Question

You are given the sample mean and the population standard deviation. Use this information to construct the 90% and 95% confidence intervals for the population mean. Interpret the results and compare the widths of the confidence intervals. From a random sample of 78 dates, the mean record high daily temperature in a certain city has a mean of 86.46°F. Assume the population standard deviation is 14.62°F. The 90% confidence interval is ( 1 ). The 95% confidence interval is (1 l). Which interval is wider?

Answer 1

The 90% confidence interval is narrower than the 95% confidence interval because a higher confidence level results in a wider interval. We use a Z-score and the known population standard deviation to calculate the error bound for the population mean.

Step-by-step explanation:

To calculate the confidence intervals for the population mean using the sample mean and population standard deviation, we apply the formula: (point estimate - EBM, point estimate + EBM), where EBM represents the error bound for a population mean. When constructing a 90% confidence interval or a 95% confidence interval, a Z-score corresponding to the confidence level is used to calculate the EBM.

The 90% confidence interval will be narrower than the 95% confidence interval, because a higher confidence level requires more area under the normal distribution, resulting in a wider interval for the same population.

To interpret, we say we are 90% or 95% confident that the true population mean falls within our calculated intervals. The higher the confidence level we desire, the wider the interval must be, indicating more uncertainty in pinpointing the exact population mean.