Dr. Mihaela Sabou wants to estimate the average length of a...

Dr. Mihaela Sabou wants to estimate the average length of a hospital stay (number of days) for patients with a certain infectious disease. From a random sample of 25 patient records, she finds that the average number of days in the hospital for such patients is 6 days, with a standard deviation of 1.8 days. a. Find the reliability factor for a $95 \%$ confidence interval estimate of the population mean length of stay. b. Find the LCL for a $99 \%$ confidence interval estimate of the population mean length of stay.

Dr. Mihaela Sabou wants to estimate the average length of a hospital stay (number of days) for patients with a certain infectious disease. From a random sample of 25 patient records, she finds that the average number of days in the hospital for such patients is 6 days, with a standard deviation of 1.8 days.
a. Find the reliability factor for a $95 \%$ confidence interval estimate of the population mean length of stay.
b. Find the LCL for a $99 \%$ confidence interval estimate of the population mean length of stay.

Key Concepts

Degrees of Freedom

Degrees of freedom, generally calculated as the sample size minus one for a single sample, determine the specific form of the t-distribution. They adjust for the fact that sample estimates, such as the sample mean and standard deviation, are used in place of their population counterparts.

Reliability Factor

The reliability factor is the critical value obtained from the t-distribution corresponding to the desired confidence level and degrees of freedom. It is used as a multiplier for the standard error to determine the margin of error in the confidence interval.

Confidence Interval Estimation

Confidence interval estimation is a statistical method used to estimate an unknown population parameter by providing a range of plausible values derived from sample data. This range is associated with a specified level of confidence, indicating that if the estimation process were repeated many times, the true parameter would lie within the interval a designated percentage of the time.

t-distribution

The t-distribution is used when drawing inferences about a population mean based on sample data, especially when the sample size is small and the population standard deviation is unknown. It accounts for extra uncertainty by having heavier tails compared to the normal distribution, and its shape depends on the degrees of freedom.

Lower Confidence Limit (LCL)

The lower confidence limit is the smallest value in a confidence interval, representing the lower bound of the estimated range for the population parameter. It is calculated by subtracting the margin of error from the sample mean, thereby providing one end of the interval estimate at the designated level of confidence.

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