A triathlon consisting of swimming, cycling, and running is...

A triathlon consisting of swimming, cycling, and running is one of the more strenuous amateur sporting events. The article "Cardiovascular and Thermal Response of Triathlon Performance" (Medicine and Science in Sports and Exercise, 1988: 385 ) reports on a research study involving nine male triathletes. Maximum heart rate (beats/min) was recorded during perfor- mance of each of the three events. For swimming, the sample mean and sample standard deviation were 188.0 and $7.2,$ respectively. Assuming that the heart-rate distribution is (approx- imately) normal, construct a 98$\%$ CI for true mean heart rate of triathletes while swimming.

A triathlon consisting of swimming, cycling, and running is one of the more strenuous amateur
sporting events. The article "Cardiovascular and Thermal Response of Triathlon Performance"
(Medicine and Science in Sports and Exercise, 1988: 385 ) reports on a research study
involving nine male triathletes. Maximum heart rate (beats/min) was recorded during perfor-
mance of each of the three events. For swimming, the sample mean and sample standard
deviation were 188.0 and $7.2,$ respectively. Assuming that the heart-rate distribution is (approx-
imately) normal, construct a 98$\%$ CI for true mean heart rate of triathletes while swimming.

Key Concepts

Confidence Interval

A confidence interval is a range of values, derived from sample data, that is likely to contain the true value of a population parameter. It reflects the uncertainty inherent in using sample statistics to estimate that parameter and is defined with a specific confidence level, such as 98% or 95%, indicating the proportion of similar intervals that would contain the true parameter if the study were repeated indefinitely.

t-Distribution

The t-distribution is a probability distribution used in inferential statistics when estimating the mean of a normally distributed population in situations where the sample size is small and the population standard deviation is unknown. It is similar to the normal distribution but has heavier tails, which provides a more conservative estimate of the parameter when variability is based solely on sample data.

Degrees of Freedom

Degrees of freedom, in the context of the t-distribution, refer to the number of independent values that are free to vary when estimating a statistical parameter. In estimating a population mean, the degrees of freedom are calculated as the sample size minus one (n - 1), which influences the critical value from the t-distribution used in constructing a confidence interval.

Standard Error

The standard error quantifies the amount of variability in the estimate of a population parameter, such as the mean, from a sample. It is calculated by dividing the sample standard deviation by the square root of the sample size, and it serves as an indicator of how much the sample mean is expected to differ from the true population mean.

Critical Value

The critical value is a factor that reflects the desired confidence level and is obtained from the t-distribution when the population standard deviation is unknown and the sample size is small. It determines how many standard errors away from the sample mean the interval must extend to capture the true population mean with the specified level of confidence.

Margin of Error

The margin of error is the product of the critical value and the standard error, and it represents the range above and below the sample mean within which the true population parameter is expected to lie. It is a key component in the construction of a confidence interval, providing a measure of the precision of the estimate.

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