A study was done that measured the maximum heart rate of men who had manually sh

Question

A study was done that measured the maximum heart rate of men who had manually shoveled snow. After removing any outliers, it was found that 11 men had a sample mean heart rate of 175 beats per minute with a standard deviation of 14.5 beats per minute. At a 99% confidence level construct a confidence interval for the true population mean. State the Margin of Error, Best point estimate and Include the written statement 5) A study was done that measured the maximum heart rate of men who had manually shoveled snow. After removing any outliers, it was found that 11 men had a sample standard deviation was 14.5 beats per minute. Construct a confidence interval for the true population standard deviation at a 99% confidence level. State the Margin of Error, Best point estimate and Include the written statement

Explanation / Answer

For both continuous and dichotomous variables, the confidence interval estimate (CI) is a range of likely values for the population parameter based on:

the point estimate, e.g., the sample mean

the investigator's desired level of confidence (most commonly 99%, but any level between 0-100% can be selected)

and the sampling variability or the standard error of the point estimate.

The Central Limit Theorem introduced in the module on Probability stated that, for large samples, the distribution of the sample means is approximately normally distributed with a mean:

and a standard deviation (also called the standard error):

For the standard normal distribution, P(-2.576 < Z < 2.576) = 0.99, i.e., there is a 99% probability that a standard normal variable, Z, will fall between -2.576 and 2.576. The Central Limit Theorem states that for large samples:

By substituting the expression on the right side of the equation:

P(-2.576<((X-)/( / n))<2.576) =0.99

Using algebra, we can rework this inequality such that the mean () is the middle term, as shown below.

P(-2.576 / n < <X+2.576 / n) =0.99

This last expression, then, provides the 99% confidence interval for the population mean, and this can also be expressed as:

X+- 2.576 / n

Thus, the margin of error is 2.576 times the standard error (the standard deviation of the point estimate from the sample), and 2.576 reflects the fact that a 99% confidence level was selected. So, the general form of a confidence interval is:

point estimate + Z SE (point estimate)

where Z is the value from the standard normal distribution for the selected confidence level (e.g., for a 99% confidence level, Z=2.576).

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