.
2. Fill in the mean and standard deviation by editing parameters.
3. Click on two-tail option in the plot itself, then you'll see that 95% confidence interval is highlighted and the edge value of this interval is the critical value from where the colour of the plot changes. If you want to change the confidence interval then click on the 0.95 text and edit there and then critical value of that will be visible.

Margin error can be calculated as Margin of error = Critical value x Standard error of the sample.

• As the confidence level increases, the critical value increases and hence the margin of error increases.This is intuitive; the price paid for higher confidence level is that the margin of errors increases.

Increasing the confidence level increases the error bound, making the confidence interval wider. Decreasing the confidence level decreases the error bound, making the confidence interval narrower.

• As the sample size decreases, the margin of error increases. This relationship is called an inverse because the two move in opposite directions.

Increasing the sample size decreases the width of confidence intervals, because it decreases the standard error.

Hope this helps!