Hannah selected a simple random sample of all adults in her town and based on this sample, consrtucted a confidence interval for the mean salary of all adults in the town. however, the distribution of salaries in the town is not exactly normal. will the confidence interval still give a good estimated of the mean salary?
It would depend on the extremeness of the skew. The more the skew, the less confidence you will have in the interval.
When constructing a confidence interval, the assumption is usually that the data follows a normal distribution. However, if the sample is random and sufficiently large, the Central Limit Theorem (CLT) comes into play. The CLT states that, regardless of the underlying distribution of the population, the sampling distribution of the mean approaches a normal distribution as the sample size increases.
So, even if the distribution of salaries in the town is not exactly normal, as long as the sample is random and sufficiently large, the confidence interval can still provide a good estimate of the mean salary of all adults in the town.
To be more certain about the appropriateness of the confidence interval, you can check for violations of the assumptions regarding normality. You can examine the skewness and kurtosis of the sample data or conduct a hypothesis test for normality, such as the Shapiro-Wilk test or the Kolmogorov-Smirnov test.
However, please note that if the sample size is small or if there are substantial deviations from normality, the confidence interval may not be as accurate or reliable. In such cases, alternative methods like bootstrapping or non-parametric confidence intervals could be considered.