Question

## Sampling distributions
Answer the questions in your own words but you may refer to notes

1. What is a sampling distribution. Describe with an example where the population is Fremont, CA and the parameter of interest is whether the household owns a Tesla.

2. Describe the concept of sampling variability.

## Explain the Central Limit Theorem
Explain what the Central Limit Theorem tells you about the sampling distribution. But don't use fancy technical terms. For example if you were researching a person's age, describe the distribution of the individual ages, and the distribution of the average age (complete with numbers)

Creating Confidence Intervals

1. What is the relationship between the Empirical Rule and a 95% confidence interval?

2. How are confidence and margin of error related? Explain with examples.

1. Explain with words and steps how to create a confidence interval for estimating the true value of the population mean of the number of cars owned by Beverly Hills households. Don't forget to write an interpretation of the confidence interval. The sample results were:
a. Sample size of 100 people.
a. Sample average was 5.2.
a. Sample standard deviation was 2.8.

Answer 1

1. A sampling distribution is a probability distribution of a statistic obtained through a large number of samples drawn from a specific population.

Example: Here parameter of interest is whether the household owns a Tesla. Thus the parameter here can be the proportion of household owning Tesla. Then the sampling distribution of the proportion is approximately normally distributed.

2. Sampling variability: It refers to the fact that how much statistical information from a sample (for a statistic) will vary as the random sampling is repeated. This depends on the sample size and as the sample size increases, sampling variability decrease.

Central Limit Theorem tells us that the sampling distribution of the sample mean will be normally distributed with mean equal to the population mean and standard deviation where is the population standard deviation and n is the sample size.

The relationship between the Empirical Rule and a 95% confidence interval is that according to empirical rule the percentage of data within 2 times the standard deviation of the mean is 95%. Thus, this interval is said to be 95% confidence interval of the mean.

The relation between confidence and margin of error: Confidence interval of a parameter is equal to the parameter estimate margin of error. e.g. confidence interval for true mean = sample margin of error.

The confidence interval for estimating the true value of the population mean of the number of cars owned by Beverly Hills households is given by the sample margin of error. Here margin of error is . Thus the 95% confidence interval is (4.6512, 5.7488).

Interpretation: we're 95% confident the true population mean of the number of cars owned by Beverly Hills households lies in the above interval.