Final answer:
To construct a confidence interval for a population mean, a t-distribution is used for small samples after checking normality, while a normal distribution is used for large samples or if the population is assumed to be normal. The interval includes the sample mean and a margin of error based on the standard deviation and a z-score or t-score.
Step-by-step explanation:
To construct a confidence interval for the population mean using the data provided, one must first determine the appropriate distribution to use. For small samples (n < 30) that do not follow a normal distribution, the t-distribution should be used after checking for normality with plots such as the histogram and boxplot. However, if the sample size is large enough, or if we can assume that the sampled population is normally distributed, the normal distribution is appropriate.
For a population mean, the confidence interval generally takes the form of the sample mean ± the margin of error. The margin of error is calculated using either the z-score or t-score multiplied by the standard deviation divided by the square root of the sample size. For a 95% confidence interval using a normally distributed large sample, you would use the z-score; for small samples, the t-score from the t-distribution would be used.
For example, to construct a 95% confidence interval for biostatistician salaries in Connecticut with a sample mean of $80,080 and sample standard deviation of $18,850 for 25 entries, you would use the t-distribution since the sample size is below 30. The same approach would apply to the context of resting heart rates for athletes and television viewing habits of statistics students.