Problem 34 Find the sample size required to... [FREE SOLUTION] (2024)

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The population mean is the average value of a characteristic in a whole population. For example, in our exercise, we want to estimate the mean pulse rate of all adult females. This mean is often denoted by the Greek letter \( \mu \), and it represents the central value around which all individual measurements tend to cluster.

To calculate this population mean exactly, we would need to measure the pulse rate of every single adult female, which is obviously impractical. Instead, we take a sample (a subset of the population) and use the sample mean to estimate the population mean. The larger and more representative the sample, the more accurate our estimate of \( \mu \) will be.

In many real-world applications, such as our body data example, we rely on statistical methods like confidence intervals and the central limit theorem to make educated guesses about the population mean based on our sample data.

A confidence interval provides a range of values within which we can be certain the population mean lies, to a specified level of confidence. In our exercise, we are looking for a 99% confidence interval. This means that we are 99% sure that the true mean pulse rate falls within our calculated range.

The formula for a confidence interval around the population mean is given by: \[ \mu = \bar{x} \pm Z_{\frac{\alpha}{2}} \frac{\sigma}{\sqrt{n}} \] where \( \bar{x} \) is the sample mean, \( Z_{\frac{\alpha}{2}} \) is the z-value corresponding to our desired level of confidence, \( \sigma \) is the population standard deviation, and \( n \) is the sample size.

For a 99% confidence level, the critical z-value (\( Z_{\frac{\alpha}{2}} \)) is 2.576. This z-value provides the number of standard deviations we would expect the interval to cover. By calculating the margin of error \( E \) and applying it to our sample mean, we produce the confidence interval for the population mean.

The standard deviation (\( \sigma \) or s for a sample) is a measure of how spread out the values in a data set are. It shows the average distance of each data point from the mean of the data set. In the context of our pulse rate data, the standard deviation gives us an idea of how much the pulse rates of different individuals vary from the average pulse rate.

To find the sample size required to estimate the population mean, we need an estimate of the standard deviation. In our exercise, we use two methods to get this estimate:

• The range rule of thumb, which approximates standard deviation as one-fourth of the range of the data
• An actual standard deviation measured directly from the data (12.5 bpm in this case)

The formula for calculating standard deviation (\( s \)) for a sample is \[ s = \sqrt{ \frac{ \sum (x_i - \bar{x})^2 }{ n - 1 } } \] where \( x_i \) represents each value in the sample, \( \bar{x} \) is the sample mean, and \( n \) is the sample size.

Knowing the standard deviation is crucial because it affects the width of our confidence intervals and the required sample size. A smaller standard deviation generally means a smaller required sample size, as seen when comparing the results of parts (a) and (b) in our exercise.