A study of the ability of individuals to walk in a straight...

A study of the ability of individuals to walk in a straight line $("$ Can We Really Walk Straight?" Amer.J. Phys. Anthropol.. $1992 : 19-27$ ) reported the accompanying data on cadence (strides per second) for a sample of $n=20$ randomly selected healthy men. $\begin{array}{llll}{.95} & {.85} & {.92} & {.95} \\ {.78} & {.93} & {.93} & {1.05}\end{array}$ $\begin{array}{cccc}{.93} & {.86} & {1.00} & {.92} \\ {.93} & {1.06} & {1.06} & {.96}\end{array}$ $\begin{array}{ll}{.85} & {.81} \\ {.81} & {.96}\end{array}$ A normal probability plot gives substantial support to the assumption that the population distribution of cadence is approximately normal. Calculate and interpret a 95$\%$ confidence interval for population mean cadence.

A study of the ability of individuals to walk in a straight line $("$ Can We Really Walk Straight?"
Amer.J. Phys. Anthropol.. $1992 : 19-27$ ) reported the accompanying data on cadence (strides per
second) for a sample of $n=20$ randomly selected healthy men.
$\begin{array}{llll}{.95} & {.85} & {.92} & {.95} \\ {.78} & {.93} & {.93} & {1.05}\end{array}$
$\begin{array}{cccc}{.93} & {.86} & {1.00} & {.92} \\ {.93} & {1.06} & {1.06} & {.96}\end{array}$
$\begin{array}{ll}{.85} & {.81} \\ {.81} & {.96}\end{array}$
A normal probability plot gives substantial support to the assumption that the population
distribution of cadence is approximately normal. Calculate and interpret a 95$\%$ confidence
interval for population mean cadence.

Key Concepts

Interpretation of the Confidence Interval

The interpretation of a 95% confidence interval is that if the same sampling procedure were repeated many times, approximately 95% of the constructed intervals would contain the true population mean. It provides a range within which one can be reasonably confident that the true mean lies, rather than giving a probability statement about any specific interval.

Degrees of Freedom

Degrees of freedom refer to the number of independent values that are free to vary when estimating a statistical parameter. In the context of a t-distribution for a mean, the degrees of freedom are typically equal to the sample size minus one, reflecting the loss of a parameter estimated from the data.

Confidence Interval for a Mean

A confidence interval for a mean is a range of values calculated from sample data that is likely to contain the true population mean with a specified level of confidence. This interval is constructed using the sample mean, the variability in the data, and the chosen confidence level, which reflects the degree of certainty in the estimation.

t-Distribution

When the population standard deviation is unknown and the sample size is relatively small, the t-distribution is used instead of the normal distribution. It accounts for additional uncertainty by having heavier tails, and it becomes more similar to the normal distribution as the sample size increases.

Standard Error

The standard error measures the variability of the sample mean by estimating the standard deviation of the sampling distribution. It is calculated as the sample standard deviation divided by the square root of the sample size, and it quantifies the precision of the sample mean as an estimate of the population mean.

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