Step 1 :Given the heart rate measurements of a patient over a week, we are asked to estimate the average heart rate and its standard deviation. The measurements are as follows: 106, 107, 113, 100, 107, 108, 111, 107, 108, 101, 112, 112, 106, 103.
Step 2 :First, we calculate the point estimate, denoted as \( \bar{x} \), which is the sample mean of the measurements. The sample mean is calculated as the sum of all measurements divided by the number of measurements.
Step 3 :Using the given measurements, the sample mean, \( \bar{x} \), is approximately 107.2143.
Step 4 :Next, we calculate the sample standard deviation, denoted as \( s \). The sample standard deviation is a measure of the amount of variation or dispersion of the measurements.
Step 5 :Using the given measurements, the sample standard deviation, \( s \), is approximately 3.9842.
Step 6 :We are also asked to determine the margin of error and a confidence interval for the average resting heart rate of this patient at an 80% confidence level. The margin of error is calculated using the formula \( E = z \cdot s \), where \( z \) is the z-score corresponding to the desired confidence level and \( s \) is the sample standard deviation.
Step 7 :The z-score for an 80% confidence level is approximately 1.2816. Therefore, the margin of error, \( E \), is approximately 5.11.
Step 8 :Finally, we calculate the confidence interval, which is given by \( (\bar{x} - E, \bar{x} + E) \).
Step 9 :The 80% confidence interval for the average resting heart rate of this patient is approximately (102.11, 112.32). Therefore, we can be 80% confident that the true average resting heart rate of this patient lies within this interval.
Step 10 :In conclusion, the point estimate, \( \bar{x} \), is approximately \(\boxed{107.2143}\) and the sample standard deviation, \( s \), is approximately \(\boxed{3.9842}\). The margin of error, \( E \), is approximately \(\boxed{5.11}\). An 80% confidence interval for the average resting heart rate of this patient is approximately \(\boxed{(102.11, 112.32)}\).