Question

To study the conversion of nitrite to nitrate in the blood, researchers injected four rabbits with a solution of radioactively labeled nitrite molecules. Ten minutes after injection, they measured for each rabbit the percentage of the nitrite that had been converted to nitrate. The results were as follows \(^{49}\) : $$ \begin{array}{llll} 51.1 & 55.4 & 48.0 & 49.5 \end{array} $$ (a) For these data, calculate the mean, median, and the standard error of the mean. (b) Assuming the percentage of nitrite converted to nitrate follows a normal distribution, construct a \(90 \%\) confidence interval for the population mean percentage. (c) Without doing any calculations, would a \(95 \%\) confidence interval be wider, narrower, or the same width as the confidence interval you found in part (b)? Why?

Short answer

Mean: 51, Median: 48.75, SEM: 1.5965; 90% CI: [49.3707, 52.6293]; A 95% CI would be wider than a 90% CI.

Step-by-step solution

Calculate the Mean

Find the sum of all the values and divide by the number of values to calculate the mean (mean = (sum of all measured percentages) / number of rabbits = (51.1 + 55.4 + 48.0 + 49.5) / 4 = 204 / 4 = 51).

Calculate the Median

Arrange the values in ascending order and find the middle value(s). Since there are four values, the median will be the average of the two middle values (median = (48.0 + 49.5) / 2 = 97.5 / 2 = 48.75).

Calculate the Sample Standard Deviation (s)

Use the formula s = sqrt[((sum of (each value - mean)^2) / (n-1))], where n is the number of values. (s = sqrt[((51.1 - 51)^2 + (55.4 - 51)^2 + (48.0 - 51)^2 + (49.5 - 51)^2) / (4-1)] = sqrt[(0.1^2 + 4.4^2 + (-3.0)^2 + (-1.5)^2) / 3] = sqrt[(0.01 + 19.36 + 9.0 + 2.25) / 3] = sqrt[(30.62) / 3] = sqrt[10.207] = 3.193).

Calculate the Standard Error of the Mean (SEM)

The standard error of the mean is the sample standard deviation divided by the square root of the number of values. (SEM = s / sqrt(n) = 3.193 / sqrt(4) = 3.193 / 2 = 1.5965).

Identify the Z-Score for a 90% Confidence Interval

A 90% confidence interval corresponds to a critical z-score (z*) roughly equal to 1.645, which you can find in a standard Z-table or by using a statistical software.

Construct the 90% Confidence Interval for the Population Mean

The confidence interval is given by (mean - z*SEM, mean + z*SEM). (confidence interval = 51 - (1.645)(1.5965), 51 + (1.645)(1.5965) = 49.3707, 52.6293).

Compare Confidence Interval Widths without Calculation

A 95% confidence interval requires a higher critical z-score than a 90% confidence interval (approximately 1.96 instead of 1.645). Since the standard error is multiplied by a larger z-score for a 95% confidence interval, the width of the confidence interval would be wider.

Key concepts

Mean Calculation

The mean, often referred to as the average, is a measure of central tendency that sums up the essence of a set of numbers. To calculate the mean, you simply add all the values together and then divide by the total number of values. In biostatistics, the mean is crucial for representing the typical value in a dataset. For example, in the exercise involving rabbits, adding the percentages of nitrite converted to nitrate for each rabbit and dividing by four (the number of rabbits) yields the mean conversion rate.

Median Calculation

The median, another measure of central tendency, represents the middle value of a dataset when it's ordered from smallest to largest. If there's an odd number of values, the median is the middle one. With an even number, as in our rabbit study, the median is the average of the two middle numbers. This makes the median distinct from the mean, as it is less affected by extreme values, which can skew the mean, making the median a valuable alternative when assessing a dataset's central tendency.

For instance, we arrange the percentages from the rabbit study in ascending order and find that the middle values are 48.0 and 49.5. To find the median, we calculate the average of these two numbers.

Standard Error of the Mean

The standard error of the mean (SEM) is a measure of how much discrepancy there is likely to be in a sample's mean compared to the true population mean. It's a form of standard deviation for the sampling distribution of the mean. To compute the SEM, take the sample standard deviation and divide it by the square root of the sample size (n).

In the context of our study, it quantifies the uncertainty in the estimate of the average conversion rate from nitrite to nitrate in the rabbits, allowing for the construction of confidence intervals around the mean.

Normal Distribution

A normal distribution, also known as a Gaussian distribution, is a bell-shaped curve that is symmetric about the mean. It is a foundational concept in biostatistics and implies most data points are near the mean, with a decrease in frequency as you move away from the mean.

In practice, biostatistical analyses often assume a normal distribution because of the Central Limit Theorem, which states that the sampling distribution of the mean tends to be normal, regardless of the shape of the original distribution, provided the sample size is large enough. Our rabbit study presumes a normal distribution to apply the concept of z-scores and construct a confidence interval for the population mean.

Z-Score

A z-score indicates how many standard deviations a data point is from the mean. It's a way of normalizing data within a normal distribution so that we can understand the chances of a given value occurring. In the context of confidence intervals, the z-score helps determine the range around the sample mean in which we expect the true population mean to fall.
For a 90% confidence interval, the z-score tells us that the true mean should fall within 1.645 standard deviations of the sample mean 90% of the time, assuming a normal distribution.

Sample Standard Deviation

The sample standard deviation is a measure of the amount of variation or dispersion in a set of values. It's calculated by taking the square root of the average squared deviation of each number from the mean of the data set. It's important because it quantifies the spread of data points around the mean, and more spread-out values yield a higher standard deviation.

When we calculate the standard error of the mean (SEM) in studies like the one involving rabbits, the sample standard deviation is pivotal because it directly influences the SEM value. The SEM, in turn, affects the width of the confidence interval calculated for the population mean.