Assume that recent wedding costs in the United States have a normal distribution
ID: 3340582 • Letter: A
Question
Assume that recent wedding costs in the United States have a normal distribution with a standard deviation of $8100. Based on a random sample of 29 recent U.S. weddings, a 95% confidence interval for the mean cost of all recent U.S. weddings id computed and found to be $22,704.5 to $29,949.3.
a) Determine if the assumptions were met to compute this confidence interval. List each assumption and show the information given to determine if each assumption is met.
b) If the assumptions are met, write the interpretation.
Explanation / Answer
a. When constructing confidence intervals the assumptions and conditions of the central limit theorem must be met in order to use the normal model.
Randomization Condition: The data must be sampled randomly.
Independence Assumption: The sample values must be independent of each other. This means that the occurrence of one event has no influence on the next event. Usually, if we know that people or items were selected randomly we can assume that the independence assumption is met.
10% Condition: When the sample is drawn without replacement (usually the case), the sample size, n, should be no more than 10% of the population.
Sample Size Condition: The sample size must be sufficiently large. Although the Central Limit Theorem tells us that we can use a Normal model to think about the behavior of sample means when the sample size is large enough, it does not tell us how large that should be. If the population is very skewed, you will need a pretty large sample size to use the CLT, however if the population is unimodal and symmetric, even small samples are ok. So think about your sample size in terms of what you know about the population and decide whether the sample is large enough. In general a sample size of 30 is considered sufficient.
Hence we see all conditions are met .
b. A 95% confidence interval is a range of values that you can be 95% certain contains the true mean of the population.
So here 95% confidence interval is $22,704.5 to $29,949.3 which you can be 95% certian contains the true mean of the population.