4. A study of the ability of individuals to walk in a straight line (“Can We Really Walk Straight?”, Amer. J. of Physical Anthro., 1992:19-27) reported the accompanying data on cadence (strides per second) for a sample of 20 randomly selected individuals.

0.95 0.82 0.92 0.95 0.93 0.86 1.00 0.92 0.85 0.81 0.78 0.93 1.05 0.93 1.06 0.96 0.81 0.96 0.92 0.93
Construct a 95% confidence interval for the true mean cadence.

To construct a 95% confidence interval for the true mean cadence, we can use the following formula:

Confidence interval = sample mean ± (critical value * standard error)

1. Calculate the sample mean:
To calculate the sample mean, we sum up all the cadence values and divide by the sample size (20):
Sample mean = (0.95 + 0.82 + 0.92 + 0.95 + 0.93 + 0.86 + 1.00 + 0.92 + 0.85 + 0.81 + 0.78 + 0.93 + 1.05 + 0.93 + 1.06 + 0.96 + 0.81 + 0.96 + 0.92 + 0.93) / 20 = 0.922

2. Calculate the standard error:
The standard error measures the variability of the sample mean. It is calculated by dividing the sample standard deviation by the square root of the sample size.
To calculate the sample standard deviation, subtract the sample mean from each cadence value, square the differences, sum them up, divide by (n-1), and take the square root:
Sample standard deviation = sqrt(((0.95 - 0.922)^2 + (0.82 - 0.922)^2 + ... + (0.93 - 0.922)^2) / (20-1)) = 0.088

Now, calculate the standard error:
Standard error = Sample standard deviation / sqrt(sample size) = 0.088 / sqrt(20) ≈ 0.0197

3. Determine the critical value:
Since we want to construct a 95% confidence interval, we need to find the critical value that corresponds to a 95% confidence level. We can refer to the t-distribution table or use statistical software. In this case, with a sample size of 20, the critical value is approximately 2.086.

4. Calculate the confidence interval:
Using the formula, we can now calculate the confidence interval:
Confidence interval = Sample mean ± (critical value * standard error)
Confidence interval = 0.922 ± (2.086 * 0.0197)
Confidence interval = 0.922 ± 0.041

Therefore, the 95% confidence interval for the true mean cadence is approximately (0.881, 0.963).